2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Cellular Neural Networks with Eﬀect from Friend of a Friend
Yoshihiro Kato† , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 Japan
Email: {kkato, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Generally, in the conventional CNN, eachcell is connected to only its neighboring cells according
to a template. In this case, the information that a cell canobtain from its neighboring cells is limited. In actual association, we tend to have a relation with another group thatis the best friend belongs. Therefore, in this study, we propose cellular neural networks with eﬀect from friend of afriend which took in the actual association to CNN.
1. Introduction
In recent years, our life teems with information bygrowth of high information society. Generally, digital circuit is used for many information processing. However,digital circuit cannot perform real time processing. Thenin 1988, Cellular Neural Networks (CNN) [1] were introduced. The idea of the CNN was inspired from the architecture of the cellular automata and the neural networks.
A diﬀerent point from the conventional neural networks isthat CNN has local connectivity property. The local connectivity property makes the CNN tailor made for VLSIimplementation. Therefore, the CNN has the features oftime continuity, spatial discreteness, nonlinearity and parallel processing capability. Furthermore, the structure ofCNN resembles that of animals’ retina. Therefore, CNN
can be used for various image processing applications [2][6]. In CNN, each cell is connected to its neighboring cellsaccording to a template. Then, each cell is influenced byneighboring cells and its value is updated.
As the proverb “Birds of a feather flock together” says,
we are likely to become friendly with someone who have
similar interests, values, purpose and so on. Hence, people with common principles and purpose organize a group.
The larger scale of the group, the group has strong power.
Therefore, large scale group is insusceptible to external influence. In job hunting, we join a company in empathy withcorporate identity, business lineup and so on. And we areworking as an employee. Although each employee has individuality at the beginning of company, they are taking on
a company style through employee some training programs
and works. In this way, on the society in which we live,
the member has a color of the group, though each one hasindividuality. Furthermore, large companies have strengthcompared with small and medium-sized enterprises.
In this study, we propose the new method in consideration of the phenomena in such actual society, and investi-gate the eﬀect in image processing. Generally, in the conventional CNN, each cell is connected to only its neighboring cells according to a template. In this case, the information that a cell can obtain from its neighboring cells islimited. In the proposed method, combinations of amongcells are decided by the value of each cell. In each cell,a cell with the nearest value of center cell is defined as“Best friend cell” in eight neighboring cells. Then, the statevalue of each cell is updated receive influences from thebest friend cell and its neighboring cells. We show somesimulation results and confirm its eﬀectiveness. Especially,
we confirm that the proposed method is eﬀective as edgepreserving smoothing.
2. Cellular Neural Networks
In this section, we explain the conventional structure ofthe CNN. The CNN has M by N processing unit circuits
called cells. The cell contains linear and nonlinear circuit
elements which are typically linear capacitors, linear resistors, linear and nonlinear controlled sources. 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 CNN is an array of cells. Each cell is connected toits neighboring cells according to a template. Usually, thetemplate is the same for all cells except for boundary cells.
When the CNN is used for image processing, each cell ofCNN is corresponded to each pixel of image.
State equation :dv xi jdt=−v xi j +j+ri+r ∑∑A(i, j;k,l) vykl (t)k=i−r l= j−r+j+ri+r ∑∑B(i, j;k,l) vukl (t) + I,(1)k=i−r l= j−r

Output equation :vyi j (t) =1(|v xi j (t) + 1| − |v xi j (t) − 1|),2(2)where v x , vy and vu represent a state, an output and an
input of cell, respectively. In the Eq. (1), A is the feedback
template and B is the control template. These and the constant bias I are collectively called general template. The
output equation is a piece-wise linear function. In fact, theoutput value of CNN is within of −1 to 1. When the CNN- 711 -is used for image processing, values of black and white aretreated as 1 and −1, respectively.
3. Cellular Neural Networks with Eﬀect from Friendof a Friend
In this section, we explain the algorithm of the cellularneural networks with eﬀect from friend of a friend (FFCNN). The FF-CNN is that the method of constituting anew combination. The algorithm is described as follows.

4. Simulation Results
In this section, we show some simulation results for image denoising, edge-preserving smoothing and divide domains by using the FF-CNN. The used templates are foundin [7]. In this study, image size is 128×128 pixels.
4.1. Image Denoising
In this subsection, we show some simulation results ofimage denoising by using the FF-CNN. Generally, in theconventional CNN, “S mall ob ject remover” template [7] isused for image denoising. However, this template can notremove a big noise and output image is converged on blackand white. In gray scale image, the conventional CNN cannot perform with maintained gray scale. As the other way
for image denoising, noisy image is defused.
In the FF-CNN, we use “Heat di f f usion” template [7]for image denoising. Boundary condition is fixed to +1.

Figure 1: Definition the best friend cell in input image.

(a)

(b)

(c)

(d)

Figure 2: Definition the best friend cell in output image.

Step 1: In each cell of input image, a cell with the nearest value of center cell is defined as “Best friend cell” ineight neighboring cells.
Step 2: The combination centering on a cell C(i, j) ischanged into a combination centering on the best friendcell.
Moreover, the element of template of the best friend andthe center cell are replaced. With this, a new combinationis constituted like Fig. 1.
Step 3: The value of each cell is updated according tothe Eqs. (1) and (2) using the new combination.
Step 4: In each cell of output image, a cell with thenearest value of center cell C(i, j) in input image is defined
as “Best friend cell” in eight neighboring cells like Fig. 2.
Step 5: Steps 2 to 4 are repeated every 0.0005 [τ].
In the proposed method, each cell in input image is used
for basis of definition the best friend. Therefore, the statevalue of each cell is updated by receiving influences fromthe best friend cell and its neighboring cells.

Figure 3: Image denoising by the conventional CNN using “S mall ob ject remover” template. (a) Binary input andinitial state images. (b) Image denoising result for binary
image. (c) Gray scale input and initial state images. (d) Image denoising result for gray scale image.

Figure 3(b) shows the image denoising result by the conventional CNN using “S mall ob ject remover” template tobinary image like Fig. 3(a). The noise of the small size isremoved. However, the noise over certain size is not removed. Moreover, the angle of the object is waned.
Figure 3(d) shows the image denoising result by the conventional CNN using “S mall ob ject remover” template togray scale image like Fig. 3(c). Similar to the above re-- 712 -Table 1: Error Evaluation
Conventional CNN FF-CNN
Figure 3(a)400.844101.078
Figure 3(c)2674.344
221.813(a)gray scale image, the error is greatly reduced since the grayobject remain in the output image.
From these results, we consider that the FF-CNN is more
eﬀective than the conventional CNN for image denoising.

(b)

4.2. Edge-preserving Smoothing(c)Figure 4: Image denoising by the FF-CNN. (a) Initial state
image. (b) Image denoising result for binary image. (c) Image denoising result for gray scale.

sult, the noise over certain size is not removed. Moreover,an output image is converged on black and white when the“S mall ob ject remover” template is used. Then, the rightside object of the value near black has disappeared.
On the other hand, Fig. 4 shows the image denoising result by the FF-CNN using “Heat di f f usion” template. Figure 4(b) shows the output image by using the FF-CNN tobinary input image and the initial state image like Fig. 3(a)and Fig. 4(a). All noise in the input image were removed.
Figure 4(c) shows the output image by using the FF-CNN
to gray scale input image and the initial state image likeFigs. 3(c) and Fig. 4(a). Similar to the above result, allnoise in the input image were removed regardless of grayscale. Moreover, there is remained with keep the value ofgray scale which is the object of the right side disappearedby the conventional CNN.
Additionally, we evaluate the performance of image denoising using the FF-CNN. The error value between theideal output and the obtained output is calculated byError =127×127∑| vmi j − vyi j |,(3)

i, j=0where vm is the value of each cell in the ideal output.
Table 1 shows the error evaluation of image denoising
using the conventional CNN and the FF-CNN. In every
case, we confirm that the FF-CNN can remove some noise
eﬀectively than the conventional CNN. Especially, for theIn this subsection, we show some simulation results ofedge-preserving smoothing by the FF-CNN. In this simulation, we confirm that the phenomenon of “Each employee has individuality, however they are taking on a company style” as discussed earlier in introduction. The same
“Heat di f f usion” template and boundary condition as subsection 4.1 are used. And Fig. 4(a) are used as initial state
image.
Figure 3(c) has white, gray objects, and a black domain of a background. Dispersion is calculated on thebasis of the value of each domain within the limits of± (1, 5, 10 and 20) [%]. And the Fig. 3(c) which gave
dispersion to each domain are used as input images.
Figure 5 shows the simulation results of edge-preserving
smoothing by the FF-CNN. For the input image with thedispersion in several percent, we confirm that image denoising and smoothing can be performed eﬀectively likeFigs. 5(a) and (b). However, in near a boundary, disorderof a value occurs when the range of dispersion becomeslarge like Figs. 5(c) and (d).
From these results, we consider that the FF-CNN is effective as edge-preserving smoothing.
4.3. Divide Domains
In this subsection, we show other application using theFF-CNN. Since the cells of the similar value generated thedomain, we thought that the FF-CNN was eﬀective as application which divides sparse domain from thick domain
in image. In this simulation, the choice of the best friend
and the construction of the new combination which is only
one time depending on an input value. To divide sparse
domain from thick domain, we use “S moothing” template[7]. Boundary condition is using mirror reflector condition.
Figure 6(a) shows the output image by the conventional
CNN using “S moothing” template. The thick domain of
the left half in the input image which remains as a black
domain. However, the sparse domain of the right half in
the input image which is still a spot pattern. On the otherhand, Fig 7(a) shows the output image by the FF-CNN using “S moothing” template. The thick domain of the left
half in the input image remains as a black domain. At thesame time, the sparse domain of the right half in the input- 713 -(a)

(b)

(a)

(b)

Figure 6: Simulation result by the conventional CNN.
(a) Input and initial state images. (b) Output image by theconventional CNN using “S moothing” template.

(c)

(d)

Figure 5: Edge-preserving smoothing by the FF-CNN.
(a) The simulation result for the input image with disproportionate rate of ± 1 [%]. (b) The simulation result for theinput image with disproportionate rate of ± 5 [%]. (c) Thesimulation result for the input image with disproportionate
rate of ± 10 [%]. (d) The simulation result for the input
image with disproportionate rate of ± 20 [%].

image remains as a white domain. Thereby, we can see thatthe thick domain and the sparse domain can be divided with
the FF-CNN.

(a)

(b)

Figure 7: Divides sparse domain from thick domain result
by the FF-CNN. (a) Initial state image. (b) Divide domains
result by the FF-CNN using “S moothing” template.

Acknowledgments
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.

5. Conclusions
In this study, we have proposed cellular neural networks
with eﬀect from friend of a friend. The FF-CNN is that thenew method in consideration of the phenomena in such actual society which likes “Birds of a feather flock together”.
From simulation results of image denoising, the noise of
the size which was not removed by the conventional CNN
is removed. Moreover, there were remained with keep thevalue of gray scale disappeared by the conventional CNN.
Furthermore, from simulation results of edge-preserving
smoothing, we have confirmed that image denoising andsmoothing could be performed eﬀectively. Moreover, fromsimulation results of divide domains, we could see that thethick domain and the sparse domain could be divided.
From some simulation results, we consider that theFF-CNN is eﬀective for strengthen similar domain andnot worry about small diﬀerence. In the future works,
we would like to investigate the phenomena about edgepreserving smoothing by the FF-CNN.

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