kalman filter partilce tracking

Kalman Filtering

Presented by

Muhammad Irfan Anjum

Introduction

Dynamical Signal Models

Scalar Kalman Filter

Vector Kalman Filter

Extended Kalman Filter

Simulation results

Outline

Introduction

• Uses a series of measurements over time, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone.

• Operates recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state.

• Two step process– Estimates of current state variables with their uncertainties.– Estimates are updated using weighted average after observing

output.• Operates on real time data, no additional past information is required.

Dynamical Signal Models

][][][ nwnAnx

][][ˆ nxnA

][][ nwAnx

Gauss Markov Process

1st order Gauss Markov Process:

Vector Gauss-Markov Model:

n

k

kn nBuAsAns0

1 ]1[]1[][

0],[]1[][ nnBunAsns

][]1[][ nunasns

n

k

kn nuasans0

1 ]1[]1[][

snansE 1])[(

1Az

][ns][nu

B

Scalar Kalman Filter

1az

(a) Dynamical Model

][ns][nu

(b) Kalman Filter

]1|[̂ nns

][nx

1az

][ˆ nu ]|[̂ nns][nK

][~ nx

][nw

][ns

Scalar Kalman Filter

][]1[][ nunasns

]1|[

]1|[][ 2

nnM

nnMnK

w

22 ]1|1[]1|[ unnMannM

]1|1[̂]1|[̂ nnsanns

][][][ nwnsnx

]1|[])[1(]|[ nnMnKnnM

])1|[̂][]([]1|[̂]|[̂ nnsnxnKnnsnns

Transmitted Signal:

Received Signal:

Minimum Prediction MMSE:

Minimum MSE:

Correction:

Kalman Gain:

Prediction:

Vector Kalman Filter

1Az

][ˆ nu ]|[̂ nns

]1|[̂ nns

][nK][~ nx

][nx

][nw

][ns

][nh

][nh

1Az

][ns][nu

B

Scalar state Vector Kalman Filter

]1|[])[][(]|[

])1|[̂][][]([]1|[̂]|[̂

][]1|[][

][]1|[][

]1|1[]1|[

]1|1[̂]1|[̂

][][][][

][]1[][

2

nnMnhnKInnM

nnsnhnxnKnnsnns

nhnnMnh

nhnnMnK

BQBnnAMnnM

nnsAnns

nwnsnhnx

nBunAsns

T

T

Tn

T

T

Transmitted Signal:

Received Signal:

Prediction:

Minimum Prediction MMSE:

Kalman Gain:

Correction:

Minimum MSE:

Vector state Vector Kalman Filter

]1|[])[][(]|[

])1|[̂][][]([]1|[̂]|[̂

][]1|[][][

][]1|[][

]1|1[]1|[

]1|1[̂]1|[̂

][][][][

][]1[][

nnMnHnKInnM

nnsnHnxnKnnsnns

nHnnMnHnC

nHnnMnK

BQBnnAMnnM

nnsAnns

nwnsnHnx

nBunAsns

T

T

T

Transmitted Signal:

Received Signal:

Prediction:

Minimum Prediction MMSE:

Kalman Gain:

Correction:

Minimum MSE:

Extended Kalman Filter

][])1[(][ nBunsans

Vector Kalman Filter

Extended Kalman Filter

][])[(][ nwnshnx

][][][][

][]1[][

nwnsnHnx

nBunAsns

]1|1[ˆ]1[|]1[

])1|1[̂(])1[(

nnsnsns

annsansa

]1|[ˆ][|][

])1|[̂(])[(

nnsnsns

hnnshnsh

]1|1[ˆ]1[|]1[

]1[

nnsnsns

anA ]1|1[ˆ][|

][][

nnsnsns

hnH

Extended Kalman Filter

]))1|[̂(][]([]1|[̂]|[̂ nnshnxnKnnsnns

TT BQBnAnnMnAnnM ]1[]1|1[]1[]1|[

])1|1[̂(]1|[̂ nnsanns

][]1|[][][

][]1|[][

nHnnMnHnC

nHnnMnK

T

T

]1|[])[][(]|[ nnMnHnKInnM

Particle Tracking using Scalar Kalman filter

MMSE in Scalar Kalman filter particle tracking

Particle Tracking using Vector Kalman filter

Particle Tracking using Extended Kalman filter