Research ArticleClosed-Loop Estimation for Randomly Sampled Measurementsin Target Tracking System

Jin Xue-bo Lian Xiao-feng Su Ting-li Shi Yan and Miao Bei-bei

College of Computer and Information Engineering Beijing Technology and Business University Beijing 100048 China

Correspondence should be addressed to Jin Xue-bo xuebojingmailcom

Received 24 October 2013 Revised 30 December 2013 Accepted 30 December 2013 Published 26 February 2014

Academic Editor Hamid Reza Karimi

Copyright copy 2014 Jin Xue-bo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Many tracking applications need to deal with the randomly sampled measurements for which the traditional recursive estimationmethod may fail Moreover getting the accurate dynamic model of the target becomes more difficult Therefore it is necessaryto update the dynamic model with the real-time information of the tracking system This paper provides a solution for the targettracking system with randomly sampling measurement Here the irregular sampling interval is transformed to a time-varyingparameter by calculating the matrix exponential and the dynamic parameter is estimated by the online estimated state with Yule-Walker method which is called the closed-loop estimation The convergence condition of the closed-loop estimation is provedSimulations and experiments show that the closed-loop estimation method can obtain good estimation performance even withvery high irregular rate of sampling interval and the developed model has a strong advantage for the long trajectory trackingcomparing the other models

1 Introduction

Target tracking is the most important preliminary step formany higher-level analysis applications Nowadays somenew sensors have been used in the tracking systems suchas the radio frequency identification (RFID) readers TheRFID stores and retrieves data through the electromagnetictransmission to an RF compatible integrated circuit Oncethe tag gets close to the readers the distance between thereaders and tags can be got and sent to the data processingcenter The measurements of RFID are randomly sampled [1]because of the data-driven measurement mechanisms Data-driven approach has been used in many applications [2] andthe irregular sampling is one of the important issues in thisapproach

In general the video tracking system has to extract thevisual information at each frame [3] which costs muchcomputing amount In [4] the target is tracked by someselected frames to reduce the calculation cost and achievethe real-time tracking which also results in the randomlysampled tracking problem If the output measurements areobtained at a set of irregular sampling times the traditional

recursive estimation from 119870 to 119870 + 1 may fail in general[5] Both the model and the estimation method should bereconsidered

Reference [6] transformed the randomly sampled mea-surement tracking to some time-varying parameters and usedthe current model to describe the processing model [7ndash11]which assumes a priori probability density of the accelerationas Rayleigh density Due to the randomly sampled measure-ment this assumption is no longer satisfied

Except the current model there were several othermodels used in the tracking such as constant-velocity (CV)model constant acceleration model (CA) and Singer model(zero mean first-order Markov model) [12 13] The CVmodels [1] emphasize that the accelerations are small Inmaneuvering target tracking the inclusion of accelerationin the state vector would degrade tracking performanceThe main attractive feature of this model is its simplicityIt is sometimes used in the maneuvering target trackingtechniques such as the so-called noise-level adjustmentwhen the maneuver is quite small or random It is also simplyreferred to as the CA model or more precisely the nearlyCA model The Singer model regards the target acceleration

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 315908 12 pageshttpdxdoiorg1011552014315908

2 Mathematical Problems in Engineering

as a first-order semi-Markov process with zero mean whichis in essence a priori model since it does not use onlineinformation on the target maneuver Again because of theirregular sampling time the priori model does not meet theactual dynamic model of the target

The approach to update the system model online hasattracted great interest of the researchers For example theinteracting multiple model (IMM) [14 15] method considersthe change of the system dynamics as aMarkovian parameterwhose transition probability is set based on the onlineestimation and then fusions several models for the trackingwhile IMM suffers heavy computational burden on conditionthat the maneuvering target has complex motion Moreoverthe complex movement can also lead to frequent switchbetween different models which can cause the trackingperformance to decline Another model [16 17] estimatedthe state of a power system where the bus voltages aretransformed to a system parameter But the works of thisclosed-loop estimation have not yet been involved in therandomly sampled tracking system

This paper will develop a joint state-and-parameter esti-mation method for the target tracking system with randomlysampled measurements where the estimation problem isreformulated as two loosely coupled linear subproblemsThis paper is organized as follows Section 2 derives thesystem dynamic model under the random sampling timeand gives the estimation method based on Kalman filterThe convergence of the algorithm is proved in Section 3The simulations and experiments are provided in Section 4Finally some concluding remarks are given in Section 5

2 System Model and Closed-Loop EstimationMethod

We begin with the continuous dynamic model of the movingtarget Let 119909 and be the target location velocityand acceleration along a generic direction and the state isexpressed as 119909 = [119909 ]

119879 Assume the nonzero meanacceleration satisfies (119905) = 119886(119905) + 119886(119905) where 119886(119905) is themean of acceleration in the interval [0 119905] and 119886(119905) is a zeromean first-order stationaryMarkov process with variance 1205752

119886

We have 119886(119905) = minus120572119886(119905) + 119908(119905) 120572 is maneuver frequencyand 119908(119905) is zero mean processing white noise with variance1205752

119908= 2120572120575

2

119886 The parameter 120572 = 1120591 is the reciprocal of the

maneuver time constant 120591 and thus depends on how longthe maneuver lasts For example for an aircraft 120591 asymp 60 s fora lazy turn and 120591 asymp 10ndash20 s for an evasive maneuver Theparameter 1205752

119886= 119864[119886

2(119905)] is the ldquoinstantaneous variancerdquo of

the accelerationThen we can obtain the acceleration satisfying (119905) =

minus120572(119905) + 120572119886(119905) +119908(119905) and the following state-space represen-tation of the continuous time model can be obtained

(119905) = 119860119909 (119905) + 119880119886 (119905) + 119861119908 (119905) (1)

where

119860 = [

[

0 1 0

0 0 1

0 0 minus120572

]

]

119880 = [

[

0

0

120572

]

]

119861 = [

[

0

0

1

]

]

(2)

and119908(119905) is process noise with the covariance matrix given by119908(119905) sim 119873(0 2120572120575

2

119886) Assume themeasurement data is obtained

at the sampling time 119905119894and the measurement equation is as

follows

119911 (119905119894) = 119867 (119905

119894) 119909 (119905119894) + V (119905

119894) 119894 = 0 1 2 (3)

where119867(119905119894) is measurement matrix and V(119905

119894) is measurement

noise with the known variance 119877 that is V(119905119894) sim 119873(0 119877)

21 Model Discretization We can get the following by thedifferential equation (1)

119909 (119905) = 119890119860(119905minus1199050)119909 (119905

0) + int

119905

1199050

119890119860(119905minus120582)

119880119886 (120582) 119889120582

+ int

119905

1199050

119890119860(119905minus120582)

119861119908 (120582) 119889120582

(4)

We can see that for any known integration interval [1199050 119905]

119909(119905) can be gotten at any time 119905 if the initial state 119909(1199050) the

parameters 119860 119880 119861 119886(119905) and 119908(119905) in [1199050 119905] are known

Consider the time interval from 119905119894minus1

to 119905119894and assume

119886 (120582) = 119886 (119905119894minus1) 119908 (120582) = 119908 (119905119894minus1)

120582 isin [119905119894minus1 119905119894]

(5)

we can have

int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119880119886 (120582) 119889120582 = int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119880119889120582119886 (119905

119894minus1)

int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119861119908 (120582) 119889120582 = int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119861119889120582119908 (119905

119894minus1)

(6)

Set th119894= 119905119894minus 119905119894minus1

we have the system matrix as119860119889(119905119894minus1) = 119890

119860th119894 119880119889(119905119894minus1) = int

119905119894

119905119894minus1119890119860(119905119894minus120582)119880119889120582 and the

noise 119908119889(119905119894minus1) = int

119905119894

119905119894minus1119890119860(119905119894minus120582)119861119889120582119908(119905

119894minus1) with the covariance

119876119889(119905119894minus1) = 119864[119908

119889(119905119894minus1)119908119879

119889(119905119894minus1)]

Because the process matrix 119860 in (2) is not a full-rankmatrix we cannot calculate the matrix exponential 119890119860th119894 bythe Lagrange-Hermite interpolation Here we use Laplacetransform and have

(119904119868 minus 119860)minus1= [

[

119904 minus1 0

0 119904 minus1

0 0 119904 + 120572

]

]

minus1

=

[[[[[[[

[

1

119904

1

1199042

1

1199042 (119904 + 120572)

01

119904

1

119904 (119904 + 120572)

0 01

119904 + 120572

]]]]]]]

]

(7)

Mathematical Problems in Engineering 3

The matrix exponential 119890119860th119894 can be gotten by the inverseLaplace transform as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572th119894minus 1 + 119890

minus120572th119894

1205722

0 11 minus 119890minus120572th119894

120572

0 0 119890minus120572th119894

]]]]]]]

]

(8)

and by the similar approach we can get the system parameter

119880119889(119905119894minus1) =

[[[[[[[

[

1

120572(minusth119894+120572 sdot th2119894

2+1 minus 119890minus120572sdotth119894

120572)

th119894minus1 minus 119890minus120572sdotth119894

120572

1 minus 119890minus120572sdotth119894

]]]]]]]

]

(9)

and the variance of 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721205752

120572[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(10)

with the parameters described as

11990211=

1

21205725[1 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894

+21205723th3119894

3minus 21205722th2119894minus 4120572 sdot th

119894119890minus120572sdotth119894]

11990212=

1

21205724[119890minus2120572sdotth119894 + 1 minus 2119890

minus120572sdotth119894

+2120572 sdot th119894119890minus120572sdotth119894 minus 2120572 sdot th

119894+ 1205722th2119894]

11990213=

1

21205723[1 minus 119890

minus2120572sdotth119894 minus 2120572 sdot th119894119890minus120572sdotth119894]

11990222=

1

21205723[4119890minus120572sdotth119894 minus 3 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894]

11990223=

1

21205722[119890minus2120572sdotth119894 + 1 minus 2120572 sdot th

119894]

11990233=1

2120572[1 minus 119890

minus2120572sdotth119894]

(11)

Thenwe get the discrete state-spacemodel of the trackingsystem as

119909 (119905119894) = 119860

119889(119905119894minus1) 119909 (119905119894minus1) + 119880119889(119905119894minus1) 119886 (119905119894minus1) + 119908119889(119905119894minus1)

119911 (119905119894) = 119867 (119905

119894) 119909 (119905119894) + V (119905

119894)

(12)

where 119909 = [119909 ]119879 is the state of the system to be

estimated and whose initial mean and covariance are knownas 1199090and 119875

0 119908119889(119905119894) and V(119905

119894) are white noise with zero

mean and independent of the initial state 1199090 119911(119905119894) is the

measurement vector 119867(119905119894) is measurement matrices and

V(119905119894) ismeasurement noise with known variance119877 Until now

the irregular sampling is turned to the varying-parametersystem We can see the same sampling interval is just aparticular case of the random sampling problem Thereforethemodel of the randomly sampling tracking is a general one

22 System Parameters Estimation Here we assume themaneuver frequency 120572 and the variance of the acceleration1205752

119886are not constant but variable and expressed as 120572

119894and 1205752

119886119894

From the processing model of (12) we have the discrete timeequation of the acceleration as

(119905119894) = 120573119894 (119905119894minus1) + (1 minus 120573

119894) 119886 (119905119894minus1) + 119908119886(119905119894minus1) (13)

where 120573119894= 119890minus120572119894th119894 and 119908119886(119905

119894minus1) is a zero mean white noise

sequence with the variance

1205752

119886119908119894= 1205752

119886119894(1 minus 120573

2

119894) (14)

120572119894is the maneuver frequency at the sampling time 119905

119894 119886(119905119894minus1)

is the mean of one interval so we have 119886(119905119894) = 119886(119905

119894minus1) Set

119886(119905119894) = (119905

119894) minus 119886(119905

119894) then we can obtain

119886 (119905119894) = 120573119894119886 (119905119894minus1) + 119908119886(119905119894minus1) (15)

Consider the estimation of acceleration 119886(119905119894) is a random

process we have

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894) (16)

where 119894 is the number of data For a first-order stationaryMarkov process (15) we have the statistics relation betweenthe autocorrelation functions 119903(0) 119903(1)with the parameters120573

119894

and 1205752119886119908119894

by the Yule-Walker method [18]

119903119894 (0) =

1

119894

119894minus1

sum

119894=0

119886 (119905119894) 119886 (119905119894)

119903119894 (1) =

1

119894

119894minus1

sum

119894=1

119886 (119905119894) 119886 (119905119894minus1)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1)

(17)

Next we can get 120572119894and 1205752

119886119894by 1205752119886119894= 1205752

119886119908119894(1 minus 120573

2

119894) 120572119894=

ln120573119894 minus th

119894 and then get the system parameters 119860

119889(119905119894minus1)

119880119889(119905119894minus1) and 119876

119889(119905119894minus1) in process function (12)

23 Algorithm Summary Now we summarize the closed-loop estimation algorithm for the randomly sampled mea-surements as follows

4 Mathematical Problems in Engineering

(1) Initialization (119894 = 0) Consider

119909 (1199050| 1199050) = 1199090

119875 (1199050| 1199050) = 1198750 1205720 1205752

1198860 119886 (1199050)

1199030(1199050) = 0sdot 0 119903

0(1199051) = 0

(18)

(2) Recursion (119894 = 119894 + 1)

(a) System update set th119894= 119905119894minus 119905119894minus1

and the systemparameter as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572119894th119894minus 1 + 119890

minus120572119894th119894

1205722

119894

0 11 minus 119890minus120572119894th119894

120572119894

0 0 119890minus120572119894th119894

]]]]]]]

]

(19)

119889(119905119894minus1) =

[[[[[[[

[

1

120572119894

(minusth119894+120572119894sdot th2119894

2+1 minus 119890minus120572119894 sdotth119894

120572119894

)

th119894minus1 minus 119890minus120572119894 sdotth119894

120572119894

1 minus 119890minus120572119894 sdotth119894

]]]]]]]

]

(20)

and the variance of the 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721198941205752

120572119894[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(21)

with parameters described as

11990211=

1

21205725

119894

[1 minus 119890minus2120572119894 sdotth119894 + 2120572

119894sdot th119894

+21205723

119894th3119894

3minus 21205722

119894th2119894minus 4120572119894sdot th119894119890minus120572sdotth119894]

11990212=

1

21205724

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2119890

minus120572119894 sdotth119894

+2120572119894sdot th119894119890minus120572119894 sdotth119894 minus 2120572

119894sdot th119894+ 1205722

119894th2119894]

11990213=

1

21205723

119894

[1 minus 119890minus2120572119894 sdotth119894 minus 2120572

119894sdot th119894119890minus120572119894 sdotth119894]

11990222=

1

21205723

119894

[4119890minus120572119894 sdotth119894 minus 3 minus 119890

minus2120572119894 sdotth119894 + 2120572119894sdot th119894]

11990223=

1

21205722

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2120572

119894sdot th119894]

11990233=

1

2120572119894

[1 minus 119890minus2120572119894 sdotth119894]

(22)

(b) State prediction consider

119909 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119909 (119905119894minus1| 119905119894minus1) + 119889(119905119894minus1) 119886 (119905119894minus1)

119875 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119875 (119905119894minus1| 119905119894minus1) 119860119879

119889(119905119894minus1) + 119876119889(119905119894minus1)

(23)

(c) State update consider

119909 (119905119894| 119905119894)

= 119909 (119905119894| 119905119894minus1) + 119870 (119905

119894) [119911 (119905

119894) minus 119867 (119905

119894) 119909 (119905119894| 119905119894minus1)]

(24)

119870(119905119894)

= 119875 (119905119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

(25)

119875 (119905119894| 119905119894) = [119868 minus 119870 (119905

119894)119867 (119905

119894)] 119875 (119905

119894| 119905119894minus1) (26)

(d) Parameter adaptation the mean of the acceleration

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894| 119905119894) (27)

When 119894 le 1198700 the maneuver frequency 120572

119894is set to 120572

0and

the covariance of the noise 1205752119886119894is gotten by the following

1205752

120572119894=

4 minus 120587

120587[119886119872minus 119909 (119905

119894| 119905119894)]2

when 119909 (119905119894| 119905119894) gt 0

4 minus 120587

120587[119909 (119905119894| 119905119894) minus 119886minus119872]2

when 119909 (119905119894| 119905119894) lt 0

a small positive constant when 119909 (119905119894| 119905119894) = 0

(28)

When 119894 gt 1198700 the parameter is updated by the following

119886 (119905119894) = 119909 (119905

119894| 119905119894) minus 119886 (119905

119894) (29)

119903119894 (1) = 119903119894minus1 (1) +

1

119894[119886 (119905119894) 119886 (119905119894minus1) minus 119903119894minus1 (1)] (30)

119903119894 (0) = 119903119894minus1 (0) +

1

119894[119886 (119905119894) 119886 (119905119894) minus 119903119894minus1 (0)] (31)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1) (32)

1205752

119886119894=

1205752

119886119908119894

1 minus 1205732

119894

120572119894=ln120573119894

minusth119894

(33)

The irregular sampling time 119905119894minus1

119905119894and the interval th

119894

reflect in the time-varying parameters of the system sowe can conclude that the Kalman filter shown in (23)ndash(33)based on system (12) with system parameters (19)ndash(22) canobtain the same estimation performance as regular samplingKalman filter

Mathematical Problems in Engineering 5

Figure 1 The video with simple background and one target

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Figure 2 The measurement of maneuvering target got from thevideo

3 Proof of the Convergence

Based on the closed-loop estimation algorithm (18)ndash(33)we can see that the parameter used to estimate state is anestimated one and similarly the estimated states to calculateparameters 120572

119894and 1205752119886119894have estimation errors tooTherefore it

is important to guarantee the convergence of the estimationof the states and parameters

From (27) (29) and (33) we know if the estimation 119909(119905119894|

119905119894) increased suddenly 119886(119905

119894) will increase greatly because the

mean changes less than 119909(119905119894| 119905119894) and 1205752

119886119908119894becomes large too

Then a very large positive 1205752119886119894will be obtained and119876

119889(119905119894)will

also contain a large number of elements (here we call it a bigmatrix) From the Riccati equation of Kalman filter

119875 (119905119894+1| 119905119894)

= 119860119889(119905119894) 119875 (119905

119894| 119905119894minus1) minus 119875 (119905

119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875 (119905119894| 119905119894minus1) 119860119879

119889(119905119894) + 119876119889(119905119894)

(34)

we find that 119875(119905119894+1

| 119905119894) will be a big matrix if 119876

119889(119905119894) is a

big one and119870(119905119894+1) will increase greatly As a result the esti-

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Te real trajectoryTe estimation trajectory

Figure 3 The real trajectory and the estimation trajectory

mation state 119909(119905119894| 119905119894) = 119909(119905

119894| 119905119894minus1) + 119870(119905

119894)[119911(119905119894) minus 119867(119905

119894)119909(119905119894|

119905119894minus1)] will be a big matrix too This trend results in positive

feedback loops whichmeans 119909(119905119894| 119905119894)will become larger and

larger and finally divergence We give the following theoremto guarantee the algorithm convergence

Theorem 1 The estimation 119909(119905119894+1

| 119905119894+1) is bounded if the

variance of the target acceleration 1205752119886119894has an upper bound that

is there is a positive 12057520satisfying 1205752

119886119894le 1205752

0

Proof We firstly consider maneuvering frequency 120572119894 From

(19) (21) and (22) we know if120572119894rarr 0 and 1205752

119886119894le 1205752

0 the target

has the constant acceleration maneuvering and the systemmodel is the constant acceleration model with the parameteras follows

119860119889(119905119894minus1) 997888rarr 119860

119889(119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876119889(119905119894minus1) 997888rarr 119876

119889(119905119894minus1) = 1205752

120572119894

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(35)

If 120572119894rarr infin and 120575

2

119886119894le 1205752

0 we can get the system

parameter matrix such as 119860119889(119905119894minus1) rarr 119860

119889(119905119894minus1) = [

1 th119894 00 1 0

0 0 1

]

and 119876119889(119905119894minus1) rarr 119876

119889(119905119894minus1) = 120575

2

120572119894[0 0 0

0 0 0

0 0 1] Therefore we can

see that 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) are the monotonic matrix with

finite value elements

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

as a first-order semi-Markov process with zero mean whichis in essence a priori model since it does not use onlineinformation on the target maneuver Again because of theirregular sampling time the priori model does not meet theactual dynamic model of the target

The approach to update the system model online hasattracted great interest of the researchers For example theinteracting multiple model (IMM) [14 15] method considersthe change of the system dynamics as aMarkovian parameterwhose transition probability is set based on the onlineestimation and then fusions several models for the trackingwhile IMM suffers heavy computational burden on conditionthat the maneuvering target has complex motion Moreoverthe complex movement can also lead to frequent switchbetween different models which can cause the trackingperformance to decline Another model [16 17] estimatedthe state of a power system where the bus voltages aretransformed to a system parameter But the works of thisclosed-loop estimation have not yet been involved in therandomly sampled tracking system

This paper will develop a joint state-and-parameter esti-mation method for the target tracking system with randomlysampled measurements where the estimation problem isreformulated as two loosely coupled linear subproblemsThis paper is organized as follows Section 2 derives thesystem dynamic model under the random sampling timeand gives the estimation method based on Kalman filterThe convergence of the algorithm is proved in Section 3The simulations and experiments are provided in Section 4Finally some concluding remarks are given in Section 5

2 System Model and Closed-Loop EstimationMethod

We begin with the continuous dynamic model of the movingtarget Let 119909 and be the target location velocityand acceleration along a generic direction and the state isexpressed as 119909 = [119909 ]

119879 Assume the nonzero meanacceleration satisfies (119905) = 119886(119905) + 119886(119905) where 119886(119905) is themean of acceleration in the interval [0 119905] and 119886(119905) is a zeromean first-order stationaryMarkov process with variance 1205752

119886

We have 119886(119905) = minus120572119886(119905) + 119908(119905) 120572 is maneuver frequencyand 119908(119905) is zero mean processing white noise with variance1205752

119908= 2120572120575

2

119886 The parameter 120572 = 1120591 is the reciprocal of the

maneuver time constant 120591 and thus depends on how longthe maneuver lasts For example for an aircraft 120591 asymp 60 s fora lazy turn and 120591 asymp 10ndash20 s for an evasive maneuver Theparameter 1205752

119886= 119864[119886

2(119905)] is the ldquoinstantaneous variancerdquo of

the accelerationThen we can obtain the acceleration satisfying (119905) =

minus120572(119905) + 120572119886(119905) +119908(119905) and the following state-space represen-tation of the continuous time model can be obtained

(119905) = 119860119909 (119905) + 119880119886 (119905) + 119861119908 (119905) (1)

where

119860 = [

[

0 1 0

0 0 1

0 0 minus120572

]

]

119880 = [

[

0

0

120572

]

]

119861 = [

[

0

0

1

]

]

(2)

and119908(119905) is process noise with the covariance matrix given by119908(119905) sim 119873(0 2120572120575

2

119886) Assume themeasurement data is obtained

at the sampling time 119905119894and the measurement equation is as

follows

119911 (119905119894) = 119867 (119905

119894) 119909 (119905119894) + V (119905

119894) 119894 = 0 1 2 (3)

where119867(119905119894) is measurement matrix and V(119905

119894) is measurement

noise with the known variance 119877 that is V(119905119894) sim 119873(0 119877)

21 Model Discretization We can get the following by thedifferential equation (1)

119909 (119905) = 119890119860(119905minus1199050)119909 (119905

0) + int

119905

1199050

119890119860(119905minus120582)

119880119886 (120582) 119889120582

+ int

119905

1199050

119890119860(119905minus120582)

119861119908 (120582) 119889120582

(4)

We can see that for any known integration interval [1199050 119905]

119909(119905) can be gotten at any time 119905 if the initial state 119909(1199050) the

parameters 119860 119880 119861 119886(119905) and 119908(119905) in [1199050 119905] are known

Consider the time interval from 119905119894minus1

to 119905119894and assume

119886 (120582) = 119886 (119905119894minus1) 119908 (120582) = 119908 (119905119894minus1)

120582 isin [119905119894minus1 119905119894]

(5)

we can have

int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119880119886 (120582) 119889120582 = int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119880119889120582119886 (119905

119894minus1)

int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119861119908 (120582) 119889120582 = int

119905119894

119905119894minus1

119890119860(119905119894minus120582)119861119889120582119908 (119905

119894minus1)

(6)

Set th119894= 119905119894minus 119905119894minus1

we have the system matrix as119860119889(119905119894minus1) = 119890

119860th119894 119880119889(119905119894minus1) = int

119905119894

119905119894minus1119890119860(119905119894minus120582)119880119889120582 and the

noise 119908119889(119905119894minus1) = int

119905119894

119905119894minus1119890119860(119905119894minus120582)119861119889120582119908(119905

119894minus1) with the covariance

119876119889(119905119894minus1) = 119864[119908

119889(119905119894minus1)119908119879

119889(119905119894minus1)]

Because the process matrix 119860 in (2) is not a full-rankmatrix we cannot calculate the matrix exponential 119890119860th119894 bythe Lagrange-Hermite interpolation Here we use Laplacetransform and have

(119904119868 minus 119860)minus1= [

[

119904 minus1 0

0 119904 minus1

0 0 119904 + 120572

]

]

minus1

=

[[[[[[[

[

1

119904

1

1199042

1

1199042 (119904 + 120572)

01

119904

1

119904 (119904 + 120572)

0 01

119904 + 120572

]]]]]]]

]

(7)

Mathematical Problems in Engineering 3

The matrix exponential 119890119860th119894 can be gotten by the inverseLaplace transform as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572th119894minus 1 + 119890

minus120572th119894

1205722

0 11 minus 119890minus120572th119894

120572

0 0 119890minus120572th119894

]]]]]]]

]

(8)

and by the similar approach we can get the system parameter

119880119889(119905119894minus1) =

[[[[[[[

[

1

120572(minusth119894+120572 sdot th2119894

2+1 minus 119890minus120572sdotth119894

120572)

th119894minus1 minus 119890minus120572sdotth119894

120572

1 minus 119890minus120572sdotth119894

]]]]]]]

]

(9)

and the variance of 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721205752

120572[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(10)

with the parameters described as

11990211=

1

21205725[1 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894

+21205723th3119894

3minus 21205722th2119894minus 4120572 sdot th

119894119890minus120572sdotth119894]

11990212=

1

21205724[119890minus2120572sdotth119894 + 1 minus 2119890

minus120572sdotth119894

+2120572 sdot th119894119890minus120572sdotth119894 minus 2120572 sdot th

119894+ 1205722th2119894]

11990213=

1

21205723[1 minus 119890

minus2120572sdotth119894 minus 2120572 sdot th119894119890minus120572sdotth119894]

11990222=

1

21205723[4119890minus120572sdotth119894 minus 3 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894]

11990223=

1

21205722[119890minus2120572sdotth119894 + 1 minus 2120572 sdot th

119894]

11990233=1

2120572[1 minus 119890

minus2120572sdotth119894]

(11)

Thenwe get the discrete state-spacemodel of the trackingsystem as

119909 (119905119894) = 119860

119889(119905119894minus1) 119909 (119905119894minus1) + 119880119889(119905119894minus1) 119886 (119905119894minus1) + 119908119889(119905119894minus1)

119911 (119905119894) = 119867 (119905

119894) 119909 (119905119894) + V (119905

119894)

(12)

where 119909 = [119909 ]119879 is the state of the system to be

estimated and whose initial mean and covariance are knownas 1199090and 119875

0 119908119889(119905119894) and V(119905

119894) are white noise with zero

mean and independent of the initial state 1199090 119911(119905119894) is the

measurement vector 119867(119905119894) is measurement matrices and

V(119905119894) ismeasurement noise with known variance119877 Until now

the irregular sampling is turned to the varying-parametersystem We can see the same sampling interval is just aparticular case of the random sampling problem Thereforethemodel of the randomly sampling tracking is a general one

22 System Parameters Estimation Here we assume themaneuver frequency 120572 and the variance of the acceleration1205752

119886are not constant but variable and expressed as 120572

119894and 1205752

119886119894

From the processing model of (12) we have the discrete timeequation of the acceleration as

(119905119894) = 120573119894 (119905119894minus1) + (1 minus 120573

119894) 119886 (119905119894minus1) + 119908119886(119905119894minus1) (13)

where 120573119894= 119890minus120572119894th119894 and 119908119886(119905

119894minus1) is a zero mean white noise

sequence with the variance

1205752

119886119908119894= 1205752

119886119894(1 minus 120573

2

119894) (14)

120572119894is the maneuver frequency at the sampling time 119905

119894 119886(119905119894minus1)

is the mean of one interval so we have 119886(119905119894) = 119886(119905

119894minus1) Set

119886(119905119894) = (119905

119894) minus 119886(119905

119894) then we can obtain

119886 (119905119894) = 120573119894119886 (119905119894minus1) + 119908119886(119905119894minus1) (15)

Consider the estimation of acceleration 119886(119905119894) is a random

process we have

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894) (16)

where 119894 is the number of data For a first-order stationaryMarkov process (15) we have the statistics relation betweenthe autocorrelation functions 119903(0) 119903(1)with the parameters120573

119894

and 1205752119886119908119894

by the Yule-Walker method [18]

119903119894 (0) =

1

119894

119894minus1

sum

119894=0

119886 (119905119894) 119886 (119905119894)

119903119894 (1) =

1

119894

119894minus1

sum

119894=1

119886 (119905119894) 119886 (119905119894minus1)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1)

(17)

Next we can get 120572119894and 1205752

119886119894by 1205752119886119894= 1205752

119886119908119894(1 minus 120573

2

119894) 120572119894=

ln120573119894 minus th

119894 and then get the system parameters 119860

119889(119905119894minus1)

119880119889(119905119894minus1) and 119876

119889(119905119894minus1) in process function (12)

23 Algorithm Summary Now we summarize the closed-loop estimation algorithm for the randomly sampled mea-surements as follows

4 Mathematical Problems in Engineering

(1) Initialization (119894 = 0) Consider

119909 (1199050| 1199050) = 1199090

119875 (1199050| 1199050) = 1198750 1205720 1205752

1198860 119886 (1199050)

1199030(1199050) = 0sdot 0 119903

0(1199051) = 0

(18)

(2) Recursion (119894 = 119894 + 1)

(a) System update set th119894= 119905119894minus 119905119894minus1

and the systemparameter as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572119894th119894minus 1 + 119890

minus120572119894th119894

1205722

119894

0 11 minus 119890minus120572119894th119894

120572119894

0 0 119890minus120572119894th119894

]]]]]]]

]

(19)

119889(119905119894minus1) =

[[[[[[[

[

1

120572119894

(minusth119894+120572119894sdot th2119894

2+1 minus 119890minus120572119894 sdotth119894

120572119894

)

th119894minus1 minus 119890minus120572119894 sdotth119894

120572119894

1 minus 119890minus120572119894 sdotth119894

]]]]]]]

]

(20)

and the variance of the 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721198941205752

120572119894[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(21)

with parameters described as

11990211=

1

21205725

119894

[1 minus 119890minus2120572119894 sdotth119894 + 2120572

119894sdot th119894

+21205723

119894th3119894

3minus 21205722

119894th2119894minus 4120572119894sdot th119894119890minus120572sdotth119894]

11990212=

1

21205724

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2119890

minus120572119894 sdotth119894

+2120572119894sdot th119894119890minus120572119894 sdotth119894 minus 2120572

119894sdot th119894+ 1205722

119894th2119894]

11990213=

1

21205723

119894

[1 minus 119890minus2120572119894 sdotth119894 minus 2120572

119894sdot th119894119890minus120572119894 sdotth119894]

11990222=

1

21205723

119894

[4119890minus120572119894 sdotth119894 minus 3 minus 119890

minus2120572119894 sdotth119894 + 2120572119894sdot th119894]

11990223=

1

21205722

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2120572

119894sdot th119894]

11990233=

1

2120572119894

[1 minus 119890minus2120572119894 sdotth119894]

(22)

(b) State prediction consider

119909 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119909 (119905119894minus1| 119905119894minus1) + 119889(119905119894minus1) 119886 (119905119894minus1)

119875 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119875 (119905119894minus1| 119905119894minus1) 119860119879

119889(119905119894minus1) + 119876119889(119905119894minus1)

(23)

(c) State update consider

119909 (119905119894| 119905119894)

= 119909 (119905119894| 119905119894minus1) + 119870 (119905

119894) [119911 (119905

119894) minus 119867 (119905

119894) 119909 (119905119894| 119905119894minus1)]

(24)

119870(119905119894)

= 119875 (119905119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

(25)

119875 (119905119894| 119905119894) = [119868 minus 119870 (119905

119894)119867 (119905

119894)] 119875 (119905

119894| 119905119894minus1) (26)

(d) Parameter adaptation the mean of the acceleration

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894| 119905119894) (27)

When 119894 le 1198700 the maneuver frequency 120572

119894is set to 120572

0and

the covariance of the noise 1205752119886119894is gotten by the following

1205752

120572119894=

4 minus 120587

120587[119886119872minus 119909 (119905

119894| 119905119894)]2

when 119909 (119905119894| 119905119894) gt 0

4 minus 120587

120587[119909 (119905119894| 119905119894) minus 119886minus119872]2

when 119909 (119905119894| 119905119894) lt 0

a small positive constant when 119909 (119905119894| 119905119894) = 0

(28)

When 119894 gt 1198700 the parameter is updated by the following

119886 (119905119894) = 119909 (119905

119894| 119905119894) minus 119886 (119905

119894) (29)

119903119894 (1) = 119903119894minus1 (1) +

1

119894[119886 (119905119894) 119886 (119905119894minus1) minus 119903119894minus1 (1)] (30)

119903119894 (0) = 119903119894minus1 (0) +

1

119894[119886 (119905119894) 119886 (119905119894) minus 119903119894minus1 (0)] (31)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1) (32)

1205752

119886119894=

1205752

119886119908119894

1 minus 1205732

119894

120572119894=ln120573119894

minusth119894

(33)

The irregular sampling time 119905119894minus1

119905119894and the interval th

119894

reflect in the time-varying parameters of the system sowe can conclude that the Kalman filter shown in (23)ndash(33)based on system (12) with system parameters (19)ndash(22) canobtain the same estimation performance as regular samplingKalman filter

Mathematical Problems in Engineering 5

Figure 1 The video with simple background and one target

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Figure 2 The measurement of maneuvering target got from thevideo

3 Proof of the Convergence

Based on the closed-loop estimation algorithm (18)ndash(33)we can see that the parameter used to estimate state is anestimated one and similarly the estimated states to calculateparameters 120572

119894and 1205752119886119894have estimation errors tooTherefore it

is important to guarantee the convergence of the estimationof the states and parameters

From (27) (29) and (33) we know if the estimation 119909(119905119894|

119905119894) increased suddenly 119886(119905

119894) will increase greatly because the

mean changes less than 119909(119905119894| 119905119894) and 1205752

119886119908119894becomes large too

Then a very large positive 1205752119886119894will be obtained and119876

119889(119905119894)will

also contain a large number of elements (here we call it a bigmatrix) From the Riccati equation of Kalman filter

119875 (119905119894+1| 119905119894)

= 119860119889(119905119894) 119875 (119905

119894| 119905119894minus1) minus 119875 (119905

119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875 (119905119894| 119905119894minus1) 119860119879

119889(119905119894) + 119876119889(119905119894)

(34)

we find that 119875(119905119894+1

| 119905119894) will be a big matrix if 119876

119889(119905119894) is a

big one and119870(119905119894+1) will increase greatly As a result the esti-

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Te real trajectoryTe estimation trajectory

Figure 3 The real trajectory and the estimation trajectory

mation state 119909(119905119894| 119905119894) = 119909(119905

119894| 119905119894minus1) + 119870(119905

119894)[119911(119905119894) minus 119867(119905

119894)119909(119905119894|

119905119894minus1)] will be a big matrix too This trend results in positive

feedback loops whichmeans 119909(119905119894| 119905119894)will become larger and

larger and finally divergence We give the following theoremto guarantee the algorithm convergence

Theorem 1 The estimation 119909(119905119894+1

| 119905119894+1) is bounded if the

variance of the target acceleration 1205752119886119894has an upper bound that

is there is a positive 12057520satisfying 1205752

119886119894le 1205752

0

Proof We firstly consider maneuvering frequency 120572119894 From

(19) (21) and (22) we know if120572119894rarr 0 and 1205752

119886119894le 1205752

0 the target

has the constant acceleration maneuvering and the systemmodel is the constant acceleration model with the parameteras follows

119860119889(119905119894minus1) 997888rarr 119860

119889(119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876119889(119905119894minus1) 997888rarr 119876

119889(119905119894minus1) = 1205752

120572119894

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(35)

If 120572119894rarr infin and 120575

2

119886119894le 1205752

0 we can get the system

parameter matrix such as 119860119889(119905119894minus1) rarr 119860

119889(119905119894minus1) = [

1 th119894 00 1 0

0 0 1

]

and 119876119889(119905119894minus1) rarr 119876

119889(119905119894minus1) = 120575

2

120572119894[0 0 0

0 0 0

0 0 1] Therefore we can

see that 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) are the monotonic matrix with

finite value elements

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The matrix exponential 119890119860th119894 can be gotten by the inverseLaplace transform as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572th119894minus 1 + 119890

minus120572th119894

1205722

0 11 minus 119890minus120572th119894

120572

0 0 119890minus120572th119894

]]]]]]]

]

(8)

and by the similar approach we can get the system parameter

119880119889(119905119894minus1) =

[[[[[[[

[

1

120572(minusth119894+120572 sdot th2119894

2+1 minus 119890minus120572sdotth119894

120572)

th119894minus1 minus 119890minus120572sdotth119894

120572

1 minus 119890minus120572sdotth119894

]]]]]]]

]

(9)

and the variance of 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721205752

120572[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(10)

with the parameters described as

11990211=

1

21205725[1 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894

+21205723th3119894

3minus 21205722th2119894minus 4120572 sdot th

119894119890minus120572sdotth119894]

11990212=

1

21205724[119890minus2120572sdotth119894 + 1 minus 2119890

minus120572sdotth119894

+2120572 sdot th119894119890minus120572sdotth119894 minus 2120572 sdot th

119894+ 1205722th2119894]

11990213=

1

21205723[1 minus 119890

minus2120572sdotth119894 minus 2120572 sdot th119894119890minus120572sdotth119894]

11990222=

1

21205723[4119890minus120572sdotth119894 minus 3 minus 119890

minus2120572sdotth119894 + 2120572 sdot th119894]

11990223=

1

21205722[119890minus2120572sdotth119894 + 1 minus 2120572 sdot th

119894]

11990233=1

2120572[1 minus 119890

minus2120572sdotth119894]

(11)

Thenwe get the discrete state-spacemodel of the trackingsystem as

119909 (119905119894) = 119860

119889(119905119894minus1) 119909 (119905119894minus1) + 119880119889(119905119894minus1) 119886 (119905119894minus1) + 119908119889(119905119894minus1)

119911 (119905119894) = 119867 (119905

119894) 119909 (119905119894) + V (119905

119894)

(12)

where 119909 = [119909 ]119879 is the state of the system to be

estimated and whose initial mean and covariance are knownas 1199090and 119875

0 119908119889(119905119894) and V(119905

119894) are white noise with zero

mean and independent of the initial state 1199090 119911(119905119894) is the

measurement vector 119867(119905119894) is measurement matrices and

V(119905119894) ismeasurement noise with known variance119877 Until now

the irregular sampling is turned to the varying-parametersystem We can see the same sampling interval is just aparticular case of the random sampling problem Thereforethemodel of the randomly sampling tracking is a general one

22 System Parameters Estimation Here we assume themaneuver frequency 120572 and the variance of the acceleration1205752

119886are not constant but variable and expressed as 120572

119894and 1205752

119886119894

From the processing model of (12) we have the discrete timeequation of the acceleration as

(119905119894) = 120573119894 (119905119894minus1) + (1 minus 120573

119894) 119886 (119905119894minus1) + 119908119886(119905119894minus1) (13)

where 120573119894= 119890minus120572119894th119894 and 119908119886(119905

119894minus1) is a zero mean white noise

sequence with the variance

1205752

119886119908119894= 1205752

119886119894(1 minus 120573

2

119894) (14)

120572119894is the maneuver frequency at the sampling time 119905

119894 119886(119905119894minus1)

is the mean of one interval so we have 119886(119905119894) = 119886(119905

119894minus1) Set

119886(119905119894) = (119905

119894) minus 119886(119905

119894) then we can obtain

119886 (119905119894) = 120573119894119886 (119905119894minus1) + 119908119886(119905119894minus1) (15)

Consider the estimation of acceleration 119886(119905119894) is a random

process we have

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894) (16)

where 119894 is the number of data For a first-order stationaryMarkov process (15) we have the statistics relation betweenthe autocorrelation functions 119903(0) 119903(1)with the parameters120573

119894

and 1205752119886119908119894

by the Yule-Walker method [18]

119903119894 (0) =

1

119894

119894minus1

sum

119894=0

119886 (119905119894) 119886 (119905119894)

119903119894 (1) =

1

119894

119894minus1

sum

119894=1

119886 (119905119894) 119886 (119905119894minus1)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1)

(17)

Next we can get 120572119894and 1205752

119886119894by 1205752119886119894= 1205752

119886119908119894(1 minus 120573

2

119894) 120572119894=

ln120573119894 minus th

119894 and then get the system parameters 119860

119889(119905119894minus1)

119880119889(119905119894minus1) and 119876

119889(119905119894minus1) in process function (12)

23 Algorithm Summary Now we summarize the closed-loop estimation algorithm for the randomly sampled mea-surements as follows

4 Mathematical Problems in Engineering

(1) Initialization (119894 = 0) Consider

119909 (1199050| 1199050) = 1199090

119875 (1199050| 1199050) = 1198750 1205720 1205752

1198860 119886 (1199050)

1199030(1199050) = 0sdot 0 119903

0(1199051) = 0

(18)

(2) Recursion (119894 = 119894 + 1)

(a) System update set th119894= 119905119894minus 119905119894minus1

and the systemparameter as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572119894th119894minus 1 + 119890

minus120572119894th119894

1205722

119894

0 11 minus 119890minus120572119894th119894

120572119894

0 0 119890minus120572119894th119894

]]]]]]]

]

(19)

119889(119905119894minus1) =

[[[[[[[

[

1

120572119894

(minusth119894+120572119894sdot th2119894

2+1 minus 119890minus120572119894 sdotth119894

120572119894

)

th119894minus1 minus 119890minus120572119894 sdotth119894

120572119894

1 minus 119890minus120572119894 sdotth119894

]]]]]]]

]

(20)

and the variance of the 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721198941205752

120572119894[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(21)

with parameters described as

11990211=

1

21205725

119894

[1 minus 119890minus2120572119894 sdotth119894 + 2120572

119894sdot th119894

+21205723

119894th3119894

3minus 21205722

119894th2119894minus 4120572119894sdot th119894119890minus120572sdotth119894]

11990212=

1

21205724

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2119890

minus120572119894 sdotth119894

+2120572119894sdot th119894119890minus120572119894 sdotth119894 minus 2120572

119894sdot th119894+ 1205722

119894th2119894]

11990213=

1

21205723

119894

[1 minus 119890minus2120572119894 sdotth119894 minus 2120572

119894sdot th119894119890minus120572119894 sdotth119894]

11990222=

1

21205723

119894

[4119890minus120572119894 sdotth119894 minus 3 minus 119890

minus2120572119894 sdotth119894 + 2120572119894sdot th119894]

11990223=

1

21205722

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2120572

119894sdot th119894]

11990233=

1

2120572119894

[1 minus 119890minus2120572119894 sdotth119894]

(22)

(b) State prediction consider

119909 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119909 (119905119894minus1| 119905119894minus1) + 119889(119905119894minus1) 119886 (119905119894minus1)

119875 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119875 (119905119894minus1| 119905119894minus1) 119860119879

119889(119905119894minus1) + 119876119889(119905119894minus1)

(23)

(c) State update consider

119909 (119905119894| 119905119894)

= 119909 (119905119894| 119905119894minus1) + 119870 (119905

119894) [119911 (119905

119894) minus 119867 (119905

119894) 119909 (119905119894| 119905119894minus1)]

(24)

119870(119905119894)

= 119875 (119905119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

(25)

119875 (119905119894| 119905119894) = [119868 minus 119870 (119905

119894)119867 (119905

119894)] 119875 (119905

119894| 119905119894minus1) (26)

(d) Parameter adaptation the mean of the acceleration

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894| 119905119894) (27)

When 119894 le 1198700 the maneuver frequency 120572

119894is set to 120572

0and

the covariance of the noise 1205752119886119894is gotten by the following

1205752

120572119894=

4 minus 120587

120587[119886119872minus 119909 (119905

119894| 119905119894)]2

when 119909 (119905119894| 119905119894) gt 0

4 minus 120587

120587[119909 (119905119894| 119905119894) minus 119886minus119872]2

when 119909 (119905119894| 119905119894) lt 0

a small positive constant when 119909 (119905119894| 119905119894) = 0

(28)

When 119894 gt 1198700 the parameter is updated by the following

119886 (119905119894) = 119909 (119905

119894| 119905119894) minus 119886 (119905

119894) (29)

119903119894 (1) = 119903119894minus1 (1) +

1

119894[119886 (119905119894) 119886 (119905119894minus1) minus 119903119894minus1 (1)] (30)

119903119894 (0) = 119903119894minus1 (0) +

1

119894[119886 (119905119894) 119886 (119905119894) minus 119903119894minus1 (0)] (31)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1) (32)

1205752

119886119894=

1205752

119886119908119894

1 minus 1205732

119894

120572119894=ln120573119894

minusth119894

(33)

The irregular sampling time 119905119894minus1

119905119894and the interval th

119894

reflect in the time-varying parameters of the system sowe can conclude that the Kalman filter shown in (23)ndash(33)based on system (12) with system parameters (19)ndash(22) canobtain the same estimation performance as regular samplingKalman filter

Mathematical Problems in Engineering 5

Figure 1 The video with simple background and one target

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Figure 2 The measurement of maneuvering target got from thevideo

3 Proof of the Convergence

Based on the closed-loop estimation algorithm (18)ndash(33)we can see that the parameter used to estimate state is anestimated one and similarly the estimated states to calculateparameters 120572

119894and 1205752119886119894have estimation errors tooTherefore it

is important to guarantee the convergence of the estimationof the states and parameters

From (27) (29) and (33) we know if the estimation 119909(119905119894|

119905119894) increased suddenly 119886(119905

119894) will increase greatly because the

mean changes less than 119909(119905119894| 119905119894) and 1205752

119886119908119894becomes large too

Then a very large positive 1205752119886119894will be obtained and119876

119889(119905119894)will

also contain a large number of elements (here we call it a bigmatrix) From the Riccati equation of Kalman filter

119875 (119905119894+1| 119905119894)

= 119860119889(119905119894) 119875 (119905

119894| 119905119894minus1) minus 119875 (119905

119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875 (119905119894| 119905119894minus1) 119860119879

119889(119905119894) + 119876119889(119905119894)

(34)

we find that 119875(119905119894+1

| 119905119894) will be a big matrix if 119876

119889(119905119894) is a

big one and119870(119905119894+1) will increase greatly As a result the esti-

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Te real trajectoryTe estimation trajectory

Figure 3 The real trajectory and the estimation trajectory

mation state 119909(119905119894| 119905119894) = 119909(119905

119894| 119905119894minus1) + 119870(119905

119894)[119911(119905119894) minus 119867(119905

119894)119909(119905119894|

119905119894minus1)] will be a big matrix too This trend results in positive

feedback loops whichmeans 119909(119905119894| 119905119894)will become larger and

larger and finally divergence We give the following theoremto guarantee the algorithm convergence

Theorem 1 The estimation 119909(119905119894+1

| 119905119894+1) is bounded if the

variance of the target acceleration 1205752119886119894has an upper bound that

is there is a positive 12057520satisfying 1205752

119886119894le 1205752

0

Proof We firstly consider maneuvering frequency 120572119894 From

(19) (21) and (22) we know if120572119894rarr 0 and 1205752

119886119894le 1205752

0 the target

has the constant acceleration maneuvering and the systemmodel is the constant acceleration model with the parameteras follows

119860119889(119905119894minus1) 997888rarr 119860

119889(119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876119889(119905119894minus1) 997888rarr 119876

119889(119905119894minus1) = 1205752

120572119894

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(35)

If 120572119894rarr infin and 120575

2

119886119894le 1205752

0 we can get the system

parameter matrix such as 119860119889(119905119894minus1) rarr 119860

119889(119905119894minus1) = [

1 th119894 00 1 0

0 0 1

]

and 119876119889(119905119894minus1) rarr 119876

119889(119905119894minus1) = 120575

2

120572119894[0 0 0

0 0 0

0 0 1] Therefore we can

see that 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) are the monotonic matrix with

finite value elements

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(1) Initialization (119894 = 0) Consider

119909 (1199050| 1199050) = 1199090

119875 (1199050| 1199050) = 1198750 1205720 1205752

1198860 119886 (1199050)

1199030(1199050) = 0sdot 0 119903

0(1199051) = 0

(18)

(2) Recursion (119894 = 119894 + 1)

(a) System update set th119894= 119905119894minus 119905119894minus1

and the systemparameter as

119860119889(119905119894minus1) =

[[[[[[[

[

1 th119894

120572119894th119894minus 1 + 119890

minus120572119894th119894

1205722

119894

0 11 minus 119890minus120572119894th119894

120572119894

0 0 119890minus120572119894th119894

]]]]]]]

]

(19)

119889(119905119894minus1) =

[[[[[[[

[

1

120572119894

(minusth119894+120572119894sdot th2119894

2+1 minus 119890minus120572119894 sdotth119894

120572119894

)

th119894minus1 minus 119890minus120572119894 sdotth119894

120572119894

1 minus 119890minus120572119894 sdotth119894

]]]]]]]

]

(20)

and the variance of the 119908119889(119905119894minus1) as

119876119889(119905119894minus1) = 119864 [119908

119889(119905119894minus1) 119908119879

119889(119905119894minus1)]

= 21205721198941205752

120572119894[

[

119902111199021211990213

119902121199022211990223

119902131199022311990233

]

]

(21)

with parameters described as

11990211=

1

21205725

119894

[1 minus 119890minus2120572119894 sdotth119894 + 2120572

119894sdot th119894

+21205723

119894th3119894

3minus 21205722

119894th2119894minus 4120572119894sdot th119894119890minus120572sdotth119894]

11990212=

1

21205724

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2119890

minus120572119894 sdotth119894

+2120572119894sdot th119894119890minus120572119894 sdotth119894 minus 2120572

119894sdot th119894+ 1205722

119894th2119894]

11990213=

1

21205723

119894

[1 minus 119890minus2120572119894 sdotth119894 minus 2120572

119894sdot th119894119890minus120572119894 sdotth119894]

11990222=

1

21205723

119894

[4119890minus120572119894 sdotth119894 minus 3 minus 119890

minus2120572119894 sdotth119894 + 2120572119894sdot th119894]

11990223=

1

21205722

119894

[119890minus2120572119894 sdotth119894 + 1 minus 2120572

119894sdot th119894]

11990233=

1

2120572119894

[1 minus 119890minus2120572119894 sdotth119894]

(22)

(b) State prediction consider

119909 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119909 (119905119894minus1| 119905119894minus1) + 119889(119905119894minus1) 119886 (119905119894minus1)

119875 (119905119894| 119905119894minus1)

= 119860119889(119905119894minus1) 119875 (119905119894minus1| 119905119894minus1) 119860119879

119889(119905119894minus1) + 119876119889(119905119894minus1)

(23)

(c) State update consider

119909 (119905119894| 119905119894)

= 119909 (119905119894| 119905119894minus1) + 119870 (119905

119894) [119911 (119905

119894) minus 119867 (119905

119894) 119909 (119905119894| 119905119894minus1)]

(24)

119870(119905119894)

= 119875 (119905119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

(25)

119875 (119905119894| 119905119894) = [119868 minus 119870 (119905

119894)119867 (119905

119894)] 119875 (119905

119894| 119905119894minus1) (26)

(d) Parameter adaptation the mean of the acceleration

119886 (119905119894minus1) =

1

119894

119894minus1

sum

119894=0

119909 (119905119894| 119905119894) (27)

When 119894 le 1198700 the maneuver frequency 120572

119894is set to 120572

0and

the covariance of the noise 1205752119886119894is gotten by the following

1205752

120572119894=

4 minus 120587

120587[119886119872minus 119909 (119905

119894| 119905119894)]2

when 119909 (119905119894| 119905119894) gt 0

4 minus 120587

120587[119909 (119905119894| 119905119894) minus 119886minus119872]2

when 119909 (119905119894| 119905119894) lt 0

a small positive constant when 119909 (119905119894| 119905119894) = 0

(28)

When 119894 gt 1198700 the parameter is updated by the following

119886 (119905119894) = 119909 (119905

119894| 119905119894) minus 119886 (119905

119894) (29)

119903119894 (1) = 119903119894minus1 (1) +

1

119894[119886 (119905119894) 119886 (119905119894minus1) minus 119903119894minus1 (1)] (30)

119903119894 (0) = 119903119894minus1 (0) +

1

119894[119886 (119905119894) 119886 (119905119894) minus 119903119894minus1 (0)] (31)

120573119894=119903119894 (1)

119903119894 (0)

1205752

119886119908119894= 119903119894 (0) minus 120573119894119903119894 (1) (32)

1205752

119886119894=

1205752

119886119908119894

1 minus 1205732

119894

120572119894=ln120573119894

minusth119894

(33)

The irregular sampling time 119905119894minus1

119905119894and the interval th

119894

reflect in the time-varying parameters of the system sowe can conclude that the Kalman filter shown in (23)ndash(33)based on system (12) with system parameters (19)ndash(22) canobtain the same estimation performance as regular samplingKalman filter

Mathematical Problems in Engineering 5

Figure 1 The video with simple background and one target

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Figure 2 The measurement of maneuvering target got from thevideo

3 Proof of the Convergence

Based on the closed-loop estimation algorithm (18)ndash(33)we can see that the parameter used to estimate state is anestimated one and similarly the estimated states to calculateparameters 120572

119894and 1205752119886119894have estimation errors tooTherefore it

is important to guarantee the convergence of the estimationof the states and parameters

From (27) (29) and (33) we know if the estimation 119909(119905119894|

119905119894) increased suddenly 119886(119905

119894) will increase greatly because the

mean changes less than 119909(119905119894| 119905119894) and 1205752

119886119908119894becomes large too

Then a very large positive 1205752119886119894will be obtained and119876

119889(119905119894)will

also contain a large number of elements (here we call it a bigmatrix) From the Riccati equation of Kalman filter

119875 (119905119894+1| 119905119894)

= 119860119889(119905119894) 119875 (119905

119894| 119905119894minus1) minus 119875 (119905

119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875 (119905119894| 119905119894minus1) 119860119879

119889(119905119894) + 119876119889(119905119894)

(34)

we find that 119875(119905119894+1

| 119905119894) will be a big matrix if 119876

119889(119905119894) is a

big one and119870(119905119894+1) will increase greatly As a result the esti-

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Te real trajectoryTe estimation trajectory

Figure 3 The real trajectory and the estimation trajectory

mation state 119909(119905119894| 119905119894) = 119909(119905

119894| 119905119894minus1) + 119870(119905

119894)[119911(119905119894) minus 119867(119905

119894)119909(119905119894|

119905119894minus1)] will be a big matrix too This trend results in positive

feedback loops whichmeans 119909(119905119894| 119905119894)will become larger and

larger and finally divergence We give the following theoremto guarantee the algorithm convergence

Theorem 1 The estimation 119909(119905119894+1

| 119905119894+1) is bounded if the

variance of the target acceleration 1205752119886119894has an upper bound that

is there is a positive 12057520satisfying 1205752

119886119894le 1205752

0

Proof We firstly consider maneuvering frequency 120572119894 From

(19) (21) and (22) we know if120572119894rarr 0 and 1205752

119886119894le 1205752

0 the target

has the constant acceleration maneuvering and the systemmodel is the constant acceleration model with the parameteras follows

119860119889(119905119894minus1) 997888rarr 119860

119889(119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876119889(119905119894minus1) 997888rarr 119876

119889(119905119894minus1) = 1205752

120572119894

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(35)

If 120572119894rarr infin and 120575

2

119886119894le 1205752

0 we can get the system

parameter matrix such as 119860119889(119905119894minus1) rarr 119860

119889(119905119894minus1) = [

1 th119894 00 1 0

0 0 1

]

and 119876119889(119905119894minus1) rarr 119876

119889(119905119894minus1) = 120575

2

120572119894[0 0 0

0 0 0

0 0 1] Therefore we can

see that 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) are the monotonic matrix with

finite value elements

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Figure 1 The video with simple background and one target

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Figure 2 The measurement of maneuvering target got from thevideo

3 Proof of the Convergence

Based on the closed-loop estimation algorithm (18)ndash(33)we can see that the parameter used to estimate state is anestimated one and similarly the estimated states to calculateparameters 120572

119894and 1205752119886119894have estimation errors tooTherefore it

is important to guarantee the convergence of the estimationof the states and parameters

From (27) (29) and (33) we know if the estimation 119909(119905119894|

119905119894) increased suddenly 119886(119905

119894) will increase greatly because the

mean changes less than 119909(119905119894| 119905119894) and 1205752

119886119908119894becomes large too

Then a very large positive 1205752119886119894will be obtained and119876

119889(119905119894)will

also contain a large number of elements (here we call it a bigmatrix) From the Riccati equation of Kalman filter

119875 (119905119894+1| 119905119894)

= 119860119889(119905119894) 119875 (119905

119894| 119905119894minus1) minus 119875 (119905

119894| 119905119894minus1)119867119879(119905119894)

times [119867 (119905119894) 119875 (119905119894| 119905119894minus1)119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875 (119905119894| 119905119894minus1) 119860119879

119889(119905119894) + 119876119889(119905119894)

(34)

we find that 119875(119905119894+1

| 119905119894) will be a big matrix if 119876

119889(119905119894) is a

big one and119870(119905119894+1) will increase greatly As a result the esti-

10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

70

80

90

100

Te real trajectoryTe estimation trajectory

Figure 3 The real trajectory and the estimation trajectory

mation state 119909(119905119894| 119905119894) = 119909(119905

119894| 119905119894minus1) + 119870(119905

119894)[119911(119905119894) minus 119867(119905

119894)119909(119905119894|

119905119894minus1)] will be a big matrix too This trend results in positive

feedback loops whichmeans 119909(119905119894| 119905119894)will become larger and

larger and finally divergence We give the following theoremto guarantee the algorithm convergence

Theorem 1 The estimation 119909(119905119894+1

| 119905119894+1) is bounded if the

variance of the target acceleration 1205752119886119894has an upper bound that

is there is a positive 12057520satisfying 1205752

119886119894le 1205752

0

Proof We firstly consider maneuvering frequency 120572119894 From

(19) (21) and (22) we know if120572119894rarr 0 and 1205752

119886119894le 1205752

0 the target

has the constant acceleration maneuvering and the systemmodel is the constant acceleration model with the parameteras follows

119860119889(119905119894minus1) 997888rarr 119860

119889(119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876119889(119905119894minus1) 997888rarr 119876

119889(119905119894minus1) = 1205752

120572119894

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(35)

If 120572119894rarr infin and 120575

2

119886119894le 1205752

0 we can get the system

parameter matrix such as 119860119889(119905119894minus1) rarr 119860

119889(119905119894minus1) = [

1 th119894 00 1 0

0 0 1

]

and 119876119889(119905119894minus1) rarr 119876

119889(119905119894minus1) = 120575

2

120572119894[0 0 0

0 0 0

0 0 1] Therefore we can

see that 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) are the monotonic matrix with

finite value elements

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Time

Hor

izon

tal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(a)

0 5 10 15 20 25 30 35 40 45 500

20406080

100

Time

Long

itudi

nal a

xis t

rack

ing

The real trajectoryThe estimation trajectory

(b)

Figure 4 The estimations of horizontal and longitudinal axis

0 5 10 15 20 25 30 35 40 45 50

0

10

20

TimeErro

r of h

oriz

onta

l axi

s tra

ckin

g

minus10

(a)

0 5 10 15 20 25 30 35 40 45 50

0

5

10

TimeErro

r of l

ongi

tudi

nal a

xis t

rack

ing

minus10

minus5

(b)

Figure 5 The location estimation errors

Then we consider the solution of Riccati equation (34)on the condition that the system parametermatrix has errorssuch as119860

119889= 119860119889+Δ119860119889and119876

119889= 119876119889+Δ119876119889 where119860

119889and119876

119889

are the actual system parameters and Δ119860119889and Δ119876

119889are the

errors of the system parameter Unlike the research about theuncertainty system here we do not know the actual systemmatrices 119860

119889and 119876

119889 but we can know the upper bound of

the system parameters 119860119889(119905119894minus1) and 119876

119889(119905119894minus1) when 1205752

119886119894le 1205752

0

such as

119860upper (119905119894minus1) = 119860119889 (119905119894minus1) =

[[[[[[

[

1 th119894

th2119894

2

0 1 th119894

0 0 1

]]]]]]

]

119876upper (119905119894minus1) = 1205752

0

[[[[[[[[[

[

th5119894

20

th4119894

8

th3119894

6

th4119894

8

th3119894

3

th2119894

2

th3119894

6

th2119894

2th119894

]]]]]]]]]

]

(36)

The perturbed discrete algebraic Riccati equation is asfollows

119875 = 119860119889(119905119894) 119875119860119879

119889(119905119894)

minus 119860119889(119905119894) 119875119867119879(119905119894) [119867 (119905

119894) 119875119867119879(119905119894) + 119877 (119905

119894)]minus1

times 119867 (119905119894) 119875119860119879

119889(119905119894) + 119876119889(119905119894)

(37)

We know that (37) is equal to

119875 = 119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894)

(38)

Then for any vector 119904 we have

119904119879119875119904

= 119904119879[119860119889(119905119894) (119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

times 119860119879

119889(119905119894) + 119876119889(119905119894) ] 119904

le 1205821(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))minus1

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

+ 119904119879119876119889(119905119894) 119904

(39)

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 002 004 006 008 01 012 014 016 018 020

2

4

6

8

10

12

14

16

18

20

The irregular rate IRrate

RMSE

2D

Figure 6 The relation between RMSE2D and IRrate

where 1205821(119883) is the maximum eigenvalue By the relation of

vector eigenvalue 120582119898(119883minus1) = 120582

minus1

119872minus119898+1(119883) where 120582

1(119883) ge

1205822(119883) ge sdot sdot sdot ge 120582

119872(119883) we have

119904119879119875119904 le

119904119879119860119889(119905119894) 119860119879

119889(119905119894) 119904

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119904119879119876119889(119905119894) 119904 (40)

That is

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

120582119872(119875minus1 + 119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894) (41)

We have

120582119872(119875minus1+ 119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

ge1

1205821 (119875)

+ 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894))

(42)

Then by (41) and (42) we have

119875 le119860119889(119905119894) 119860119879

119889(119905119894)

(11205821 (119875)) + 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

=1205821 (119875)119860119889 (119905119894) 119860

119879

119889(119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876119889(119905119894)

le

1205821 (119875)119860upper (119905119894) 119860

119879

upper (119905119894)

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))+ 119876upper (119905119894)

(43)

Next by the relation ofHermitematrix and its eigenvaluewe have

1205821 (119875) le

1205821 (119875) 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

1 + 1205821 (119875) 120582119872 (119867

119879 (119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821(119876upper (119905119894))

(44)

Then we have

1205822

1(119875) 120582119872 (119867

119879(119905119894) 119877 (119905119894)119867 (119905

119894))

+ 1205821 (119875) [1 minus 1205821 (119860upper (119905119894) 119860

119879

upper (119905119894))

minus120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))]

minus 1205821(119876upper (119905119894)) le 0

(45)

Assume that 120582119872(119867119879(119905119894)119877(119905119894)119867(119905119894)) gt 0 and set

1 minus 1205821(119860upper (119905119894) 119860

119879

upper (119905119894))

minus 120582119872(119867119879(119905119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894)) = 1199011

(46)

We have the following solution of (45)

1205821 (119875)

le

minus1199011+ radic1199012

1+ 4120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894)) 1205821(119876upper (119905119894))

2120582119872(119867119879 (119905

119894) 119877 (119905119894)119867 (119905

119894))

(47)

Therefore we can conclude that themaximum eigenvalueof estimation covariance119875has the upper bound shown as (47)if 1205752119886119894le 1205752

0

If one step predictive covariance is boundedthat is |119875(119905

119894| 119905119894minus1)| le 119875

0 then we know 119875(119905

119894+1| 119905119894)

must be bounded by (47) with the fact that |119876119889(119905119894)| le 119876

0

And based on (25) we know 119870(119905119894+1) must be a bounded

matrix and 119909(119905119894+1| 119905119894+1)must be bounded too

4 Simulations and Experiments

41 The Estimation by Different Extraction Rate and IrregularRate The method here is applied to a two-dimensional

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 20 40 60 80 100 1200

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

minus10

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(b)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(c)

0 20 40 60 80 100 1200

102030405060708090

100

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(d)

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(e)

0 20 40 60 80 100 1200

102030405060708090

100

minus20

The real trajectoryBy CV modelBy CA modelBy Singer model

By current modelBy IMMBy adaptive model

(f)

Figure 7 The tracking results for videos

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0

05

1

0

02

04

06080

50

100

150

EXrateIRrate

By CV modelBy CA modelBy Singer modelBy current model I

By current model IIBy current model IIIBy IMMBy adaptive model

Figure 8 RMSE2D under different EXrate and IRrate

Figure 9The tracking results in number 1 27 40 65 74 97 128 129158 181 189 and 226 frames

planar video tracking Here as a tracking problem we justuse the simple background and one target The video gottenby the Image Capture Test Bed is shown in Figure 1

We control the car maneuvering on the test bed and catchthe images of target movement by a stationary camera Forevery image of the video the target is extracted based on thecolor and then we get the measurement data of maneuveringtarget on the Image Capture Test Bed like Figure 2

We know that the camera catches the image under thesame interval and that will produce large amounts of imagedata If we can use some of images in the video for trackingthe image storage and computation cost will greatly reduceBut ldquousing some of imagesrdquo means that the measurements

no longer have the same sampling interval Here define theExtraction Rate as

EXrate

=extracted number of images from the video

total number of images in the videotimes 100

(48)

to describe the image compression rate And define theIrregular Rate to measure the sampling interval as

IRrate =sum119873

119894=1

10038161003816100381610038161003816th119894minus sum119873

119894=1th119894

10038161003816100381610038161003816

119873 (49)

The state for the target in the 2D space is 119909(119896) =

[119909(119896) (119896) (119896) 119910(119896) 119910(119896) 119910(119896)] The initial state esti-mate 119909

0and covariance 119875

0are assumed to be 119909

0=

[119909(0) 0 0 119910(0) 0 0]119879 and 119875

0= diag(10 10 10 10 10 10)

We extract 243 images from a video with 491 imageswhere EXrate = 4949 and IRrate = 01043 and by thealgorithm developed with the initial parameters 120572

0= 120

1205752

1198860= 10 119886

0= 0 120572

119872= 3 119870

0= 3 we get the

estimation of trajectory with estimation covariance 100881along the horizontal axis and 81660 along the vertical axisshown in Figure 3 The estimation trajectories of horizontaland longitudinal axis is shown in Figure 4 and the estimationerror are shown in Figure 5

To illustrate how the irregular rate affects estimationperformance the algorithm is used to estimate the target

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 The different irregular rate for 10 cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10001 006 008 009 010 013 014 014 018 019

trajectory under different Irregular Rate (shown in Table 1)with the same Extraction Rate EXrate = 4986 TheRMSE position is defined as RMSE

2D = radicRMSE2119867+ RMSE2

119871

where RMSE119867and RMSE

119871are the root-mean square errors

(RMSE) of position for horizontal and longitudinal axisrespectively The relation between RMSE

2D and IRrate isshown in Figure 6 We can see that the Irregular Rateaffects the estimation performance very little The IrregularRate changes 21 times almost from 00088 in Case 1 to01928 in Case 10 but RMSE

2D is about 14 for all IRrateWe can conclude that IRrate does not affect the trackingperformance when with the same EXrate

42 The Performance with Different Models Next we com-pare the model developed here with other dynamics modelsuch as CV model [12] CA model [12] Singer model [13]current model [8] and IMM [14] We set the process noisecovariance as 119876 = 1 for the CV and CA model and 1205902

119908= 1

and 120572 = 120 for Singer model Because the current model isvery sensitive to the priori parameters we give several systemparameters such as 120572 = 130 120572max = 3 (current model I) 120572 =120 120572max = 30 (current model II) and 120572 = 120 120572max = 3(current model III) After 100 Monte Carlo simulation runsRMSE

2D are calculated For different trajectory with differentEXrate and IRrate the estimation results are shown in Figures7(a)ndash7(f) where in order to show clearly we use the blackldquoOrdquo to describe the actual trajectory at the sampling time inFigures 7(e) and 7(f)

Table 2 and Figure 8 show RMSE2D under the different

IRrate and EXrateWe can see that themodel here can get thebetter estimation performance than CV CA Singer modelcurrentmodel and IMM for almost all EXrate and IRrateWealso note that the currentmodel needs the right parameter orelse the performance will become worse

We note that in Figure 7(f) the tracking error of thedeveloped model is larger than current models II III andIMM even CA We find that there is a big estimation errorat 5th second The reason is that there are not enough datagotten to update the parameter at 119870

0= 4 Therefore the

estimation error is bigger But we also note that the estimationerror declined quickly so the developed model has a strongadvantage for the long trajectory tracking comparing theother models

Another fact we also noticed is that though IRrate almostdoes not affect the tracking performance it is obvious thatlow EXrate can decline the tracking performance This isbecause the lower EXrate means less measured data gottenand less useful information that can be provided thereforethe estimate is more inaccurate

As to the sampling interval th119894 the lower EXrate means

larger th119894 If the sampling interval th

119894is large enough to break

Shannon Sampling Theorem the estimation performancewill decline

43 The Estimation of Video Target At last we use thedeveloped method to track a target in real scene In orderto decrease the calculation cost we select some frames fromthe video according to the characteristics of the movementThat is if we find that the target is stationary or moves slowlythen we discard these frames We use a threshold to testwhether a target makes a big maneuver or not Obviously alarge threshold canmake the calculation cost lower but lowerEXrate will make the performance decrease too

So the threshold should be carefully selected to balancethe calculation cost and performance Here we select 95frames from 245 frames EXrate and IRrate are 3877 and01367 respectively Figure 9 gives the tracking results ofnumber 1 27 40 65 74 97 128 129 158 181 189 and226 frames in the video The estimation of target is markedby ldquoblackrdquo dot The estimation covariance of RMSE

2D as1034mm is obtained (the tracking area is 300 lowast 300mm2)

5 Conclusions

The main contribution of this paper is to model the real-time system dynamics at the random sampling points (1)By calculating the matrix exponential with inverse Laplacetransform the irregular sampling interval is transformed totime-varying parameters matrix of the system (2) Based onthe statistics relation between the autocorrelation functionand the covariance ofMarkov random processing the systemmodel with online parameter is developedThe proof and theexperimental results show that the developedmethod can getgood tracking performance

As an example the developed method is used for thevideo tracking problem According to the motion character-istics of the target some frames are selected for the trackingpurposeThe tracking results show that good tracking perfor-mance is obtained by a smaller amount of calculation

Disclosure

The authors declare that they have no financial or personalrelationships with other people or organizations that caninappropriately influence their work and there is no pro-fessional or other personal interest of any nature in anyproduct service andor company that could be construed asinfluencing the position presented in this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table2Th

eestim

ationcovaria

ncea

ndirr

egular

ratein

Figure

7

Thev

ideo

(fram

es)

EXrate

IRrate

RMSE

2D

Total

Extracted

Them

odelhere

CVmod

elCA

mod

elSing

ermod

elCu

rrentm

odelI

120572=130

120572max=3

Currentm

odelII

120572=120

120572max=30

Currentm

odelIII

120572=120

120572max=3

IMM

(a)

381

339

8898

00261

1072

2954

1699

3376

9084

1156

1014

2188

(b)

241

188

7801

00514

1276

5416

2103

5137

1210

1201

1301

2451

(c)

221

150

6787

00711

1439

6622

3725

6768

1447

1730

1764

2935

(d)

231

121

5238

01087

1669

3914

2010

3794

9485

1384

1582

3090

(e)

371

107

2884

02124

2334

3101

2388

2712

4961

2073

2010

3743

(f)

211

281327

04635

4093

6985

2592

4974

7378

2314

2606

3814

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Acknowledgments

This work is partially supported by NSFC under Grant nos61273002 and 60971119 and the Importation and Develop-ment of High-Caliber Talents Project of Beijing MunicipalInstitutions no CITampTCD201304025

References

[1] S Srinivasan and H Ranganathan ldquoRFID sensor network-based automation system for monitoring and tracking of san-dalwood treesrdquo International Journal of Computational Scienceand Engineering vol 8 no 2 pp 154ndash161 2013

[2] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[3] F Dornaika and F Chakik ldquoEfficient object detection andtracking in video sequencesrdquo Journal of the Optical Society ofAmerica A vol 29 no 6 pp 928ndash935 2012

[4] J Xue-Bo D Jing-Jing and B Jia ldquoFast tracking for video targettrackingrdquo Applied Mechanics and Materials vol 303-306 pp2245ndash2248 2013

[5] H ZhangM V Basin andM Skliar ldquoIto-Volterra optimal stateestimation with continuous multirate randomly sampled anddelayed measurementsrdquo Institute of Electrical and ElectronicsEngineers vol 52 no 3 pp 401ndash416 2007

[6] J Xue-Bo D Jing-Jing and B Jia ldquoTarget tracking of a lineartime invariant system under irregular samplingrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 11 pp 1ndash12 2012

[7] S Vasuhi V Vaidehi and T Rincy ldquoIMM estimator formaneuvering target tracking with Improved current statisticalmodelrdquo in Proceedings of the International Conference on RecentTrends in Information Technology (ICRTIT rsquo11) pp 286ndash290June 2011

[8] W-S Liu Y-A Li and L Cui ldquoAdaptive strong trackingalgorithm for maneuvering targets based on current statisticalmodelrdquo Systems Engineering and Electronics vol 33 no 9 pp1937ndash1940 2011

[9] WWang andH-LHou ldquoAn improved current statisticalmodelfor maneuvering target trackingrdquo in Proceedings of the4th IEEEConference on Industrial Electronics and Applications (ICIEArsquo09) pp 4017ndash4020 May 2009

[10] H Li and C Li ldquoMissile-borne radar data filtering algorithmbased on the ldquocurrentrdquo statistical modelrdquo Advanced MaterialsResearch vol 433-440 pp 6965ndash6973 2012

[11] Y-L Liu and X-H Gu ldquoCurrent statistical model trackingalgorithm based on improved auxiliary particle filterrdquo SystemsEngineering and Electronics vol 32 no 6 pp 1206ndash1209 2010

[12] X R Li and V P Jilkov ldquoSurvey of maneuvering target trackingPart I dynamic modelsrdquo IEEE Transactions on Aerospace andElectronic Systems vol 39 no 4 pp 1333ndash1364 2003

[13] X Chen Y Pang Y Li and D Li ldquoAUV sensor fault diagnosisbased on STF-Singer modelrdquo Chinese Journal of ScientificInstrument vol 31 no 7 pp 1502ndash1508 2010

[14] T-J Ho ldquoA switched IMM-extended Viterbi estimator-basedalgorithm formaneuvering target trackingrdquoAutomatica vol 47no 1 pp 92ndash98 2011

[15] R W Osborne and W D Blair ldquoUpdate to the hybrid con-ditional averaging performance prediction of the IMM algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 47 no 4 pp 2967ndash2974 2011

[16] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estima-tion of state and parameter with synchrophasors-Part I statetrackingrdquo IEEE Transactions on Power Systems vol 26 no 3pp 1196ndash1208 2011

[17] X Bian X R Li H Chen D Gan and J Qiu ldquoJoint estimationof state and parameter with synchrophasors-Part II parametertrackingrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1209ndash1220 2011

[18] E Wensink and W J Dijkhof ldquoOn finite sample statistics forYule-Walker estimatesrdquo Institute of Electrical and ElectronicsEngineers vol 49 no 2 pp 509ndash516 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of