ECCTD’03 - European Conference on Circuit Theory and Design, September 1-4, 2003, Cracow, Poland

Small-World Cellular Neural Networks
for Image Processing Applications
Kazuya Tsuruta ∗Zonghuang Yang∗

Yoshifumi Nishio∗

Akio Ushida∗

Abstract — In this paper, we propose a Small-World
Cellular Neural Network (SWCNN) model, which isconstructed by introducing some random couplings
between cells to adding to the original Chua-Yang
CNN. Applying this SWCNN model to some applications of image processing such as small object
remover, edge detecttiom, etc. are investigated. We
found from numerical simulations that the SWCNN
can improve the results of the image output in acertain content.

between regular and random networks by introducing the concept of connection topology of smallworld network in CNN. By applying the SWCNN
to the existing CNN image processing applications
such as small object remover, grey-scale to binary
encoding and edge detection of grey-scale image,some interesting results are given.

1In this section, we describe the connection topology of the SWCNN composed of a two-dimensional
M by N array structure. Basically, the SWCNNis obtained by adding some random couplings between cells in the original Chua-Yang CNN. Here,
we introduce a probability pc , which means whatpercentage of the cells in the CNN occur to random
coupling to another cell. We assume that, besides
its local couplings, each cell in the array is only
permitted a maximum of one random coupling toanother cell, moreover, the coupling is bidirectional.
Thereby, for a M × N array, it has a maximum Pcof M × N/2 pairs of cell couplings. Figure 1 showsa sketch map of the SWCNN consisting of 4 × 4cells. Obviously, when pc = 0, the SWCNN is completely the same with the original CNNs, and themaps correponding to 0 < pc < 1 and pc = 1 arerespectively shown in the middle part and the righthand side of the Fig.1. The state equation of eachcell c(i, j) of the SWCNN is formulated by Eq. (1),and the output equation is defined by Eq. (2).

INTRODUCTION

2SMALL-WORLD CNN STRUCTURE

Studies of network map are very important, because they help us to understand the basic featuresand requirements of various systems. So far many
connection topologies of network assumed to be either completely regular or completely random havebeen studied in the past. Cellular Neural Network(CNN) model invented by Chua and Yang in 1988
[1] is a typical of those completely local connectivities, which is presented as a preferred implementation of locally and regularly coupled neuralnetworks. The CNN has been successfully used forvarious high-speed parallel signals processing applications such as image processing, pattern recognition as well as modeling of various phenomena in
nonlinear systems [1]–[6]. However, in many casesin real life, many network topologies such as biological, technological and social networks are known to
be not completely random nor completely local but
somewhere in between. This was modeled in an interesting work by Watts and Strogatz in 1998 [7]as the small-world model. The model is a networkẋij (t) = −x
ij (t) + I
consisting of many local links and fewer long range+ c(k,l)∈Nr (i,j) A(i, j; k, l)ykl (t)(1)‘short cuts’. Therefore, it has a high clustering coef+ c(k,l)∈Nr (i,j) B(i, j; k, l)ukl (t)ficient like regular lattices and a short characteristic+ Wc M (i, j; p, q)ypq (t)path length of typical random networks. Interesting examples are shown by collaboration of movie
1(2)yij (t) = (|xij (t) + 1| − |xij (t) − 1|)stars, connectivity of internet web pages or neu2
ral nets, etc. Up to the present, many phenomenaof synchronous chaos in networks with samll-world
i = 1, 2, ..., M, j = 1, 2, ..., N.
interactions have been investigated [8] – [10].
In this paper, we propose a Small-World CNN where Nr (i, j) denotes the neighbor cells of radius(SWCNN) model to investigate the middle ground r of a cell c(i, j); A, B, and I are real constantscalled as feedback template, control template and∗ Department
of Electrical and Electronic Engi- bias current, respectively; xij , yij , uij denotes theneering, Tokushima University, 2-1 Minami-Josanjima,state, input and output of the cell, respectively;
Tokushima, Japan, e-mail: [tsuruta, yangzh, nishio,ushida]@ee.tokushima-u.ac.jp, tel: +81 88 656 7470, fax: M (i, j; p, q) describes the small-world map that is+81 88 656 7471.
randomly created by program with indicating theI-225

probability pc in advance, if there is a coupling be- well as shown in Fig. 2(c). The randomly coupledtween one cell c(i, j) and another cell c(p, q), then weight wc = 2.0, the other template parameters
the M (i, j; p, q) is equal to 1, otherwise is zero; and hold unchanged.
Wc stands for the coupling weight between the randomly coupled cells.
Now, we apply the SWCNN to invesigate its abilities in image processing applications.

(a)Figure 1: A two-dimensional SWCNN architecturewith different probability pc .
3

APPLYING TO IMAGE PROCESSINGAPPLICATIONSImage processing is one of important applications ofthe CNN. As well known, many templates and algorithms used for this purpose have been developed.
In this section, we investigate the effect of applying
the SWCNN to existing image processing application, such as small object remover, grey-scale tobinary encoding and grey-scale edge detection.
3.1

Small object remover

1 1A= 1 21 1101 , B =  01002000 ,0I=0(3)The simulation result is shown in Fig. 2, whereFig. 2(a) is the input image and Fig. 2(b) is theresult for the original CNN. In this simulation, theboundary condition is set as zero-flux and the initial state of the network is initialized by the input
image. We can see that some bigger objects remain in the output image yet. If we applying theSWCNN with the probabilty pc = 1 to the above
problem, those bigger objects can be deleted as(c)Figure 2: An simulation example for small object
remover in binary image, where (a) is the input andthe initial state image, (b) is the output result of
the SWCNN with pc = 0, and (c) is the output
result of the SWCNN with pc = 1.0.
3.2This task is to delete noise in binary image. Here,we consider a SWCNN array consisting of 64 × 64
cells. If we use the SWCNN with the randomly
coupled probability pc = 0 (i.e. the SWCNN isthe same with the Chua-Yang original CNN), thenthe SWCNN can filter those isolated black/white
dots or lines with one pixel width by adopting thefollowing template (3).
(b)Grey-scale to binary encoding

In general, this task is defined as changing agrayscale image into a binary image. The black(white) pixels in the output image correspond to
the locations in the gray-scale image where the average of pixel intensities over the local couplingradius r feedback convolution window is positive(negative). If we do not consider the average concept, then the task can be simply obtained byadopting the following template (4).
0A= 00010
000 , B= 000040

00 ,0I=0(4)The simulation result is shown in Fig. 3, whereFig. 3(a) is the input gray-scale image and Fig.
3(b) is the output binary image for the original
CNN. Now, we investigate the effect by appling theSWCNN to this problem with the various different
probabilities pc . Figures 3(c)-(f) show the simulation results that respectively correspond to theI-226

probability pc adopted by 0.1, 0.3, 0.5 and 1.0. In
all these cases, the coupling weight is
wc = 1.0.

(5)These simulation results show that, by using theSWCNN, the binary output images are to use thedensity of black pixels in certain area to represent
the gray-scale of the input image. Comparing withthe result Fig. 3(b). The quality of their output images is optimized by human visualisation. With theprobability increasing, the density increases. The
better visual effect is shown in Fig. 3(d) obtainedby pc = 0.3. The SWCNN does the work as theHalftoning CNN does in the original CNN model.
Moreover, we confirmed that the speed of the convergence is improved.
3.3

(a)

(b)

(c)

(d)

(e)

(f)

Gray-scale edge detection

The edge detection extracts edges of objects in abinary image where each black pixel with at least
one white nearest neighbor is defined to be an edge
cell. The template (5) is designed to work well forbinary iuput images only.
0
A= 00
−1 −1 −10 08 −1  ,1 0 , B = −1−1 −1 −10 0
I = −1.

(6)

If the input image is a gray-scale image, the output may not be a binary image, and will in general be gray-scale where black pixels correspond tosharp edges, near-black pixels correspond to fuzzyedges, and near-white pixels correspond to noise.
The simulation result is shown in Fig. 4(b). Figure 4(a) is the input gray-scale image. Similarly,
we apply the SWCNN model with different random coupling probabilities to this problem. Under
the above template (6) and the random couplingweight wc = −0.5, the simulation results are shownin Figs. 3(c)-(f), which correspond to the coupling
probabilities pc = 0.1, 0.3, 0.5 and 1.0, respectively.
we observe from the simulations that theSWCNN can detect some edges such as the pillar in
the left hand side of the image, the pattern of thehat, the face characters, etc. that can not be detected by the original CNN by using the SWCNN.
Moreover, the ability of the edge detection become
higher with the probability increases. The better
result shown in Fig. 4(f) is obtained by pc = 1.0.
Therefore, we can conclude from these simulation
results that the application of the SWCNN to thegray-scale edge detection problem could improvethe efficiency of the original CNN.

Figure 3: An example for gray-scale image to binary encoding by using SWCNN. (a) is the input
and initial state image, (b) is the binary output image for the original CNN, (c)-(d) are the binary output images obtained by using the SWCNN with theprobability p = 0.1, 0.3, 0.5 and 1.0, respectively.
4

CONCLUSIONSIn this study, we have proposed a SWCNN model,
and preliminary discussed the image processing applications of the SWCNN such as small object remover, gray-scale to binary encoding and grey-scaleedge detection. Although these problems can besolved by the original CNN, we found from our simulation results that the application of the SWCNN
to these problems permit to improve the quality ofthese results with the same parameters of the templates A, B and I. Due to introducing the random
coupling between the cells in the SWCNN, we will
try to analysis the SWCNN in statistical method
and find more applications in the future works.

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(f)

Figure 4: An example for gray-scale edge detection by using SWCNN. (a) is the input and initialstate image, (b) is the edge output image for theoriginal CNN, (c)-(d) are the binary output images
obtained by using the SWCNN with the probability
p = 0.1, 0.3, 0.5 and 1.0, respectively.
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