IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Cellular Neural Networks with Delay Output for Half-Toning
Taro Nakano, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {taro, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—Cellular Neural Network (CNN) is constructed thegrating coupling of analog circuits. The performance of the CNN
depends on the parameters called the template. Template can usefor various image processing applications. In this study, we apply
the CNN with delay output (D-CNN) to image processing. In ageneral way, the D-CNN is used by motion picture processing.
However, we apply the D-CNN to image processing and obtaineffective results. From simulation results, we confirm that theproposed model is effective for half-toning processing.

an array of cells. Cells are arranged in a reticular pattern to M
line and N row. Each cell is coupled only to its neighboring
cells according to a template. Usually, the template is the samefor all cells except for boundary cells. The array of the CNN
is shown in Fig. 1. Also, Fig. 2 shows a block diagram of theconventional CNN.

I. I NTRODUCTION
In recent years, our life teems with information by growthof high information society. Generally, digital circuit is usedfor many information processing. However digital circuit cannot perform real time processing. In 1988, Cellular NeuralNetworks (CNN) were introduced by L. O. Chua and L. Yang[1]. The idea of the CNN was inspired from the architecture ofthe cellular automata and the neural networks. The CNN hasfeatures of time continuity, spatial discreteness, nonlinearityand parallel processing capability. Furthermore, the structureof the CNN resembles that of the human’s nervous system [2][6]. Therefore, the CNN has been successfully used for various
high-speed parallel signals processing applications such asimage processing, pattern recognition and so on [7]. The
performance of the CNN depends on the parameters called
the template. The template of the CNN consists three elements
as the feedback template A, the control template B, and theconstant bias T . Generally, the conventional CNN depends onan input image because of space invariant template. In generalimages, the complex portions of input image are not processedwell.
In this study, we apply the CNN with delay output (D-CNN)
to image processing. The D-CNN depend on the past information as well as the current information in the processing. In ageneral way, the D-CNN is applied motion picture processing.
On the other hand, we applied it to image processing ofhalf-toning processing. Additionally, we investigate how theD-CNN influenced image processing. From some simulation
results, we obtain good results in the image processing byusing the D-CNN.

Fig. 1. The structure of the conventional CNN.

Fig. 2.

The block diagram of the CNN.

State Equation :dvxij
dt= −vxij +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) + T.

(1)

1(|vxij (t) + 1| − |vxij (t) − 1|).
2

(2)

k=i−r l=j−r

Output Equation :II. C ELLULAR N EURAL N ETWORKSIn this section, we explain the structure of the CNN. TheCNN has M by N processing unit circuits called cells.
These circuit elements are constructed linear capacitors, linearresistors, linear and nonlinear controlled sources. The CNN isvyij(t)=Where vx , vy and vu represent a state, an output and aninput values of cell. In the state equation, A is the feedback- 67 -

template, B is the control template and T is the constantbias. State and output equation of the conventional CNN canbe shown as Eqs. (1) and (2). The r-neighborhood of a cellC(i, j) in the CNN defined byOutput Equation :vyij (t)1(|vxij (t) + 1| − |vxij (t) − 1|).
2=(5)Nr (i, j) = {C(i, j) | (max |k − i|, |l − j| ) ≤ r,1 ≤ k ≤ M ; 1 ≤ l ≤ N }.

(3)

Where r is the positive integer number, and called as theneighborhood radius. The output equation can be shown bypiecewise linear function. The piecewise linear function isshown in Fig. 3.
Fig. 4. The block diagram of the D-CNN.

In Fig. 4, the D-CNN contains the output with time delay. In
the conventional CNN, it can perform image processing only
depending on current information. On the other hand, the DCNN depends on the past information as well as the current
information in the processing. In a general way, the D-CNN
is useful of motion picture processing. However, in this study,
we use the D-CNN in the image processing of half-toning
processing . D template can substitute the value with various
characteristics in the D-CNN.
IV. S IMULATION R ESULTIn this section, we show some simulation results for halftoning by using the D-CNN. Moreover, from simulation results, we compare the difference of result in the conventional
CNN and the D-CNN. In the conventional CNN, it is moredifficult to detect indistinct parts in the input image well.
Templates of the half-toning are shown as follows.

Fig. 3. The piecewise linear function of the CNN.

III. C ELLULAR N EURAL N ETWORKS WITH
D ELAY O UTPUT

Half-toning template :In this section, we explain the structure of the D-CNN.
Figure 4 shows the block diagram of the D-CNN. In the block
diagram, A is the feedback template, B is the control template,
T is the bias and D is the delay type feedback template. Thestate and output equation of the D-CNN are shown as follows.
State Equation :dvxij
dt= −vxij +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)k=i−r l=j−r+i+r∑j+r∑

D(i,j;k,l) vykl (t − τ ) + T. (4)k=i−r l=j−r- 68 -
A =B=


−0.07 −0.1 −0.07
 −0.12−0.1  ,−0.07 −0.1 −0.07
0.07 0.1 0.07
 0.1 0.32 0.1  , T = 0.
0.07 0.1 0.07

Half-toning of the D-CNN :−0.07 −0.1 −0.07
2−0.1  ,A =  −0.1
−0.07 −0.1 −0.07

0.07 0.1 0.07
B =  0.1 0.32 0.1  ,0.07 0.1 0.07
0−10
D =  −1 3.25 −1  , T = 0.
−0 −10(6)(7)

logical difference from τ = 10 to 5 in the D-CNN. FromFig. 5(f), even if delay time is changed, simulation results are
similar visually.
V. C ONCLUSION(a)

In this study, we investigate the D-CNN to image processing. In the half-toning processing, we compare the results ofthe conventional CNN and the D-CNN. The D-CNN adds apast output value by the D template. The D-CNN depend on
the past information as well as current information in theprocessing. From some simulation results, the D-CNN candetect indistinct parts of the input image. In the future works,
we would like to research other templates in the D-CNN.

(b)

R EFERENCES

(c)

(d)

(e)

(f)

[1] L. O. Chua and L. Yang, “Cellular Neural Networks: Theory,” IEEETrans. Circuits Syst., vol. 35, pp. 1257-1272, Oct. 1988.
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pp. 229-232, Sep. 2003.
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Circuits Syst., vol. 42, no. 10, pp. 583-601, Oct. 1995.
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vol. E85-A, no. 9, pp. 2052-2060, Sept. 2002.

Fig. 5. Simulation results. (a) Input image. (b) Simulation result of

the conventional CNN. (c) Simulation result of the D-CNN (τ = 5).
(d) Simulation result of the D-CNN (τ = 10). (e) Simulation resultof the logical difference from the D-CNN to the conventional CNN.
(f) Simulation result of the logical difference from τ = 10 to 5 in
the D-CNN.

Figure 5 shows simulation results of the half-toning. Figure 5(a) shows an input image. Figure 5(b) shows the result of
the conventional CNN. In Fig. 5(b), the conventional CNN can
not process half-toning well. It is expressed only with blackor white when it exceed a certain constant pixel level in thehalf-toning. Because of this, the parts of lips and the moatof eyes express almost all with white or black. Figures 5(c)and (d) show simulation results of the D-CNN (τ = 5 and10). In Fig. 5(c) and (d), the D-CNN can process the halftoning well. In the D-CNN, the parts of lips and the moatof eyes express color strength by using white and black.
Figure 5(e) is the simulation result of the logical difference
from the D-CNN to the conventional CNN. In Fig. 5(e), the DCNN can process half-toning well comparing the conventional
CNN. Moreover, we investigate how processing change in the
different delay time. Figure 5(f) is the simulation result of the- 69 -