2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

Cellular Neural Networks with Hopfield Neural NetworksConsidering the Confidence Degree
Yasuhiro Ueda† , Masakazu Kawahara† , Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email:{yasuhiro, kawahara, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we propose cellular neural networks (CNN) with Hopfield neural networks (HopfieldNN) considering the confidence degree. The Hopfield NN
works as an associative memory to retrieve one of the embedded patterns from the local data of the input images.
The confidence degree means the diﬀerence between theretrieved pattern and the input image. When the diﬀerenceis small the confidence degree is set to be large. The confidence degree works to enhance the CNN operation. By
computer simulations, we investigate the basic property ofthe proposed method and confirm its eﬀectiveness for example of pattern detection.

operation. By computer simulations, we investigate the basic property of the proposed method and confirm its eﬀectiveness for for example of pattern detection.
In Sec. 2, we review the basic of the standard CNN. In
Sec. 3, we review the basic of the standard Hopfield neural network. In Sec. 4, we describe the structure of thespace-varying CNN designed by Hopfield NN considering
the confidence degree. In Sec. 5, simulation results of binary image processing task are shown. Section 6 concludes
this article.

1. Introduction

In this section, we describe the basic structure of CNN.
CNN has M by N processing unit circuits called cells. Cells
are arranged in a reticular pattern to M line by 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 and nonlinear circuit elements. CNN is an array of cells. Each cell is connected toits neighborhood cells according to a template. Usually, thesame template is used for all the cells except for boundarycells. CNN has the features of time continuity, spatial discreteness, nonlinearity and parallel processing capability.
The state equation and the output equation of the cellC(i, j) are shown as follows.
State equation:

Cellular Neural Networks (CNN) [1] were introduced by
Chua and Yang in 1988. The idea of CNN was inspired
from the architecture of the cellular automata and the neural networks. A diﬀerent point from the conventional neural networks is that CNN has local connectivity property.
Since the structure of CNN resembles the structure of animals’ retina, CNN can be used for various image processing applications [2]-[5]. CNN is composed of the basic
circuit units called cells. The cell contains linear and nonlinear circuit elements which are typically linear capacitors,
linear resistors, linear and nonlinear controlled sources.
In the previous study, we have proposed a space-varying
CNN designed using the Hopfield NN. We prepare theHopfield NN as associative memory to which some patterns are embedded. In that method, the templates of CNN
was chosen from some memorized templates according to
the retrieved pattern. By using this method, we can performbetter image processing depending on the local informationof input images. For example, when the retrieved patternis like white and black image we may use edge template,while for relatively smooth retrieved pattern we may use
diﬀusion template. By this setting, the CNN is expected
to detect the edges of the input image with diﬀusion forsmooth area.
In this study, in order to improve the above method, weconsider the confidence degree which means the diﬀerence
between the retrieved pattern and the input image. Whenthe diﬀerence is small the confidence degree is set to belarge. The confidence degree works to enhance the CNN

2. Cellular Neural Networks [1]dv xi jdt=−v xi j +j+ri+r ∑∑A(i, j;k,l) vykl (t)k=i−r l= j−rj+ri+r ∑∑B(i, j;k,l) vukl (t) + I.

(1)

1(|v xi j (t) + 1| − |v xi j (t) − 1|).
2(2)+k=i−r l= j−r

Output equation:
vyi j (t) =where v x , vy and vu represent a state, an output and an inputof the cell, respectively. In the Eq. (1), A is the feedback
template and B is the control template. These templates andthreshold value I are collectively called the general template.

- 19 -The r-neighborhood of C(i, j) in CNN is defined byNr(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
the value of r as 1.
3. Hopfield Neural Network Working as AssociativeMemory
In this section, we describe the basic structure of Hopfield Neural Network (Hopfield NN). Associative memory
is a system which returns a stored pattern or its reversed
pattern that is similar to an inputted pattern. Noisy patterns
can be associated or distorted patterns can be recognized
by a well-constructed associative memory.
The Hopfield NN is used as an associative memory by
exploiting the property that the network has multiple stable states. Namely, if the parameters of the network canbe decided in such a way that the patterns to be stored become stable states of the network, the network produces astored pattern that is similar to an input pattern. The energyfunction of the Hopfield NN with N neurons and P stored
binary patterns is defined by the following equation.

E=−N NN∑1 ∑∑wi j xi x j +θi xi ,
2 i=1 j=1
i=1

(4)

where wi j is the weight between i-th neuron and j-th neuron, and θi is the threshold of the i-th neuron. The weight
wi j is given as follows.

wi j ={ ∑P

p=1 xi(p)x j (p)0.

(i , j)(i = j)ability of the associative memory of Hopfield NN. In general, the design of space-varying systems is not easy. However, we can set one of prepared existing templates oneach cell of CNN according to the retrieved pattern byHopfield NN to which some typical local image structures
are embedded. Namely, we need only some existing templates and their associated patterns. The design method
is described as follows. Firstly, we prepare some twodimensional arbitrary patterns for the memory of Hopfield
NN and some existing templates related with these patterns, respectively. Secondly, Hopfield NN memorize thesome two-dimensional arbitrary patterns. Thirdly, Hopfield
NN compare between memorized the some arbitrary patterns and each pixel and its two neighborhood pixels of ainput image. Hence, from some arbitrary patterns, Hopfield
NN associate the certain arbitrary pattern which is the mostsimilar to the pattern of each pixel and its two neighborhood pixels. Fourth, from the prepared some existing templates, the template of each cell in CNN is designed basedon the associated pattern. Finally, input image is processedby using the space-varying CNN designed by Hopfield NN.
Each templates are applied to each cell by these steps.
In this study, we propose the developing system by considering the confidence degree to space-varying CNN. We
calculate the Hamming diﬀerence between the retrieved
pattern and the input image. When the Hamming diﬀerence is small the confidence degree is set to be large. The
confidence degree works to enhance the CNN operation.
By using this system, it is possible to detect the some objects by changing the threshold value when several similar
objects exist in the input image. The threshold value Icd
is defined as following in Eq. 8 and the characteristics ofthe threshold value Icd with the Hamming distance and theconfidence degree is shown in Fig. 1.

(5)

The weight wi j is 0 when i = j, because all the units havecombined with all the units of the others except themselves
The states of the neurons are asynchronously updated dueto the following diﬀerence equation:∑xi (t + 1) = sgn( wi j x j (t)),
(6)

Icd =

13 × Con f idence Degree[%]100(1 + exp(HD))(8)

In Eq. (8), HD is calculated by Eq. (9), andCon f idence Degree[%] is a parameter to change thecharacteristics of the threshold value Icd .
HD =−1 × Hamming Distance + 13
5(9)

i, j

where sgn is an output function as follows:
5. Simulation Results{sgn(a) =1 (a ≥ 0)
−1. (a < 0)

(7)

4. Space-Varying Cellular Neural Networks Considering the Confidence Degree
In the previous study, we have proposed space-varying
cellular neural networks which is designed by using theIn order to confirm the eﬀectiveness of this proposed
method, the CNN application of the object detection is simulated. In this simulation, we use “Logic AND” template.
This template reads out the black when input value and initial value are black. The others situation, the output becomes white. Therefore, when the input image and the initial state are same, the output is same with input image.
However, if the threshold value of “Logic AND” template- 20 -5Threshold Value Icd

43

250 %

170 %
80 %

Figure 3: Input binary image to CNN and Hopfield NN.

0012345678

910111213

Hamming distance

Figure 1: Change of the threshold value Icd by the confidence degree.

Moreover, by changing the confidence degree which is50 [%], 70 [%] and 80 [%], the extra threshold value Icd
is decided like Eq. 8 and Fig. 1. State equation of CNN
updated is described as follows.

is changed smaller than -2.1, CNN outputs white regardlessof the input image and the initial state.
In this section, we show some simulation results using
the proposed CNN. The input binary image is 4 and its sizeis 64 by 64. First, we describe the memorized patterns andrelated templates. And templates are found in [8].

010
0 
0  , I = −1.
0“White filler” template: 0 0 0 A =  0 2 0  , B = 0, I = −4.


0 0 0

=−v xi j +j+ri+r ∑∑An(i, j;k,l) vykl (t)k=i−r l= j−r+j+r
i+r ∑∑Bn(i, j;k,l) vukl (t) + I − Icd . (12)k=i−r l= j−r

By using the extra threshold value Icd , simulation results
are changed as shown in Fig. 4 (a), (b) and (c), respectively.
Therefore, some points of the object are detected by changing the confidence degree. Additionally, some points whichhave same pattern with memorized pattern is detected irrespectively the confidence degree.

Figure 2: Memorized patterns.
“Logic AND” template:


 0
 0 0 0 
A =  0 2 0  , B =  0
00 0 0dv xi jdt

(10)

(a)

(b)

(c)

(11)
Figure 4: Output image by using proposed system.
(a) Confidence degree is 50 [%]. (b) Confidence degree
is 70 [%]. (c) Confidence degree is 80 [%].

This template of Eq. (11) changes all outputs into -1.
The memorized patterns are set as shown in Fig. 2. Four
patterns which are Fig. 2 from the left are related with
“Logic AND” template. These patterns corresponds to thepart of objects in the input image. The rightmost pattern isrelated with “White Filler” template. This pattern describe
the uniform region. By using these patterns, wiring weightsof Hopfield NN is decided and each templates are appliedto each cells.

By using this system, we detected some points of eachobjects. Additionally, by changing the confidence degree,
the number of the detected points are changed. Moreover,
we carry out “Recall” template with conventional CNN toFig. 4.

- 21 -

110

pattern and the input image. When the diﬀerence was small
the confidence degree was set to be large. The confidence
degree worked to enhance the CNN operation. By computer simulations, we investigated the basic property of theproposed method and confirmed its eﬀectiveness for several examples.

The Number of Detected Points

100908070605040

References30[1] L.O. Chua and L. Yang, “Cellular Neural Networks:Theory,” IEEE Trans. Circuits Syst., vol. 32,pp. 1257-1272, Oct. 1988.

201000102030

405060708090100110

Confidence Degree [%]Figure 5: Change of detected points by Confidence degree.

“Recall” template: 0 0 0 
 0.3 0.3 0.3 
A =  0.3 4 0.3  , B =  0 5.1 0  , I = 0.

0 0 00.3 0.3 0.3(13)Simulation results are shown as follows (Fig. 6).

[2] 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.
[3] M. Namba and Z. Zhang, “Cellular Neural Network for
Associative Memory and Its Application to Braille Image Recognition,” Proc. of IJCNN’06, pp. 4716-4721,
Jul. 2006.
[4] H. Koeppl and L.O. Chua, “An Adaptive CellularNonlinear Network and its Application,” Proc. ofNOLTA’07, pp. 15-18, Sep. 2007.
[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. 707712, 1995.
[6] K. Sumitomo and A. Ushida, “A Design Method ofCellular Neural Networks using Fuzzy Inference,” IEICE Technical Report, NLP, vol. 98, no. 443, pp. 3946, Dec, 1998.

(a)(b)(c)Figure 6: Output image by applying the “Recall” templatewith conventional CNN to Fig. 4. (a) Recall the input image in Fig. 4 (a). (b) Recall the input image in Fig. 4 (b).
(c) Recall the input image in Fig. 4 (c).

From this simulation results, we can change the type of thedetected objects by changing the confidence degree. Additionally, we could detect the object which is including thesame pattern with the memorized pattern.

[7] J. J. Hopfield, “Neurons with Graded Response Have
Collective Computational Properties like Those of
Two-State Neurons,” Proc. Natl. Acad. Sci. USA, vol.
81, pp. 3088-3092, 1984.
[8] Cellular Sensory Wave Computers Laboratory Computer and Automation Research Institute Hungarian
Academy of Sciences, “Cellular wave Computing Library (Template, Algorithms, and Programs) Version
2.1”

6. Conclusions
In this study, we proposed cellular neural networks(CNN) with Hopfield neural networks (Hopfield NN) considering the confidence degree. The Hopfield NN worked
as an associative memory to retrieve one of the embedded
patterns from the local data of the input images. The confidence degree meant the diﬀerence between the retrieved- 22 -