Two-Layer Cellular Neural Networks
with Layer of Delay Output
Takahisa Ando, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {ando,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—Cellular Neural Networks (CNN) has been developed as a high-speed parallel signal processing platform by L.
O. Chua and L. Yang in 1988. Then, generalized two-layer CNN
model has been proposed. Two-layer CNN efficiently behavescompared to single-layer CNN for many applications of image
processing. In this study, we propose a new two-layer CNN
system including layer of delay output. In the proposed system,
second-layer CNN is added a delay output value by the delay
type feedback template. In addition, first-layer CNN receives
information including delay output from second-layer CNN. We
apply the proposed system to edge detection and investigate its
performance.

I. I NTRODUCTION
In recent years, our life teems with information by growthof high information society. In order to conduct parallel signaland flexible processing like human, Neural Network has beenproposed. Neural Network is based on the nervous system ofhuman. Additionally, Neural Network can perform expressionof nonlinear operating characteristics. Then, Cellular NeuralNetworks (CNN) was introduced by L. O. Chua and L. Yangin 1988 [1]. The idea of CNN was inspired from the architecture of the cellular automata and Neural Network. CNN
has local connectivity property and its structure resembles
the retina. CNN has been successfully used for various highspeed parallel signal processing applications such as image
processing, pattern recognition and so on. CNN is composedof the basic analog circuit units called cell. The performance ofCNN depends on the parameters which are called the template.
Each cell is connected to its neighboring cells according
to the template. Additionally, various applications for image
processing and pattern recognition have been reported [2]-[5].
In image processing of CNN, it is difficult to processcomplex parts of the input image; edge, background, etc. In
order to process complex parts of the input image, Cellular
Neural Networks with delay output (DCNN) has been proposed [6]. The feature of DCNN is added a delay outputvalue by the delay type feedback template. DCNN depends
on the past information as well as the current information in
the processing. However, noise effect can be observed in theoutput image of using DCNN.
In this study, we propose a new CNN system including
layer of delay output. The proposed system is based on
the structure of two-layer CNN. In the proposed system,
second-layer CNN is added a delay output value by the delay

type feedback template. In addition, first-layer CNN receives
information including delay output from second-layer CNN.
We consider that noise effect can be reduced by receiving
only important information of the input image. Therefore,
two layers are connected by only one coupling template. We
apply the proposed system to edge detection and investigate
its performance.
II. C ONVENTIONAL S YSTEM
A. Cellular Neural Networks
CNN has M by N processing unit circuits calld cells. Thecell consists of linear element and nonlinear element. Cells
are arranged in a reticular pattern to M line N row. In Fig 1,
we show the structure of CNN. In image processing of CNN,
each cell is corresponded to each pixel of the input image.

Fig. 1. The structure of CNN.

Each cell is connected to its neighboring cells according
to the template. Generally, the template is given for all cells
except for boundary cells. The template perform the influenceamang cells. The state and output equation of cell are shownas follows.
State Equation :dvxij
dt=−vxij +i+r
X

j+r
X

A(i,j;k,l) vykl (t)k=i−r l=j−r+i+r
X

j+r
X

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

k=i−r l=j−r(|i − k| ≤ r , |j − l| ≤ r).

(1)

Takahisa Ando, Yoko Uwate and Yoshifumi Nishio, “Two-Layer Cellular Neural Networks with Layer of Delay Output” in Proceedings of the 16th International Workshop on Cellular Nanoscale Networks and their Applications (CNNA) and the 6th Memristor and Memristive Symposium (MMS) August 27-30,
2018, Budapest, Hungary. P. Szolgay Ed. Budapest: Pázmány University ePress, 2018, pp 71– 74.

71

Output Equation :vyij (t)1(|vxij (t) + 1| − |vxij (t) − 1|).
2=(2)vx , vy and vu are state value, output value and input value.
In Eq. (1), A, B and I are feedback template, feedforward
template and threshold. These parameters determine the performance of CNN.

C. Two-Layer Cellular Neural Networks
In Fig 3, we show the block diagram of the conventional
two-layer CNN. The conventional two-layer CNN is constructed by two conventional single-layer CNN. Two conventional single-layer CNN are coupled by two coupling templates
C12 and C21 . They are used to transfer output values between
both layers.

B. Cellular Neural Networks with Delay Output (DCNN)
DCNN is added a delay output value by D template to
the conventional single-layer CNN. In the conventional CNN,
it can perform image processing only depending on current
information. On the other hand, DCNN depends on the past
information as well as the current information. In Fig. 2, weshow the block diagram of DCNN. In the block diagram, D
is the delay type feedback template and τ is delay time. The
state and output equation of DCNN are shown as follows.

III. P ROPOSED S YSTEMIn Fig. 4, we show the block diagram of the proposedsystem. The structure of the proposed system is based onconventional two-layer CNN. In the proposed system, twoconventional single-layer CNN are coupled by only one coupling template C12 . In addition, second-layer CNN includes
the output with time delay [6]. Second-layer CNN depends on
the past information as well as the current information likethe processing of DCNN. The state equation and the output
equation of each layer are shown as follows.

Fig. 2. Block diagram of DCNN.

State Equation (t ≤ τ ) :dvxij
dt= −vxij +j+r
Xi+rX

Fig. 3. Block diagram of two-layer CNN.

A(i,j;k,l) vykl (t)k=i−r l=j−r+i+r
Xj+rX

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

k=i−r l=j−r(|i − k| ≤ r , |j − l| ≤ r).

(3)

State Equation (t > τ ) :
dvxij
dt

=−vxij +i+r
Xj+rX

A(i,j;k,l) vykl (t)k=i−r l=j−r
i+r
X+j+r
X

Fig. 4. Block diagram of proposed system.

B(i,j;k,l) vukl (t)State equation of first-layer CNN :k=i−r l=j−ri+r
X+j+r
X

dv1xijdt

D(i,j;k,l) vykl (t − τ ) + I

k=i−r l=j−r(|i − k| ≤ r , |j − l| ≤ r).

=−v1xij +j+rX

A1(i,j;k,l) v1ykl (t)k=i−r l=j−r(4)+Output Equation :vyij (t) =i+r
X

i+r
X

j+r
X

B1(i,j;k,l) v1ukl (t)k=i−r l=j−r1(|vxij (t) + 1| − |vxij (t) − 1|).
2(5)+i+rX

j+rX

C12(i,j;k,l) v2ykl (t) + I1

k=i−r l=j−r(|i − k| ≤ r, |j − l| ≤ r).

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(6)

State equation of second-layer CNN (t ≤ τ ) :dv2xijdt=−v2xij +i+r
Xj+rX

A2(i,j;k,l) v2ykl (t)k=i−r l=j−r+j+r
X

i+r
X

B2(i,j;k,l) v2ukl (t) + I2

k=i−r l=j−r(|i − k| ≤ r, |j − l| ≤ r).

(7)

(a)

(b)

(c)

(d)

State equation of second-layer CNN (t > τ ) :dv2xijdti+r
X−v2xij +=j+r
X

A2(i,j;k,l) v2ykl (t)k=i−r l=j−r+i+r
X

j+r
X

B2(i,j;k,l) v2ukl (t)k=i−r l=j−r+i+r
Xj+rX

D(i,j;k,l) v2ykl (t − τ ) + I2

k=i−r l=j−r(|i − k| ≤ r, |j − l| ≤ r).

(8)

Output equation of first-layer CNN :1(|v1xij (t) + 1| − |v1xij (t) − 1|).
2
Output equation of second-layer CNN :v1yij (t) =v2yij (t) =1(|v2xij (t) + 1| − |v2xij (t) − 1|).
2(9)

(10)

IV. S IMULATION R ESULTS
We show some simulation results for edge detection byusing the proposed system. In this simulation, templates foredge detection in the proposed system are shown as follows[7].
Edge detection template :
0 0 0=  0 1 0 ,0 0 0
−1 −1 −1=  −1 8 −1  ,−1 −1 −1−0.12 −0.12 −0.120.1−0.12  ,=  −0.12−0.12 −0.12 −0.12A1 = A2

B1 = B2D
I1 = I2 = −1.

(11)

Coupling template :C12=(e)Fig. 5. Simulation results 1. (a) Input image. (b) Simulation result of theconventional CNN. (c) Simulation result of DCNN. (d) Simulation result offirst-layer CNN in the proposed system. (e) Simulation result of second-layer
CNN in the proposed system.

0−0.050 −0.05 0.49 −0.05  .
0−0.050(12)In Fig. 5, we show the input image and simulation resultsby the conventional CNN, DCNN and the proposed systemof each layer. The defocused part of the input image is theoutline of woman’s face and hands. It is difficult to detect
the edge lines of that part clearly. In Fig. 5(b), the defocusedpart is not detected by the conventional CNN. On the other
hand, in Fig. 5(c), the defocused part is detected by DCNN.
However, some noise remains in the output image. In theproposed system, the edge lines such as the defocused partcan be detected clearly. In addition, noise is reduced comparedto the method of DCNN. We consider that first-layer CNN
receives the characteristic of woman’s outline from secondlayer CNN. Therefore, amount of characteristic in the input
image is detected more effectively than the conventional CNN.
We have confirmed that the proposed system could perform
edge detection for general image. The proposed system could
detect characteristic of the input image including the defocusedparts. Then, we evaluate the performance of the proposedsystem for edge detection. We compare the proposed system
to the conventional system with input images which containa gradient region. In Fig. 6, we show input images. In each
input image, boundary of two parts become defocused as itgoes top of the image.

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TABLE I
D ETECTION R ATE [%].

(a)

(b)

(c)

(d)

Figure 6(a)Figure 6(b)Figure 6(c)Figure 6(d)Conventional CNN
25.00
33.59
33.59
33.59

DCNN
80.61
88.92
88.38
91.16

Proposed system
55.51
66.40
63.84
67.68

Fig. 6. Input images. (a) Gradient (white to whitish gray) and white.
(b) Gradient (black to blackish gray) and black. (c) Gradient (gray to whitishgray) and gray. (d) Gradient (gray to blackish gray) and gray.

(a)

(b)

(c)

(d)

Fig. 7. Simulation results using the conventional CNN. (a) Output imagefor Fig. 6(a). (b) Output image for Fig. 6(b). (c) Output image for Fig. 6(c).
(d) Output image for Fig. 6(d).

(a)(b)(c)(d)Fig. 8. Simulation results using DCNN. (a) Output image for Fig. 6(a).
(b) Output image for Fig. 6(b). (c) Output image for Fig. 6(c). (d) Output
image for Fig. 6(d).

boundary of two regions. From Tab. 1, the detection rate byusing DCNN is high and it can detect boundary line in eachcase. However, in image processing of general image, noise
effect can be observed by using DCNN. Then, detection rate
will decrease considering this point. On the other hand, wehave confirmed that detection rate of the proposed system isimproving about two times compared with the conventional
CNN.
V. C ONCLUSIONSIn this study, we have proposed a new CNN system
including layer of delay output. In the proposed system,
second-layer CNN depends on the past information by thedelay type feedback template D and first-layer CNN receives
that information. We applied the proposed system to edge
detection. From the simulation results, the proposed system
could detect edge of the defocused part in the input image.
Therefore, the proposed system is more effective than theconventional CNN for edge detection. In the future works,
we would like to investigate the performance of the proposedsystem for another image processing. In addition, we consider
that some one-way coupling templates are switched by using
the output values of first-layer CNN and second-layer CNN.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
R EFERENCES

(a)

(b)

(c)

(d)

Fig. 9. Simulation results using the proposed systyem. (a) Output image forFig. 6(a). (b) Output image for Fig. 6(b). (c) Output image for Fig. 6(c).
(d) Output image for Fig. 6(d).

Figure 7 shows simulation results of using the conventionalsystem. In any cases, it is difficult to detect boundary. Figure 8
shows simulation results of using DCNN. The boundary of twoparts is detected better than using the conventional CNN. In
each case, high detection rate can be observed by including
delay output. On the other hand, Fig. 9 shows simulation
results of using the proposed system. The proposed system
can detect the boundary better than using the conventional
CNN.
In Tab. 1, we show detection rate of each input image
using the conventional CNN, DCNN and the proposed system.
We define that the range from the bottom to the top of theboundary is 0[%] to 100[%]. Ideal edge is assumed as the74[1] L. O. Chua and L. Yang, “Cellular Neural Networks: Theory,”
IEEE Trans. Circuits Syst., vol. 35, pp. 1257-1272, Oct. 1988.
[2] Z. H. Yang, Y. Nishio and A. Ushida, “Image Processing of TwoLayer CNNs -Applications and Their Stability-,” IEICE Trans. Fundamentals, vol. E85-A, no. 9, pp. 2052-2060, 2002.
[3] M. Hänggi and G. S. Moschytz, “Cellular Neural Networks Analysis, Design and Optimization.” Kluwer Academic Publishers.
[4] F. Dirk and T. Ronald, “Coding of Binary Image Data Using Cellular Neural Networks and Iterative Annealing,” Proc. of ECCTD’03, vol. 1, pp. 229-232, Sep. 2003.
[5] T. Kozek, K. R. Crounse, T. Roska and L. O. Chua, “Smart Image Scanning Algorithms for the CNN Universal Machine,”
Proc. of NOLTA’95, vol. 2, pp. 707-712, 1995.
[6] T. Nakano, Y. Uwate and Y. Nishio. “Edge Detection by Using Switching System with Cellular Neural Networks and Delayed Cellular Neural Networks,” SJCIEE, pp. 3, 2015.
[7] L. Kek, K. Karacs and T. Roska, “Cellular Wave Computing Library(Templates, Algorithms, and Programs), Version 2.1, ” CellularSensory Wave Computers laboratory, Hungarian Academy of Sciencesk,Budapest, Hungary, 2007.