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Hole Filing IFCNN Simulation by Parallel RK(5,6)
Techniques(Hole Filing by Parallel RK(5,6))
Sukumar Senthilkumar*
Universiti Sains Malaysia
School of Mathematical Sciences11800 USM Pulau Pinang
MALAYSIA
E-mail: [email protected]
Abd Rahni Mt Piah
Universiti Sains Malaysia
School of Mathematical Sciences11800 USM Pulau Pinang
MALAYSIA
E-mail: [email protected]
Abstract — This paper concentrates on employing different
parallel RK(5,6) techniques for hole-filing via unique
characteristics of improved fuzzy cellular neural network(IFCNN) simulation to improve the performance of an
image or handwritten character recognition. Results are
presented according to the range of template selected for
simulation.
Keywords- Parallel 5-order 6-stage numerical integration
techniques, Improved fuzzy cellular neural network, Hole filing,
Simulation, Ordinary differential equations.
I. I NTRODUCTION
Parallel computing techniques are used to carry out
computations simultaneously, operating on the principle that
large problems are often can be divided into smaller ones,
which can then be solved concurrently. It is a simultaneous
process of multiple computing resources to solve a
computational problem easily and quickly. In real time it is practically believed by researchers that a possible way of
solving many significant computationally intensive problems
in science and engineering is by employing parallel algorithmseffectively.
From the literature, it is observed that most of the real time
problems are solved by adapting Runge-Kutta (RK) methods
which in turn are applied to compute numerical solutions for various problems, which are modeled in terms of initial value
problems as in Alexander and Coyle [3], Evans [4], Hung [5],Shampine and Watts [6] and Shampine and Gordon [7].Shampine and Watts [6] developed mathematical codes for
Runge-Kutta fourth order method to solve many numerical
problems. Runge-Kutta formula of fifth order has been
developed by Butcher [8-10] to solve many computational
problems. Evans and Sanugi [11] developed parallel
integration techniques of Runge-Kutta for step by step solutionof ordinary differential equations to obtain results.
Ponalagusamy and Ponammal [12-14] developed new parallel
fifth order algorithm to solve robot arm model, time varying
network for first order initial value problems and new
generalised plasticity equation for compressible powder metallurgy materials with results on stability region for test
equation. Keyes et al. [15] provided a survey towards
applications requiring memories and processing rates of large-scale parallelism, leading algorithmicist applications of
parallel numerical algorithms. Further, focused on practical
medium-granularity parallelism, approachable through
traditional programming languages. Gear [16] gave the
potentiality behavior for parallelism in solving real time problems using ordinary differential equations. A survey of potential for parallelism in Runge-Kutta techniques and
parallel numerical techniques for initial value problems for
ordinary differential equations are demonstrated by Norsettand Jackson [17] and Jackson [18]. Using fourth order explicit
Runge-Kutta method, a parallel mesh chopping algorithm for a
class of initial value problem is illustrated by Katti and
Srivastava [19]. Harrer et al. [20] introduced explicit Euler,
predictor-corrector and fourth-order Runge-Kutta algorithmsfor simulating cellular neural networks. The RK-Butcher
algorithm has been introduced by Bader [21, 22] for finding
truncation error estimates, intrinsic accuracies and earlydetection of stiffness in coupled differential equations that
arises in theoretical chemistry problems. Senthilkumar and
Piah [23] implemented parallel Runge-Kutta arithmetic mean
algorithm to obtain a solution to a system of second order
robot arm. In this paper a new attempt has been made toemploy parallel RK(5,6) algorithm for hole filing problem
under IFCNN environment. Oliveira [24] introduced a popular
sequential RK-Gill algorithm to evaluate effectiveness factor
of immobilized enzymes.
This research work is carried out by the first author under a post doctoral
fellow scheme at the School of Mathematical Sciences, Universiti Sains
Malaysia, 11800 USM Pulau Pinang, MALAYSIA.
*Corresponding Author.
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Computing value is easy in case of implementing VLSICNN chips, thereby making real-time operations possible.
Roska [28] and Roska et al. [29] have presented the first
widely used simulation system which allows simulation of alarge class of CNN and is especially suited for image
processing applications. It also includes signal processing,
pattern recognition and solving ordinary and partial
differential equations, as in Gonzalez et al. [30]. The existingRK-Butcher fifth order method hole filing problem has beenstudied by Murugesh and Badri [32] via CNN simulation
model. Similarly, hole filing problem has been analyzed by
Murugesan and Elango [50] by means of existing RK fourthorder method under CNN simulation. Dalla Betta et al. [46]
implemented CMOS implementation of an analogy
programmed cellular neural network. Anguita et al. [31]
discussed in detail about parameter configurations for hole
extraction in cellular neural networks.
Zadeh [35] and Zadeh et al. [36] introduced the concept of
fuzzy sets (FSs) theory. Different notions of higher-order FSs
have been proposed by different researchers. Recently, fuzzycellular neural network (CNN) model [43-45] has attracted a
great deal of interest among researchers from different
disciplines. A locally interconnected, regularly repeated,
analogue (continuous- or discrete-time) circuits with a one-or-
two-or three-dimensional grid architecture called CNNsintroduced by Chua and Yang [25-26] and Chua [27]. Each
cell (neuron) in CNN is a non-linear dynamic system coupled
only to its nearest neighbors. Because of this localinterconnection property, CNNs have been considered
specifically suitable for very-large-scale integration
implementations. Shitong et al. [37] proposed improved fuzzy
cellular networks to incorporate the novel fuzzy status
containing the useful information beyond a white blood cellinto its state equation, resulting in enhancing the boundary
integrity. Laiho et al. [38] proposed template design for CNNs
with 1-bit weights.
This paper is ordered as follows. A brief introduction on
improved fuzzy cellular non-linear network is presented in
section 2. Section 3 deals with the performance of hole-filler
template design and simulation results. Section 4 discusses
parallel RK(5,6) numerical integration techniques. Finally,
concluding remarks is presented in section 5.
II. A BRIEF OVERVIEW OF IFCNN
The capability of the conventional cellular neural network
to solve different kinds of image processing problems and the
capability of fuzzy logic to cope with uncertainty in imagesare the inherent features of FCNN [37]. Moreover, it also has
inbuilt connections with mathematical morphology. The
unique characteristic of IFCNN is incorporating novel fuzzystatus with feed-forward and feedback templates in FCNN
such that the useful information beyond the region can be
sufficiently utilized. FCNN is a locally connected network [37] and the output of a neuron is connected to the inputs of
every neuron/cell in its r × r neighborhood, and similarly the
inputs of a neuron are only connected to the outputs of every
neuron in its r × r neighborhood. It is apparent that feedback
(not recurrent) connections are presented in detail. Thearchitecture of IFCNN is shown in Figure 1.
Figure 1. Architecture of IFCNN
The state equation of IFCNN is given by,
( , ) ( , )
( , ) ( , )
min( , ) ( , )
max( , ) ( , )
min( , ) ( , )
( , )
1( , ; , )
( , ; , )
( ( , ; , ) )
( ( , ; , ) )
( ( , ; , ) )
r
r
r
r
r
r
ij
ij kl
c k l N i j x
kl
c k l N i j
ij f klc k l N i j
f klc k l N i j
f klc k l N i j
c k l N
dxc x A i j k l y
dt R
B i j k l u
I A i j k l y
A i j k l y
B i j k l u
∈
∈
∈
∈
∈
∈
−= + +
+ + ∧ + +
∨ + +
+ ∧ +
∨
∑
∑
%
%
%
%max
( , )
min( , ) ( , )
max( , ) ( , )
( ( , ; , ) )
( ( , ; , ) )
( ( , ; , ) )
r
r
f kli j
f klc k l N i j
f klc k l N i j
B i j k l u
F i j k l x
F i j k l x
∈
∈
+
∧ +
∨
%
%
(1)
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and the input equation of Cij is given by,
0,ij iju E = ≥ (2)
1 ≤ i ≤ M ; 1 ≤ j ≤ N .
the output equation of C ij is given by,
1( ) 1 1 ,
2ij ij ij ij y f x x x⎡ ⎤= = + − −⎣ ⎦ (3)
1 ≤ i ≤ M ; 1 ≤ j ≤ N .
The constraints /conditions are given by
Afmax(i, j;k ,l) = Afmin(k ,l;i, j);
Afmax(i, j;k ,l) = Af max(k ,l;i, j);
F fmax(i, j;k ,l) = F fmin(k ,l;i, j);
F fmax(i, j;k ,l) = F f max(k ,l;i, j);
1 ≤ i ≤ M ; 1 ≤ j ≤ N . (4)
1)0( ≤ij x ;1 ≤ i ≤ M ; 1 ≤ j ≤ N .
1)0( ≤iju ;1 ≤ i ≤ M ; 1 ≤ j ≤ N .
),;,(),;,( jilk Alk ji A =
From the above Eqs. (1) - (4),~
∧ ,~
∨ , N r (i, j), and A areidentical as in FCNN. Comparing (4) with FCNN, the only
one discrepancy between the equation is the novel fuzzy
status.
~
min( , )
~
max
( , )
( ( ( , ; , ) )
( ( , ; , ) )
))
kl r
kl r
f klc N i j
f kl
c N i j
kl
F i j k l x
F i j k l x
x
∈
∈
+ +
+ +
∧
∨ (5)
is adhered to Eq. (1), which obviously reflects the required
information where F fmin(i, j;k ,l) and F fmax(i, j;k ,l) indicates theconnected weights between cell C ij and C kl respectively.
Hence, the complete template determines the connection
between cell and its neighbors, consists of (2r × 1) and (2r ×1) matrices A, B, F fmin and F fmax. The symmetric matrices are
considered in the above template to congregate the IFCNN’ssymmetric requirements.
III. A BRIEF SKETCH ON HOLE-FILLER AND SIMULATION
RESULTS
In a bipolar image, all the holes are filled and remains
unaltered outside the holes, in case of hole filing IFCNN
simulation [46-50]. Allow 1,1 == C R x and take +1 to
represent the black pixel and –1 for the white pixel. If the
bipolar image is input with ijuU = into IFCNN and images
having holes are enclosed by the black pixels, then initial state
values are set to be 1)0( =ij x . The output values are obtained
as N j M i yij ≤≤≤≤= 1,1,1)0( from equation (1).
Consider the templates A, B and independent current source I
as
,
00
00
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
a
aba
a
A a > 0, b > 0
(6)
,
000
040
000
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
= B I = -1
where the template parameters a and b are to be determined. In
order to make the outer edge cells become the inner ones,
normally auxiliary cells are added along the outer boundary of
the image and their state values are set to be zeros by circuit
realization resulting in zero output values. The state equation
(1) can then be rewritten as
min( , ) ( , )
max( , ) ( , )
( ( , ; , ) )
( ( , ; , ) ) 4 ( ) .
r
r
ij
ij f klc k l N i j
f kl ijc k l N i j
dx x A i j k l y
dt
A i j k l y u t I
∈
∈
= − + ∧ + +
∨ + + −
%
%
(7)
For instance, here the cells C (i+1, j), C (i-1, j), C (i, j+1) andC (i, j-1) are non-diagonal cells. Designing of hole-filler template [31] and its various sub-problems are discussed using
CNN simulations [46-50]. Figures 2 and 3 show the hole filing
of an image (before and after) by employing a parallel
RK(5,6) type-III technique. The settling time Ts andcomputation time Tc for different step sizes are considered for
the purpose of comparison. The settling time Ts is the time
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Figure 2(a). Original image and hole filed image
Figure 2(b). Original image and hole filed image
from start of computation until the last cell leaves the interval
[-1.0, 1.0]
Figure 2. Hole filing before and after adapting type-III parallel RK(5,6)technique
Figure 3. Hole filing before and after employing type-III parallel RK(5,6)
technique
Figure 4. Range of the template
which is based on a specified limit (e.g., |d x/dt |< 0.01). The
computation time Tc is the time taken for settling the network
and adjusting the cell for proper position once the network issettled. The simulation shows the desired output for every
neuron/cell. Specifically, note that +1 and -1 indicate the black
and white pixels, respectively. The marked selected template
parameters a and b are restricted to the shaded area, as shownin figure 4 for the simulation.
IV. PARALLEL R UNGE-K UTTA FIFTH ORDER TECHNIQUES: A
BRIEF OVERVIEW
A. Parallel Runge-Kutta 5-Order 6-Stage Type-I Technique
A parallel Runge-Kutta 5-order 6-stage type-I technique [12-14] is one of the simplest method used to solve ordinary
differential equations. It is an explicit formula which adapts
the Taylor’s series expansion in order to obtain the
approximation. A parallel Runge-Kutta 5-order 6-stage type-I
technique is used to determine y j and , 1, 2, 3,.... j y j m=&such
that
]90
7
90
32
90
2
90
32
90
7[ 654311 k k k k k y y nn +++++=+
(8)
Thus, the corresponding parallel Runge-Kutta 5-order 6-stage
type –I technique of Butcher array represents
0
5
2
5
2
4
1
64
11
64
5
2
1
16
3
16
5
4
3
32
9
32
27−
4
3
16
9
128
9−
28
350
7
12−
7
8
90
70
90
32
90
2
90
32
90
7
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The 6 stage 5th order algorithm with 5 parallel and 2 processors, by selecting a43 = 0 to evaluate k 3 and k 4
simultaneously is given by
. ))((1 nij
ijt xtf k Δ= ,
2 12 2( ( ) )5 5
ij ij
ij nt k tf x t k Δ= Δ + + ,
3 1 2
11 5( ( ) )
4 64 64
ij ij ij
ij n
t k tf x t k k
Δ= Δ + + + =
*
3
ijk ,
4 1 2
3 5( ( ) )
4 16 16
ij ij ij
ij n
t k tf x t k k
Δ= Δ + + + =
*
4
ijk ,
5 1 2 3 4
9 275 3 9( ( ) )
2 32 32 4 16
ij ij ij ij ij
ij n
t k tf x t k k k k
Δ= Δ + − + +
=*
5
ijk ,
6 1 2 4 5
3 9 35 12 8( ( )
4 28 28 7 7
ij ij ij ij ij
ij nk tf x t t k k k k = Δ + Δ − + − +
=*
6 .ijk (9)
Therefore, the final integration is a weighted sum of the five
calculated derivatives which is given as
1
1 3 4 5 6( ( )) [7 32 12 32 7 ].
90
n
n
t
ij ij ij ij ij
t
t f x t dt k k k k k
+ Δ= + + + +∫
(10)
B. Parallel Runge-Kutta 5-Order 6-Stage Type-II Technique
A parallel Runge-Kutta 5-order 6-stage type-II technique [12-
14] is also one of the simplest method used to solve ordinarydifferential equations. It is an explicit formula which adapts
the Taylor’s series expansion in order to obtain the
approximation. A parallel Runge-Kutta 5-order 6-stage type-II
technique determines y j and m j y j ,....3,2,1, =& such that
1 1 3 5 6
23 125 81 125[ ].90 192 192 192
n n y y h k k k k + = + + − +
(11)
Thus, the corresponding parallel Runge-Kutta 5-order 6-stage
technique of type-II Butcher array represents
0
3
1
3
1
5
2
. 25
4
25
6
2
1
4
1-3
4
15
3
2
81
6
81
90−
81
50
81
8
5
4
75
6−
75
36
75
10
75
8
192
230
192
1250
192
812−
192
125
Therefore, the final integration is a weighted sum of four calculated derivatives per time step which is given by
]192
125
192
81
192
125
90
23[ 65311 k k k k h y y nn +−++=+ .
(12)
The 6 stage 5th order algorithm with 5 parallel and 2
processors by selecting a65 = 0 to evaluate k 5 and k 6 simultaneously is given by
. ))((1 nij
ijt xtf k Δ= ,
ij
nij
ij k t
t xtf k 123
1))
3(( +
Δ+Δ= ,
ijij
nij
ijk k
t t xtf k 213
25
6
25
4))
3
2(( ++
Δ+Δ= ,
4
153
4)
2( 3
21
4
ij
ij
ij
n
ij k k
k t t tf k +−+
Δ+Δ=
ijijij
nij
ij kk k k t t xtf k 321581
8
81
50
81
90
81
6)
3
2(( +−−+Δ+Δ=
=*
5
ijk
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6 1
2 3 4
4 6( ( ) )
5 75
36 10 8
75 75 75
ij ij
ij n
ij ij ij
t k tf x t k
k k k
Δ= Δ + − +
+ +
=*
6 .ijk (12)
Therefore, the final integration is a weighted sum of the five
calculated derivatives which is given by1
1 3 5 6( ( )) [23 125 81 125 ].192
n
n
t
ij ij ij ij
t
t f x t dt k k k k
+ Δ= + − +∫
(13)
C. Parallel Runge-Kutta 5-Order 6-Stage Type-III Technique
A parallel Runge-Kutta 5-order 6-stage type-III technique [12-
14] is another simple method used to solve ordinarydifferential equations. It is also an explicit formula which
adapts the Taylor’s series expansion for an approximation. A
parallel Runge-Kutta 5-order 6-stage type-III technique
determines y j and , 1, 2, 3,.... j y j m=&such that
1 1 3
4 5 6
17 250[306 153
442 8192 31].
255 9945 234
n n y y h k k
k k k
+ = + − +
+ +
(14)
Thus, the corresponding parallel Runge-Kutta 5-order 6-stage
technique of type-III Butcher array represents
0
5
1
5
1
5
2.
160
39
32
5
2
1
24
1
24
5−
3
2
16
3
8
1
16
3−
4
1
114
9−
14
15
7
8
7
12
7
8
306
17−0
153
250−
255
442
9945
8192
234
31
Therefore, the final integration is a weighted sum of fivecalculated derivatives per time step which is given by
1 1 3
4 5 6
17 250[306 153
442 8192 31
].255 9945 234
n n y y h k k
k k k
+ = + − +
+ +
(15)
The 6 stage 5th order algorithm with 5 parallel and 2 processors by selecting a54 = 0 to evaluate k 5 and k 4
simultaneously is given by
. ))((1 nij
ijt xtf k Δ= ,
ij
nij
ij k t
t xtf k 125
1))
5(( +
Δ+Δ=
ijij
nij
ij k k t
t xtf k 21332
5
160
39))
5
2(( ++
Δ+Δ= ,
3
2
24
5
24)
2( 321
4
ijijij
n
ij k k k t t tf k +−+
ΔΔ= =
*
4
ijk
ijijij
nij
ij k k k t t xtf k 32154
1
16
3
8
1)
16
3(( −−+Δ+Δ= =
*
5
ijk
6 1
2 3 4 5
9
( ( ) ) 14
15 8 12 8.
14 7 7 7
ij ij
ij n
ij ij ij ij
k tf x t t k
k k k k
= Δ + Δ − +
+ − +
(16)
Therefore, the final integration is a weighted sum of the fivecalculated derivatives which is given by
1
31
5 64
25017( ( )) [
306 153
8192 31442
].255 9945 234
n
n
t ijij
t
ij ijij
k k f x t dt t
k k k
+
= Δ − − +
+ +
∫(17)
V. CONCLUDING REMARKS
In this paper, hole filing problem is addressed under IFCNNmodel using parallel RK(5,6) techniques and its validity is
illustrated by simulation results. It is observed that the hole is
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filled and the outside image remains unaffected, that is, theedges of the images are preserved and are intact. The
templates of the cellular neural network are not unique and
this is important in its implementation. The significance of thiswork is to improve the performance of handwritten character
recognition because in many language scripts, numerals and in
images etc., there are many holes and the CNN described
above can be used in addition to the connected componentdetector. It is also noticed that IFCNN preserves the boundaryintegrity.
ACKNOWLEDGMENT
The first author would like to extend his sincere gratitude
to Universiti Sains Malaysia for supporting this work under its
post doctoral fellowship scheme. Much of this work was
carried out during his stay at Universiti Sains Malaysia in
2011. He wishes to acknowledge Universiti Sains Malaysia’sfinancial support.
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Senthilkumar was born in Neyveli Township,
Cuddalore District, Tamilnadu, India on 18th July
1974. He received his B.Sc in Mathematics from
Madras University in 1994, M.Sc in Mathematics
from Bharathidasan University in 1996, M.Philin Mathematics from Bharathidasan University in
1999 and M.Phil in Computer Science &
Engineering from Bharathiar University in 2000.
He also has a PGDCA and PGDCH in Computer
Science and Applications and Computer Hardware from BharathidasanUniversity which he obtained in 1996 and 1997, respectively. He has a
doctoral degree in Mathematics and Computer Applications from National
Institute of Technology [REC], Tiruchirappalli, Tamilnadu, India. Currently,he is a post doctoral fellow at the School of Mathematical Sciences, Universiti
Sains Malaysia, 11800 USM Pulau Pinang, Malaysia. Prior to this
appointment, he was a lecturer/assistant professor in the Department of Computer Science at Asan Memorial College of Arts and Science, Chennai,
Tamilnadu, India. He has published many good research papers in
international conference proceedings and peer-reviewed/refereed international journals with high impact factor. He has made significant and outstanding
contributions to various activities related to research work. He is also an
associate editor, editorial board member, reviewer and referee for many
scientific international journals. His current research interests includeadvanced cellular neural networks, advanced digital image processing,
advanced numerical analysis and methods, advanced simulation and
computing and other related areas.
Abd Rahni Mt Piah was born in Baling, Kedah Malaysia on 8th May 1956. Hereceived his B.A. (Cum Laude) in Mathematics
from Knox College, Illinois, USA in 1979. He
received his M.Sc in Mathematics from
Universiti Sains Malaysia in 1986. He obtainedhis Ph.D in Approximation Theory from the
University of Dundee, Scotland UK in 1993. Hehas been an academic staff member of the School
of Mathematical Sciences; Universiti Sains
Malaysia since 1981 and at present is an
Associate Professor. He was a program chairman and deputy dean in theSchool of Mathematical Sciences, Universiti Sains Malaysia for many years.
He has published various research papers in refereed national and international
conference proceedings and journals. His current research areas include
Computer Aided Geometric Design (CAGD), Medical Imaging, Numerical
Analysis and Techniques and other related areas.
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