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Int J. Bifurcation and Chaos Submission Style

EXPERIMENTS ON AN ELECTRICAL NONLINEAR OSCILLATORS

NETWORK

S. Morfu∗

Laboratoire d’Electronique, Informatique et Image (LE2i)

UMR C.N.R.S 5158, Aile des Sciences de l’Ing´ enieur,

BP 47870,21078 Dijon Cedex, France,†smorfu@u-bourgogne.fr‡

http://<www.le2i.com>

J. Bossu

Laboratoire d’Electronique, Informatique et Image (LE2i)

UMR C.N.R.S 5158, Aile des Sciences de l’Ing´ enieur,

BP 47870,21078 Dijon Cedex, j.bossu@enesad.fr

P. Marqui´ e

Laboratoire d’Electronique, Informatique et Image (LE2i)

UMR C.N.R.S 5158, Aile des Sciences de l’Ing´ enieur,

BP 47870,21078 Dijon Cedex, marquie@u-bourgogne.fr

Received (Day Month Year)

Revised (Day Month Year)

We have recently proposed a Cellular Nonlinear Network (CNN) based on nonlinear oscillator proper-

ties to perform image processing tasks. We present here the electronic implementation of the elemen-

tary cell of this CNN. We experimentally verify the main property of the CNN, that is the possibility to

enhance a weak difference of initial condition between two specific cells of the CNN at a given time.

For this optimal time, a contrast enhancement of a weak contrasted gray scale is possible.

Keywords: nonlinear signal processing; nonlinear circuits.

1. Introduction

Since the pioneer work of L. Chua concerning Cellular Nonlinear Networks (CNN) [Chua

(1998)], it has become clear that nonlinear media can be regarded as parallel multiproces-

sors systems dedicated to signal processing. In these nonlinear systems, the signal ampli-

tudeconstitutes an additionaldimensionwhichoffersa rich varietyof propertiesnot shared

by linear systems. Image processing with nonlinear networks [Adamatzky et al. (2002);

∗Laboratoire d’Electronique, Informatique et Image (LE2i) UMR C.N.R.S5158, Aile des Sciences de l’Ing´ enieur,

BP 47870,21078 Dijon Cedex, France.

†France

‡smorfu@u-bourgogne.fr.

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S. Morfu, J. Bossu and P. Marqui´ e

Rambidi et al. (2002); Perona and Malik (1990); Juli´ an et al. (2002); Morfu and Comte

(2004)], signal detection/transmission via the nonlinear stochastic resonance phenomenon

[Gammaitoni et al. (1998); Godivier et al. (1999); Morfu et al. (2003)] are few examples

of a nonrestrictive list where taking into account nonlinearity allows to transcend the limi-

tation of classical linear processes.

Note that until now, most of the electronically implemented CNN are based on reaction-

diffusion systems and perform different tasks such as edge extraction in image process-

ing field [Comte et al. (2001)], or noise filtering in signal processing area [Marqui´ e et al.

(1998)]. However, it is also possible to use the properties of nonlinear inertial systems to

develop image processing tools. Indeed, recently, we have theoretically and numerically

introduced a CNN based on the properties of nonlinear oscillators to perform a contrast

enhancement of a weak contrasted image [Morfu and Comte (2004)]. This paper is mainly

devoted to the electronic implementation of such a CNN. We first present the electrical

lattice made of uncoupled nonlinear oscillators. Then, we show theoretically and exper-

imentally that two oscillators with a slight difference of initial amplitude can present a

maximum amplitude difference at a given time. This main property is experimentally con-

firmed and allows, as widely discussed in [Morfuand Comte (2004)],to performa contrast

enhancement of an image.

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Fig. 1. Sketch of the 1D CNN and its elementary cell.

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EXPERIMENTS ON AN ELECTRICAL NONLINEAR OSCILLATORS NETWORK

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2. The nonlinear lattice

We consider a lattice of N elementary cells consisting of a nonlinear circuit and a 1N4148

diode as represented in figure 1. Each nonlinear circuit includes a polynomial source

P(U) = K(U −m+α)(U −m)(U −m−α)/100 made with AD633JN analog multipliers

and standard amplifier circuit with amplification −K to balance the scale factor 1/10V−1

of the multipliers. The worth of the zeros m+α, m−α and m is adjusted, for all cells, with

only three external voltage sources to ensure a strict homogeneity in the lattice. Moreover,

a double integrator realized with classical operational amplifier circuit, resistor R and ca-

pacitorC assumes a feedbackbetween the input/outputof the polynomialsource (figure 1).

According to the sketch of figure 1 and setting K′= K/10RC, the voltage Uiat the diode

cathode of the ithcell obeys to:

d2Ui

dt2= −K

Furthermore, the initial condition Ui,0of the ithcell is introduced at the diode anode, by

adding an offset voltage equal to the diode thresholdVT= 0.7V. The distribution of initial

condition is considered to be linear and growing versus the cell number i such that Ui,0=

(i−1)×0.5/(N −1). Note that in image processing context, this distribution of initial

conditions could be considered as a discrete gray scale.

Setting xi=Ui−m, eq. (1) can be normalized as

d2xi

dt2= −K

Solutions of eq.(2) for a zero initial velocity are given by the following Jacobian elliptic

functions [Abramowitz and Stegun (1970)]:

xi(t) = xi,0cn(ωit,ki),

where xi,0, ωiand 0 ≤ ki≤ 1, correspond respectively to the oscillations amplitude, the

pulsation and the modulus of the Jacobian elliptic function cn.

Using the properties of the cn function and deriving twice equation (3), we get

d2xi

dt2= −2 kiω2

x2

i,0

which provides, after identification with eq.(2), the pulsation ωiand modulus kiof the cn

function

ωi(xi,0) = K′?

Writing the initial condition Ui,0= xi,0+m, solutions of eq. (1) can be straightforwardly

deduced from eqs. (3) and (5), namely

Ui(t) = m+(Ui,0−m)cn(ωit,ki),

′2(Ui−m+α)(Ui−m−α)(Ui−m).

(1)

′2xi(xi−α)(xi+α).

(2)

(3)

i

xi[x2

i−2 ki−1

2ki

x2

i,0],

(4)

x2

i,0−α2

and

ki(xi,0) = x2

i,0/2(x2

i,0−α2).

(5)

(6)

with

ωi(Ui,0) = K′?

(Ui,0−m)2−α2

and

ki(Ui,0) =1

2

(Ui,0−m)2

(Ui,0−m)2−α2

(7)

Both parameters ωiand kiof the ithcell appear then as driven by the initial condition Ui,0

applied to that cell.

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S. Morfu, J. Bossu and P. Marqui´ e

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Fig. 2. Time evolution of the cells 1 (a) and N (b) and of their oscillations difference (c). Parameters: α = 1.02 V,

m = 2.58 V, K = 10, U1,0= 0 V, UN,0= 0.5 V, R = 10 KΩ, C = 10 nF. (+): Experimental results, solid line:

Theoretical results obtained with Eqs. (6-7). The time to reach a phase opposition is topt= 0.66 ms.

3. Behavior of the nonlinear electrical chain

To illustrate the behavior of the chain, we have reported fig. 2 the oscillations of the cells

corresponding to the two ends of the lattice, namely cells i = 1 and i = N with respective

initial conditions Ui,0= 0 V and UN,0= 0.5 V. The oscillations take place in the range

[0 V; 5.16 V] for cell 1 and in [0.5 V; 4.66 V] for the cell N as predicted by the range

[Ui,0; 2m−Ui,0] for i ∈ {1,N} deduced from eq. (6). Moreover, the two oscillators quickly

achieve a phase opposition when UN= 4.66V andU1= 0 V, involving a maximum oscil-

lation differenceUN−U1= 4.66V at the time topt= 0.66 ms (dotted line in figure 2). The

weak initial amplitude difference ε =UN,0−U1,0= 0.5 V is then strongly increased at the

time topt= 0.66 ms.

In order to obtain the dynamics of the whole chain, we have plotted the voltage Ui(topt)

reached by each cell at the time toptversus its initial condition Ui,0(figure 3). Due to the

nonlinearity P(U) of the system, this curve is not linear, which means that all initial ampli-

tudes are not increased by the same scale factor.

This property can be extended to the field of image processing when a 2-dimensional net-

work is considered [Morfu and Comte (2004)]. Indeed the initial conditionUi,0would cor-

respond to the gray level of a pixel of a weak contrasted image, and Ui(topt) would be its

gray level after a processing time topt. Therefore, in image processing context, the horizon-

tal axis of fig. 3 would represent the initial gray scale of the weak contrasted image while

the vertical axis would correspond to the gray scale of the processed image (the dynamics

of both gray scales being defined by 0V for black and 4.66V for white).

Such a 2-dimensional system is described by the set of N ×M differential equations:

d2Vi,j

dt2

= −K

′2(Vi,j−m+α)(Vi,j−m−α)(Vi,j−m),

(8)

where N ×M is the image size, i = 1,2...N, j = 1,2...M andVi,jrepresent respectively the

pixel coordinates and its corresponding gray level. Under these conditions, the behavior

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EXPERIMENTS ON AN ELECTRICAL NONLINEAR OSCILLATORS NETWORK

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Fig. 3. State of the lattice at the optimal time topt= 0.66 ms versus its initial state. Parameters: m = 2.58 V,

α = 1.02 V, K = 10, R = 10 KΩ, C = 10 nF. (+): Experimental results; solid line: Theoretical results obtained

with Eqs (6-7).

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Fig. 4. a: Weak contrasted picture of the famous Coliseum and its histogram. b: Enhanced image and its histogram

obtained with a direct simulation of Eq. (8). Parameters: α = 1.02V, m = 2.58 V, K = 10, N = M = 128, time of

simulation topt= 0.66×10−3s, R = 10 KΩ, C = 10 nF.

of the 2D electronic CNN can be predicted loading the weak contrasted image of figure

4.a as initial condition and simulating eq. (8) with a fourth order Runge-Kutta algorithm.

Moreover, the range of initial conditions is still [0, 0.5] as shown by the histogram of the

weak contrasted picture of figure 4.a.

After a simulation time topt= 0.66×10−3, the range of gray levels of the resulting image

is strongly increased and becomes [0, 4.66] as for the 1D-network (histogram of figure

4.b).Therefore,for the specific processingtimetopt=0.66×10−3, a contrast enhancement

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S. Morfu, J. Bossu and P. Marqui´ e

allows to reveal the famous Coliseum (figure 4.b) hidden in figure 4.a. As for the 1D-

network, it is important to note that the nonlinear processing, performed by the system

described by eq. (8), does not correspond to a multiplication of all gray levels by the same

scale factor.

4. Conclusion:

In this paper, we have electronically realized an elementary cell of a CNN which was

previously introduced for image processing purpose in ref. [Morfu and Comte (2004)].

We have shown experimentally that, in a 1D network, two elementary cells with a weak

difference of initial conditions can present a maximum amplitude difference at an optimal

time. For this optimal time, the response of the whole network is in good agreement with

the theoretical prediction, which allows to validate the experimental device. Moreover, a

contrastenhancementofanimageis possibleforthis specificprocessingtime as reportedin

[Morfu and Comte (2004)]. Therefore, this experimental network constitutes a framework

for further studies and applications of nonlinear science in signal and image processing

areas. The influence of linear and/or nonlinear coupling between cells is actually under

investigation and may lead to new interesting properties of this CNN.

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