Output Characteristics of Cellular Neural Networks
Using Mixture Template
Takashi Inoue, Masaru Nakano, Yoshifumi Nishio

Abstract— In this research, we propose cellular neural networks using mixture template as an example of space-varying
cellular neural networks. As the ﬁrst step of the investigationof such complex nonlinear circuit networks, we propose two
mixing methods of the templates and investigate the output
characteristics of the simple image processing with a binary
image and a grayscale image by computer simulations.

I. I NTRODUCTIONELLULAR Neural Networks (CNN) [1] were introduced by Chua and Yang in 1988. The idea of the CNN
was inspired from the architecture of the cellular automata
and the neural networks. Unlike the conventional neuralnetworks, the CNN has local connectivity property. Since
the structure of the CNN resembles the structure of animals’
retina, the CNN can be used for various image processing
application [2]-[4] including character extractions [5][6].
Further, the CNN can be utilized to produce some kinds ofpattern generation [7][8].
Wiring weights of the cells are established by parameters
called the template. The performance of the CNN is decided
by the template. If the templates of all the cells in the CNN
are identical, the system is called space-invariant, while if
the templates of all the cells in the CNN are not identical,
the system is called space-varying. Typically, space-invariant
system is mainly used in the studies of the CNN, because
the implementation cost is cheap. However, on the ﬂip side,
the performance of the system is considered to be limited.
In this research, we propose the CNN with mixture template as an example of space-varying CNN. By mixing two
different templates, we expect that the proposed CNN possesses the both effects of the two templates. It may be usefulto reduce the number of the successive precessing for somekinds of image processing tasks. Further, we are interested
in the complex behaviors of such a new class of nonlinear
circuit networks. They can be utilized for future engineering
applications. As the ﬁrst step of the investigation of thisconcept, in this research, we propose two mixing methods ofthe templates and investigate the output characteristics of thesimple image precessing with a binary image and a grayscale
image.
In the Sec. 2, we review the basic of the standard CNN.
In the Sec. 3, we show how to mix the templates and show
some computer simulated results. In the Sec. 4, noise removal
performance of the proposed mixture templates is evaluated.
In the Sec. 5, simulation results for a grayscale image areC

Department of Electrical and Electronic Engineering, Tokushima University, 2-1 Minami-Josanjima, Tokushima 770-8506, Japan (email: {takashi,
nakano, nishio}@ee.tokushima-u.ac.jp).

shown. In the Sec. 6, simulation results for a texture image
are shown. Section 7 concludes the article.
II. C ELLULAR NEURAL NETWORKS [1]In this section, we describe the basic structure of the CNN.
The CNN has M by N processing unit circuits called cells
as Fig. 1. Cells are arranged in a reticular pattern to M
line N row. We represent a cell C(i, j) using a variable
i which denotes vertical position and a variable j which
denotes horizontal position. The cell contains linear andnonlinear circuit elements. The CNN is an array of cells.
Each cell is connected to its neighboring cells according toa template. Usually, the template is the same for all cells
except for boundary cells. The CNN has the features oftime continuity, spatial discreteness, nonlinearity and parallel
processing capability.
The state equation and the output equation of the cell are
shown as follows.
State equation:
dvxij
dt

i+r


j+r


A(i,j;k,l) vykl (t)k=i−r l=j−r

+

i+r


j+r


B(i,j;k,l) vukl (t) + I(1)

k=i−r l=j−r

Output equation:
1(|vxij (t) + 1| − |vxij (t) − 1|) (2)2where vx , vy and vu represent a state, an output and an inputof cell, respectively. In the equation (1), A is the feedback
template and B is the control template. These and bias I arecollectively called general template.
The r − neighborhood of the cell C(i, j) in the CNN isdeﬁned by
vyij (t) =N r(i, j) = {C(k, l) | max { |k − i| , |l − j| } ≤ r,
1 ≤ k ≤ M; 1 ≤ l ≤ N}

(3)

where r is a positive integer number. In our study, we ﬁx thevalue of r as 1.
III. M IXTURE TEMPLATE
Usually, it is desirable that all of the templates at allcells are identical, namely, space-invariant. However, we areinterested in what kind of phenomena can be observed fromspace-varying CNN, because such systems are new class ofnonlinear circuit networks and they may have a possibility
for novel engineering applications.

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c978-1-4244-1821-3/08/$25.002008IEEE= −vxij +In this research, we propose the CNN with two kinds of
mixture templates, matrix mixing and element mixing, as an
example of space-varying CNN,
A. Matrix mixing
In the ﬁrst method of the mixing, we prepare two different
CNN templates and choose one of them for each cell at
random.

(a)

(b)

(c)

Fig. 3. The performance of Patch Maker. (a) Input image. (b) Transitional
image. (c) Output image.

(a)

(b)

(c)

Fig. 1. Examples of matrix mixing. Black and white cells correspond totwo different CNN templates. (a) white : black = 75 : 25. (b) white :black = 50 : 50. (c) white : black = 25 : 75.

Figure 1 shows structure examples of the matrix mixing.
In this ﬁgure, black cells and white cells have different
templates. Though we assume that the mixing is completely
random in this article, the mixing method can be modiﬁed tosome kinds of regular mixing method like alternate patterns.
Also the ratio of black cells and white cells is decided
optionally.
We carry out computer simulations of the matrix mixing
using “Small Object Remover” template and “Patch Maker”
template.
Small Object Remover :
⎡
⎤
⎡
1 1 1
0
A = ⎣ 1 2 1 ⎦, B = ⎣ 0
1 1 1
0

0
0
0

(a)

⎤
0
0 ⎦ , I = −1. (4)0(b)

Fig. 2.
The performance of Small Object Remover. (a) Input image.
(b) Output image.

Figure 2 shows how the Small Object Remover template
works. By using this template, large objects remain and small
objects are removed from input binary images.
P atch M aker:
⎡
⎤
⎡
0 1 0
0
A = ⎣ 1 2 1 ⎦, B = ⎣ 0
0 1 0
0

0
1
0

⎤
0
0 ⎦ , I = 4.5. (5)0Figure 3 shows how the Patch Maker template works. By
using this template, black objects are expanded without limit.

Figure 4 shows the simulation results using only Small
Object Remover template for an input image with relatively
large spot noises. We can see that the Small Object Removercannot remove such noises at all.

(a)

(b)

Fig. 4. Simulation results using only Small Object Remover template.
(a) Input image. (b) Output image.

Figure 5 shows the simulation results for the same input
image using the mixture template with the matrix mixing.
From the output images, we can say that the CNN withthe mixture templates possesses the both effects of the two
templates, namely black regions are expanded by the effectof the Patch Maker template and the shrinking spot noises are
removed by the effect of the Small Noise Remover template.
Although the effect of the noise removal is enhanced byincreasing the ratio of the Patch Maker template to the Small
Object Remover template, it makes the shape of the large
objects broken and the output becomes completely black for
Figs. 5(d) and (e).
B. Element mixing
In the second method of the mixing, we prepare twodifferent CNN templates as the matrix mixing, but choose
the element of them for each element of the matrices andthe threshold I at random as shown in Figs. 6 and 7.
Some examples of the mixture templates obtained fromthe Small Object Remover template and the Patch Maker
template using the element mixing are as follows.
Example 1:
⎡
1
A=⎣ 1
0

1
2
1

⎤
⎡
0
0
1 ⎦, B = ⎣ 0
102008 International Joint Conference on Neural Networks (IJCNN 2008)

0
1
0

⎤
0
0 ⎦ , I = 4.5.
0

(6)

4125

The above templates are obtained by mixing the elementswith even ratio.
Figure 8 shows the simulation results for the same input image as in the previous subsection using the mixture
template with the element mixing of different mixing ratios.
From the output images, we can say that this element mixing
method has similar effect to the previous matrix mixing
method.
(a)

(b)

(a)

(b)

(c)

(d)

(c)
Fig. 5. Simulation results for the same input image in Fig. 4(a) using
mixture template with matrix mixing. (a) small object remover : patchmaker= 80:20. (b) small object remover : patch maker= 70:30. (c) small
object remover : patch maker= 60:40.

Fig. 6.

Wiring weights of two templates.

(e)Fig. 8. Simulation results for the same input image in Fig. 4(a) using
mixture template with element mixing. (a) small object remover : patchmaker= 80:20. (b) small object remover : patch maker= 70:30. (c) small
object remover : patch maker= 60:40. (d) small object remover : patchmaker= 50:50. (e) small object remover : patch maker= 40:60.
Fig. 7.

Wiring weights of element mixing.

IV. N OISE REMOVAL PERFORMANCE
Example 2:
⎡
0
A=⎣ 1
0

1 1
0
2 1 ⎦, B = ⎣ 0
1 1
0

0
0
0

0
0 ⎦ , I = −1.
0(7)

Example 3:
⎡
0
A=⎣ 1
0

1
2
1

⎤
⎡
1
0
1 ⎦, B = ⎣ 0
1
0

0
0
0

⎤
0
0 ⎦ , I = 4.5.
0

(8)

4126

⎤

⎡

⎤

In order to compare the two mixing methods quantitatively,
we examine the noise removal performance investigating thedenoising level of the small objects and the distortion levelof the large objects of the input image. The input image have
40 noises. The Large objects of input image consist of 11514
pixels.
The denoising level of the small objects are evaluatedby the number of denoising, namely, the number of thevanishing spot noises. While the distortion level of the large
objects are evaluated by the number of the cells in the2008 International Joint Conference on Neural Networks (IJCNN 2008)

TABLE IN OISE REMOVAL PERFORMANCE OF MIXTURE TEMPLATES .

template (4) : (5)= 80:20
template (4) : (5)= 70:30
template (4) : (5)= 60:40
template (4) : (5)= 50:50
template (4) : (5)= 40:60

Matrix mixing
number of denoising15
distortion level
783
number of denoising26
distortion level
1015
number of denoising33
distortion level
1438
number of denoising40
distortion level
11514
number of denoising40
distortion level
11514

inputted large objects whose colors change from white toblack, namely the difference in pixels between the original
object images and the output object images.
The results are summarized in Table 1 for various mixing
ratios. The both mixing methods remove the noise as theratio of the Patch Maker increases. However, the distortionlevel becomes large. We can see that the effect of the matrix
mixing seems to be larger than that of the element mixing.

Element mixing
number of denoising4
distortion level
663
number of denoising10
distortion level
808
number of denoising20
distortion level
1033
number of denoising26
distortion level
1394
number of denoising30
distortion level2116

20 : 80. Figure 11 shows the converged output images withchanging the mixing ratio of (9) and (10). As a result, if
we can choose suitable combination of multiple templates,
then the image processing with both characteristics, namelysmoothing and diffusion, can be realized,

V. S IMULATION RESULTS FOR GRAYSCALE IMAGE
In this section, we apply CNN with mixture template of
the matrix mixing to a grayscale image in Fig. 9. In thissimulation. we use the Smoothing Operation template andthe Diffusion template.
Smoothing Operation:
⎡
⎤
⎡
⎤
0 1 0
0 0 0
A = ⎣ 1 2 1 ⎦ , B = ⎣ 0 0 0 ⎦ , I = 0.
0 1 0
0 0 0

(a)

(b)

(c)

(d)

(9)

Dif f usion:⎡⎤⎡⎤
0.1 0.15 0.1
0 0 00.15 ⎦ , B = ⎣ 0 0 0 ⎦A = ⎣ 0.15 0
0.1 0.15 0.1
0 0 0
I = 0.

(10)

Fig. 10. Variation of output when mixing ratio of (9) and (10) is 20 : 80.
(a) t =1. (b) t =5. (c) t =15. (d) t =20.

VI. S IMULATION REUSLTS FOR TEXTURE IMAGE

Fig. 9.

Grayscale input image.

Figure 10 shows the transient variation of the output
images when the mixing ratio of (9) and (10) is ﬁxed as

In this section, We apply CNN with mixture template of
the matrix mixing to a texture image.
First, texture image is processed by SPOT template[9].
Figure 12 shown simulation results using SPOT template.
By using this template, periodic pattern is emphasized.
Second, template that detects four directions is mixed. (ex.
“Vertical Line Detect Template” and “Horizon Line Detect
Template” is mixed.) Firure 13 shown how the template that
detects four directins works. Figure 14 shown simulation
results using mixture template with matrix mixing. The
mixing rate is 50:50. As a results, when two types templates
that detect one direction are mixed, two directions can bedetected.

2008 International Joint Conference on Neural Networks (IJCNN 2008)

4127

(a)

V ertical Line Detect T emplate:⎡
0 0 0 0⎢ 0 0 0 0⎢
A = ⎢⎢ 0 0 2 0
⎣ 0 0 0 0
0 0 0 0⎡−1 0.5 1⎢ −1 1 1⎢
B = ⎢⎢ −1 −1 5⎣ −1 1 1−1 0.5 1
I = −13.

(b)

(c)

(d)

Fig. 11. Converged output images with changing mixing ratio of (9) and(10). (a) (9) : (10) = 0 : 100. (b) 5 : 95. (c) 20 : 80. (d) 100 : 0.

(a)Fig. 12.

(b)(a) Texture image. (b) Simulation results using SPOT template.

SP OT T emplate:⎡−0.25 −1
⎢ −1
2.5
⎢
7
A = ⎢
⎢ −1.5⎣ −1
2.5−0.25 −1⎡
0 0 0 0⎢ 0 0 0 0⎢
B = ⎢⎢ 0 0 1 0
⎣ 0 0 0 0
0 0 0 0I = 0.

4128

−1.5
7
−23.25
7
−1.5
⎤
0
0 ⎥
⎥
0 ⎥
⎥
0 ⎦
0

−1
2.5
7
2.5
−1

⎤
−0.25
−1 ⎥
⎥
−1.5 ⎥
⎥
−1 ⎦
−0.25

⎤
0
0 ⎥
⎥
0 ⎥
⎥
0 ⎦
0
0.5
1
−1
1
0.5

Horizon Line Detect T emplate:⎤⎡
0 0 0 0 0
⎢ 0 0 0 0 0 ⎥
⎥
⎢
⎥
A = ⎢
⎢ 0 0 2 0 0 ⎥⎣ 0 0 0 0 0 ⎦0 0 0 0 0⎡
−1 −1 −1 −1⎢ 0.5 1 −1 1
⎢
1
5
1
B = ⎢
⎢ 1
⎣ 0.5 1 −1 1
−1 −1 −1 −1
I = −13.

⎤
−1
−1 ⎥
⎥
−1 ⎥
⎥
−1 ⎦
−1
(12)

⎤
−1
0.5 ⎥
⎥
1 ⎥
⎥
0.5 ⎦
−1
(13)

Diagonally Right U p Line Detect T emplate:⎤⎡
0 0 0 0 0
⎢ 0 0 0 0 0 ⎥
⎥
⎢
⎥
A = ⎢
⎢ 0 0 2 0 0 ⎥⎣ 0 0 0 0 0 ⎦0 0 0 0 0⎤⎡
−1 −1 −1 0.5 1⎢ −1 −1 1
1 0.5 ⎥
⎥
⎢
−1
1
5
1
−1 ⎥
B = ⎢
⎥
⎢
⎣ 0.5 1
1 −1 −1 ⎦1 0.5 −1 −1 −1
I = −13.

(14)

Diagonally Lef t U p Line Detect T emplate:⎤⎡
0 0 0 0 0
⎢ 0 0 0 0 0 ⎥
⎥
⎢
⎥
A = ⎢
⎢ 0 0 2 0 0 ⎥⎣ 0 0 0 0 0 ⎦0 0 0 0 0⎤⎡
1 0.5 −1 −1 −1⎢ 0.5 1
1 −1 −1 ⎥⎥⎢⎢5
1 −1 ⎥
B = ⎢ −1 1⎥
⎣ −1 −1 1
1 0.5 ⎦
−1 −1 −1 0.5 1
I = −13.

(15)

(11)

2008 International Joint Conference on Neural Networks (IJCNN 2008)

(a)

(b)

(a)

(b)

(c)

(d)

(c)

(d)

(e)

(f)

Fig. 13. Simulation results for the input image in Fig. 12(b). (a) Using
Vertical Line Detect. (b) Using Horizon Line detect. (c) Using Diagonally
Right Up Line Detect. (d) Using Diagonally Left Up Line Detect.

VII. C ONCLUSIONSIn this research, we have proposed the CNN using mixture
template with two kinds of mixing methods, the matrix
mixing and the element mixing. By computer simulationsof simple image processing tasks for a binary image anda grayscale image and texture image, we investigated thebasic output characteristics of the proposed systems. As we
expected, we could conﬁrm the effects of the both templates
by mixing them to one mixture template. At the moment,
we do not say that the proposed method exhibited a superior
performance than the original CNN at the moment. However,
we feel that we obtained some results to open a new directionof the research on the CNN.
R EFERENCES

Fig. 14. Simulation results for the input image in Fig. 12(b) using mixture
template with matrix mixing. (a) Vertical Line Detecter : Horizon Line
Detect=50:50. (b) Diagonally Right Up Line Detect : Diagonally Left Up
Line Detect=50:50. (c) Diagonally Right Up Line Detect : Vertical Line
Detect=50:50. (d) Horizon Line Detect : Diagonally Right Up Line Detect=50:50. (e) Vertical Line Detect : Diagonally Left Up Line Detect=50:50.
(f) Horzon Line Detect : Deiagonally Left Up Line Detect=50:50.

[9] Taisuke NISHIO and Yoshifumi NISHIO, “Image Processing Using
Periodic Pattern Formation in Cellular Neural Networks,” Proceedingsof European Conference on Circuit Theory and Design (ECCTD’05),
vol. 3, pp. 85-88, Aug. 2005.

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