# Nonlinear Systems for Image Processing

**Abstract**

Nonlinear Systems for Image Processing

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CHAPTER

3

Nonlinear Systems for Image

Processing

Saverio Morfu*, Patrick Marquié*, Brice Nofiélé*,

and Dominique Ginhac*

Contents

I Introduction 79

II Mechanical Analogy 83

A Overdamped Case 84

B Inertial Systems 90

III Inertial Systems 95

A Image Processing 95

B Electronic Implementation 103

IV Reaction-Diffusion Systems 108

A One-Dimensional Lattice 108

B Noise Filtering of a One-Dimensional Signal 111

C Two-Dimensional Filtering: Image Processing 119

V Conclusion 133

VI Outlooks 134

A Outlooks on Microelectronic Implementation 134

B Future Processing Applications 135

Acknowledgments 141

Appendix A 142

Appendix B 143

Appendix C 144

Appendix D 145

References 146

I. INTRODUCTION

For almost 100 years, nonlinear science has attracted the attention of AQ1

researchers to circumvent the limitation of linear theories in the expla-

nation of natural phenomenons. Indeed, nonlinear differential equations

can model the behavior of ocean surfaces (Scott, 1999), the recurrence

of ice ages (Benzi et al., 1982), the transport mechanisms in living cells

* Laboratoire LE2I UMR 5158, Aile des sciences de l’ingénieur, BP 47870 21078 Dijon, Cedex, France

Advances in Imaging and Electron Physics,Volume 152, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)00603-4.

Copyright © 2008 Elsevier Inc. All rights reserved.

79

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Saverio Morfu et al.

(Murray, 1989), the information transmission in neural networks

(Izhikevich, 2007; Nagumo et al., 1962; Scott, 1999), the blood pressure

propagation in arteries (Paquerot and Remoissenet, 1994), or the excitabil-

ity of cardiac tissues (Beeler and Reuter, 1977; Keener, 1987). Therefore,

nonlinear science appears as the most important frontier for a better

understanding of nature (Remoissenet, 1999).

In the recent ﬁeld of engineering science (Agrawal1, 2002; Zakharov and

Wabnitz, 1998), considering nonlinearity has allowed spectacular progress

in terms of transmission capacities in optical ﬁbers via the concept of soli-

ton (Remoissenet, 1999). More recently, nonlinear differential equations

in many areas of physics, biology, chemistry, and ecology have inspired

unconventional methods of processing that transcend the limitations of

classical linear methods (Teuscher and Adamatzky, 2005). This growing

interest for processing applications based on the properties of nonlinear

systems can be explained by the observation that fundamental progress

in several ﬁelds of computer science sometimes seems to stagnate. Novel

ideas derived from interdisciplinary ﬁelds often open new directions of

research with unsuspected applications (Teuscher and Adamatzky, 2005).

On the other hand, complex processing tasks require intelligent sys-

tems capable of adapting and learning by mimicking the behavior of

the human brain. Biologically inspired systems, most often described by

nonlinear reaction-diffusion equations, have been proposed as convenient

solutions to very complicated problems unaccessible to modern von Neu-

mann computers. It was in this context that the concept of the cellular

neuralnetwork (CNN)was introducedby ChuaandYang asa novelclass of

information-processing systems with potential applications in areas such

as image processing and pattern recognition (Chua and Yang, 1988a, b). In

fact, CNN is used in the context of brain science or the context of emer-

gence and complexity (Chua, 1998). Since the pioneer work of Chua, the

CNN paradigm has rapidly evolved to cover a wide range of applica-

tions drawn from numerous disciplines, including artiﬁcial life, biology,

chemistry, physics, information science, nonconventional methods of com-

puting (Holden et al., 1991), video coding (Arena et al., 2003; Venetianer

et al., 1995), quality control by visual inspection (Occhipinti et al., 2001),

cryptography (Caponetto et al., 2003; Yu and Cao, 2006), signal-image pro-

cessing (Julian and Dogaru, 2002), and so on (see Tetzlaff (2002), for an

overview of the applications).

In summary, the past two decades devoted to the study of CNNs

have led scientists to solve problems of artiﬁcial intelligence by com-

bining the highly parallel multiprocessor architecture of CNNs with the

properties inherited from the nonlinear bio-inspired systems. Among the

tasks of high computational complexity routinely performed with non-

linear systems are the optimal path in a two-dimensional (2D) vector

ﬁeld (Agladze et al., 1997), image skeletonization (Chua, 1998), ﬁnding

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Nonlinear Systems for Image Processing

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the shortest path in a labyrinth (Chua, 1998; Rambidi and Yakovenchuk,

2001), or controlling mobile robots (Adamatzky et al., 2004). However, the

efﬁciency of these nonlinear systems for signal-image processing or pat-

tern recognition does not come only from their biological background.

Indeed, the nonlinearity offers an additional dimension lying in the signal

amplitude, which gives rise to novel properties not shared by linear sys-

tems. Noise removal with a nonlinear dissipative lattice (Comte et al., 1998;

Marquié et al., 1998), contrast enhancement based on nonlinear oscillators

properties (Morfu and Comte, 2004), edge detection exploiting vibration

noise (Hongler et al.,2003), optimizationby noise of nonoptimum problems

orsignal detectionaided bynoise via thestochastic resonancephenomenon

(Chapeau-Blondeau, 2000; Comte and Morfu, 2003; Gammaitoni et al.,

1998) constitute a nonrestrictive list of examples in which the properties

of nonlinear systems have allowed overcoming the limitations of classical

linear approaches.

Owing to the rich variety of potential applications inspired by nonlin-

ear systems, the efforts of researchers have focused on the experimental

realization of such efﬁcient information-processing devices. Two different

strategies were introduced (Chua and Yang, 1988a; Kuhnert, 1986), and

today, the fascinating challenge of artiﬁcial intelligence implementation

with CNN is still being investigated.

The ﬁrst technique dates from the late 1980s with the works of

Kuhnert, who proposed taking advantage of the properties of Belousov–

Zhabotinsky-type media for image-processing purposes (Kuhnert, 1986;

Kuhnert et al., 1989). The primary concept is that each micro-volume

of the active photosensitive chemical medium acts as a one-bit proces-

sor corresponding to the reduced/oxidized state of the catalyst (Agladze

et al., 1997). This feature of chemical photosensitive nonlinear media

has allowed implementation of numerous tools for image processing.

Edge enhancement, classical operations of mathematical morphology, the

restoration of individual components of an image with overlapped com-

ponents (Rambidi et al., 2002), the image skeletonization (Adamatzky

et al., 2002), the detection of urban roads, or the analysis of medical images

(Teuscher and Adamatzky, 2005) represent a brief overview of processing

tasks computed by chemical nonlinear media. However, even consider-

ing the large number of chemical “processors,” the very low velocity of

trigger waves in chemical media is sometimes incompatible with real-time

processing constraints imposed by practical applications (Agladze et al.,

1997). Nevertheless, the limitations of these unconventional methods of

computing in no way dismiss the efﬁciency and high prospects of the pro-

cessing developed with active chemical media (Adamatzky and de Lacy

Costello, 2003).

By contrast, analog circuits do not share the weakness of the previous

strategy of integration. Therefore, because of their real-time processing

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Saverio Morfu et al.

capability, electronic hardware devices constitute the most common way

to implement CNNs (Chua and Yang, 1988a). The ﬁrst step to electroni-

cally develop a CNN for image-processing purposes consists of designing

an elementary cell. More precisely, this basic unit of CNNs usually con-

tains linear capacitors, linear resistors, and linear and nonlinear controlled

sources (Chua and Yang, 1988b; Comte and Marquié, 2003). Next, to

complete the description of the network, a coupling law between cells

is introduced. Owing to the propagation mechanism inherited from the

continuous-time dynamics of the network, the cells do not only inter-

act with their nearest neighbors but also with cells that are not directly

connected. Among the applications that can be electronically realized

are character recognition (Chua and Yang, 1988), edge ﬁltering (Chen

et al., 2006; Comte et al., 2001), noise ﬁltering (Comte et al., 1998; Julián

and Dogaru, 2002; Marquié et al., 1998), contrast enhancement, and gray-

level extraction with a nonlinear oscillators network (Morfu, 2005; Morfu

et al., 2007).

The principle of CNN integration with discrete electronic components

is closely related to the development of nonlinear electrical transmission

lines (NLTLs) (Remoissenet, 1999).Indeed, undercertain conditions (Chua,

1998), the parallel processing of information can be ruled by nonlinear

differential equations that also describe the evolution of the voltage at

the nodes of an electrical lattice. It is then clear that considering a one-

dimensional(1D) latticeallowssignal ﬁltering,whileextending theconcept

to a 2D network can provide image processing applications.

The development of NLTLs was motivated mainly by the fact that

these systems are quite simple and relatively that inexpensive experimen-

tal devices allow quantitative study of the properties of nonlinear waves

(Scott, 1970). In particular,since thepioneering works by Hirotaand Suzuki

(1970) and Nagashima and Amagishi (1978) on electrical lines simulating

the Toda lattice (Toda, 1967), these NLTLs, which can be considered as

analog simulators, provide a useful way to determine the behavior of exci-

tations inside the nonlinear medium (Jäger, 1985; Kuusela, 1995; Marquié

et al., 1995; Yamgoué et al., 2007).

This chapter is devoted primarily to the presentation of a few particular

nonlinear processing tools and discusses their electronic implementation

with discrete components.

After a brief mechanical description of nonlinear systems, we present a

review of the properties of both purely inertial systems and overdamped

systems. The following sections show how taking advantage of these pro-

perties allows the development of unconventional processing methods.

Especiallyconsidering the featuresofpurelyinertialsystems, we showhow

it is possible to perform various image-processing tasks, such as contrast

enhancement of a weakly contrasted picture, extraction of gray levels, or

encryption of an image. The electronic sketch of the elementary cell of this

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Nonlinear Systems for Image Processing

83

inertial CNN is proposed, and the nonlinear properties that allows the

previous image processing tasks are experimentally investigated.

The thirdpart of this chapter is devoted exclusively to the ﬁltering appli-

cations inspired by reaction-diffusion media—for example, noise ﬁltering,

edge detection, or extraction of interest regions in a weakly noisy con-

trasted picture. In each case, the elementary cell of the electronic CNN is

developed and we experimentally investigate its behavior in the speciﬁc

context of signal-image processing. We conclude by discussing the possi-

ble microelectronic implementations of the previous nonlinear systems. In

addition, the last section contains some perspectives for future develop-

ments inspired by recent properties of nonlinear systems. In particular, we

present a paradoxical nonlinear effect known as stochastic resonance (Benzi

et al., 1982; Chapeau-Blondeau, 1999; Gammaitoni et al., 1998), which is

purported to have potential applications in visual perception (Simonotto AQ:2

et al., 1997).

We trust that the multiple topics in this contribution will assist readers

in better understanding the potential applications based on the properties

of nonlinear systems. Moreover, the various electronic realizations pre-

sented constitute a serious background for future experiments and studies

devoted to nonlinear phenomena. As it is written for an interdisciplinary

readership of physicist and engineers, it is our hope that this chapter will

encourage readers to perform their own experiments.

II. MECHANICAL ANALOGY

In order to understand the image-processing tools inspired by the pro-

perties of nonlinear systems, we present a mechanical analogy of these

nonlinear systems. From a mechanical point of view, we consider a chain

of particles of mass M submitted to a nonlinear force f deriving from a

potential and coupled with springs of strength D.IfW

n

represents the

displacement of the particle n, the fundamental principle of the mechanics

is written as

M

d

2

W

n

dt

2

+ λ

dW

n

dt

=−

d

dW

n

+ R

n

, (1)

whereM

d

2

W

dt

2

representsthe inertia termand λ

dW

dt

correspondstoa friction

force. Furthermore, the resulting elastic force R

n

applied to the n

th

particle

by its neighbors can be deﬁned by:

R

n

= D

j∈Nr

W

j

− W

n

, (2)

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Saverio Morfu et al.

where Nr is the neighborhood, namely, Nr ={n −1, n + 1} in the case of

a 1D chain.

We propose to investigate separately the purely inertial case, that is

M

d

2

W

dt

2

>> λ

dW

dt

, and the overdamped one deduced when M

d

2

W

dt

2

<<

λ

dW

dt

.

A. Overdamped Case

In this section, an overdampedsystem is presentedby neglectingthe inertia

term of Eq. (1) compared to the friction force. We speciﬁcally consider

λ = 1 and the case of a cubic nonlinear force

f (W) =−W(W − α)(W − 1), (3)

deriving from the double-well potential (W) =−

W

0

f (u)du as repre-

sented in Figure 1 for different values of α. The roots of the nonlinear

force 0 and 1 correspond to the positions of the local minima of the poten-

tial, namely, the well bottoms, whereas the root α represents the position of

the potential maximum. The nonlinearity threshold α deﬁnes the potential

barrier between the potential minimum with the highest energy and the

potential maximum. To explain the propagation mechanism in this chain,

it is convenient to deﬁne the excited state by the position of the potential

minimum with the highest energy, and the rest state by the position corre-

sponding to the minimum of the potential energy. As shown in Figure 1a,

0.04

20.2

0.02

0

0 0.2 0.4 0.6 0.8 1 1.2

0.06

␣ 5 0.8

␣ 5 0.4

␣ 5 0.2

␣ 5 0.3

␣ 5 0.7

␣ 5 0.6

W

0

20.2

20.02

20.04

0 0.2 0.4 0.6 0.8 1 1.2

0.02

W

F(W )

F(W )

(a) (b)

FIGURE 1 Double-well potential deduced from the nonlinear force (3). (a) For

α<1/2 the well bottom with highest energy is located at W = 0, the potential

barrier is given by =

α

0

f (u)du = φ(α) − φ(0). (b) For α>1/2 the symmetry of

the potential is reversed: W = 1 becomes the position of the well bottom of highest

energy, and the potential barrier is =

α

1

f (u)du = φ(α) − φ(1).

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Nonlinear Systems for Image Processing

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the excited state is 0 and the rest state is 1 when the nonlinearity threshold

α<1/2. In the case α>1/2, since the potential symmetry is reversed, the

excited state becomes 1 and the rest state is 0 (Figure 1b). The equation that

rules this overdamped nonlinear systems can be deduced from Eq. (1).

Indeed, when the second derivative versus time is neglected compared to

the ﬁrstderivative and when λ = 1, Eq.(1) reduces to the discreteversion of

Fisher’s equation, introduced in the 1930s as a model for genetic diffusion

(Fisher, 1937):

dW

n

dt

= D(W

n+1

+ W

n−1

− 2W

n

) + f (W

n

).(4)

1. Uncoupled Case

We ﬁrst investigate the uncoupled case, that is, D = 0 in Eq. (4), to deter-

mine the bistability of the system. The behavior of a single particle of

displacement W and initial position W

0

obeys

dW

dt

=−W(W − α)(W − 1). (5)

The zeros of the nonlinear force f , W = 1 and W = 0 correspond to stable

steady states, whereas the state W = α is unstable. The stability analy-

sis can be realized by solving Eq. (5) substituting the nonlinear force

f =−W(W − α)(W − 1) for its linearized expression near the considered AQ:3

steady states W

∗

∈{0, 1, α}.Iff

W

(W

∗

) denotes the derivative versus W of

the nonlinear force for W = W

∗

, we are led to solve:

dW

dt

= f

W

(W

∗

)(W − W

∗

) + f (W

∗

). (6)

The solution of Eq. (6) can then be easily expressed as

W(t) = W

∗

+ Ce

f

W

(W

∗

)t

−

f (W

∗

)

f

W

(W

∗

)

(7)

where C is a constant depending on the initial condition—the initial

position of the particle. The solution in Eq. (7), obtained with a linear

approximation of the nonlinear force f , shows that the stability is set by

the sign of the argument of the exponential function.

Indeed, for W

∗

= 0 and W

∗

= 1, the sign of f

W

(W

∗

) is negative, involv-

ing that W(t →∞) tends to a constant. Therefore, the two points W

∗

= 0

and W

∗

= 1 are stable steady states. Conversely, for W

∗

= α, f

W

(W

∗

) is

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86

Saverio Morfu et al.

positive, inducing a divergence for W(t →∞). W

∗

= α is an unstable

steady state.

We now focus our attention on the particular case α = 1/2 since it will

allow interesting applications in signal and image processing.

This case is intensively developed in Appendix A, where it is shown

that the displacement of a particle with initial position W

0

can be

expressed by

W(t) =

1

2

⎛

⎜

⎝

1 +

W

0

−

1

2

(W

0

−

1

2

)

2

− W

0

(W

0

− 1)e

−

1

2

t

⎞

⎟

⎠

. (8)

This theoretical expression is compared in Figure 2 to the numerical

results obtained solving Eq. (5) using a fourth-order Runge–Kutta algo-

rithm with integrating time step dt = 10

−3

. As shown in Figure 2, when

the initial condition W

0

is below the unstable state α = 1/2, the particle

evolves toward the steady state 0. Otherwise, if the initial condition W

0

exceeds the unstable state α = 1/2, the particle evolves toward the other

steady state 1. Therefore, the unstable states α = 1/2 acts as a threshold

and the system exhibits a bistable behavior.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Displacement x (normalized units)

t (normalized units)

dis

p

lacement W

F(W )(10

23

)

0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

01234567891016 8 0

Stable

state

Stable

state

Unstable

state

FIGURE 2 Bistable behavior of the overdamped system in the case α = 1/2. Left:

Evolution of a particle for different initial conditions in the range [0; 1]. The solid line

is plotted with the analytical expression in Eq. (8), whereas the (o) signs correspond to

the numerical solution of Eq. (5) for different initial conditions W

0

∈[0; 1]. The

potential φ obtained by integrating the nonlinear force (3) is represented at the right

as a reference.

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Nonlinear Systems for Image Processing

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2. Coupled Case

We now consider the coupled case (D = 0). In such systems ruled by

Eq. (4), the balance between the dissipation and the nonlinearity gives

rise to the propagation of a kink (a localized wave) called a diffusive soli-

ton that propagates with constant velocity and proﬁle (Remoissenet, 1999).

To understand the propagation mechanism, we ﬁrst consider the weak

coupling limit and the case α<1/2. The case of strong coupling, which

corresponds to a continuous medium, is discussed later since it allows

theoretical characterization of the waves propagating in the medium.

a. Weak Coupling Limit. As shown in Figure 3a, initially all particles of

the chain are located at the position 0—the excited state. To initiate a kink,

an external forcing allows the ﬁrst particle to cross the potential barrier

in W = α and to fall in the right well, at the rest state deﬁned by the

position W = 1. Thanks to the spring coupling the ﬁrst particle to the

second one, but despite the second spring, the second particle attempts

to cross the potential barrier with height (α) =−

α

4

12

+

α

3

6

(Morfu, 2003)

(see Figure 3b).

D

D

D

D

D

D

W

4

W

3

W

2

W

1

(W

n

)

Δ(␣)

W

n

(t 5 0)

0

␣

1

(a)

(b)

W

4

W

3

W

2

D

?

W

1

(W

n

)

Δ(␣)

W

2

W

n

(t . 0)

0

␣

1

FIGURE 3 Propagation mechanism. (a) Initially all particles of the chain are in the

excited state 0, that is, at the bottom of the well with highest energy. (b) State of the

chain for t > 0. The first particle has crossed the potential barrier and attempts to

pull the second particle down in its fall.

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Saverio Morfu et al.

According to the value of the resulting force applied to the second

particle by the two springs compared to the nonlinear force f between

[0, α[, two behaviors may occur:

1. If the resulting elastic force is sufﬁciently important to allow the second

particle to cross the potential barrier (α), then this particle falls in

the right well and pulls the next particle down in its fall. Since each

particle of the chainsuccessively undergoes a transition from the excited

state 0 to the rest state 1, a kink propagates in the medium. Moreover,

its velocity increases versus the coupling and as the barrier decreases

(namely, as α decreases).

2. Otherwise, if the resulting force does not exceed a critical value (i.e., if

D < D

∗

(α)), the second particle cannot cross the potential barrier and

thus stays pinned at a position w in [0; α[: it is the well-known propaga-

tion failure effect (Comte et al., 2001; Erneux and Nicolis, 1993; Keener,

1987; Kladko et al., 2000).

The mechanical model associated with Eq. (4) shows that in the weak

coupling limit the characteristics of the nonlinear system are ruled by the

coupling D and the nonlinear threshold α. Moreover, the propagation of a

kink is due to the transition from the excited state to the rest state and is

only possible when the coupling D exceeds a critical value D

∗

(α).

b. Limit of Continuous Media. The velocity of the kink and its proﬁle can be

theoreticallyobtainedin thelimitof continuousmedia—whenthe coupling

D is large enough compared to the nonlinear strength.

Then, in the continuous limit, the discrete Laplacian of Eq. (4) can be

replaced by a second derivative versus the space variable z:

∂W

∂t

= D

∂

2

W

∂z

2

+ f (W). (9)

This equation, introduced by Nagumo in the 1940s as an elementary

representation of the conduction along an active nerve ﬁber, has an impor-

tant meaning in understanding transport mechanism in biological systems

(Murray, 1989; Nagumo et al., 1962).

Unlike the discrete Equation (4), the continuous Equation (9) admits

propagative kink solution only if

1

0

f (u)du = 0, which reduces to α = 1/2

in the case of the cubic force (3) (Scott, 1999).

Introducing the propagative variable ξ = z − ct, these kinks and anti-

kinks have the form (Fife, 1979; Henry, 1981)

W(ξ) =

1

2

1 ± tanh

1

2

√

2D

(ξ − ξ

0

)

, (10)

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Nonlinear Systems for Image Processing

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where ξ

0

is the initial position of the kink for t = 0 and where the kink

velocity is deﬁned by c =±

D/2(1 − 2α).

When α<1/2, the excited state is 0, and the rest state is 1. Therefore, the

rest state 1 spreads in the chain, which set the sign of the velocity according

to the proﬁle of the kink initiated in the nonlinear system:

1. If the proﬁle is given by W(ξ) =

1

2

1 − tanh

1

2

√

2D

(ξ − ξ

0

)

, a kink

propagates from left to right with a positive velocity c =

D/2(1 − 2α)

(Figure 4a, left).

2. Otherwise, if the proﬁle is set by W(ξ) =

1

2

1 + tanh

1

2

√

2D

(ξ − ξ

0

)

,

a kink propagates from right to left with a negative velocity c =

−

D/2(1 − 2α) (Figure 4a, right).

When α>1/2, since the symmetry of the potential is reversed, the

excited state becomes 1 and the rest state is 0. The propagation is then due

to a transition between 1 and 0, which provides the following behavior:

1. If W(ξ) =

1

2

1 − tanh

1

2

√

2D

(ξ − ξ

0

)

, a kink propagates from right to

left with a negative velocity c =

D/2(1 − 2α) (Figure 4b, left).

215 2505 15

0

0.2

1

0.6

Z

215 2505 15

Z

W

W(z)

0

0.2

1

0.6

W(z)

1

0.5

0

0.5 21.5 23.5

F(W ) 10

2

(a)

215 2505 15

0

0.2

1

0.6

Z

215 2505 15

Z

W

W(z)

0

0.2

1

0.6

W(z)

1

0.5

0

420

F(W ) 10

2

(b)

FIGURE 4 Propagative solution of the continuous Nagumo Equation (9) with D = 1.

Spatial representation of the kink for t = 0 in dotted line and for t = 20 in solid line.

The arrow indicates the propagation direction, the corresponding potential is

represented at the right end to provide a reference. (a) α = 0.3, (b) α = 0.7.

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Saverio Morfu et al.

2. Else if W(ξ) =

1

2

1 + tanh

1

2

√

2D

(ξ − ξ

0

)

, a kink propagates from left

to right with a positive velocity c =−

D/2(1 − 2α) (Figure 4b, right).

B. Inertial Systems

In this section, we neglect the dissipative term of Eq. (1) compared to the

inertia term and we restrict our study to the uncoupled case. Moreover, in

image-processing context, it is convenient to introduce a nonlinear force f

under the form

f (W) =−ω

2

0

(W − m)(W − m − α)(W − m + α),(11)

where, m and α<m are two parameters that allow adjusting the width

and height = ω

2

0

α

4

/4 of the potential (Figure 5):

(W) =−

W

0

f (u)du. (12)

The nonlinear differential equation that rules the uncoupled chain can

be deduced by inserting the nonlinear force (11) into Eq. (1) with D = 0.

first particle: W

1

0

second particle: W

2

0

0

21

22

23

24

25

26

27

28

0 0.5 1 1.5 2 2.5 3 3.5 4 54.5

W

2

0

m 2 ␣ m 1 ␣

W (Arb.Unit)

2m 2W

2

0

Potential energy

!

W

2

!

W

2

FIGURE 5 Double-well potential deduced from the nonlinear force (11) represented

for m =2.58, α =1.02, and ω

0

=1. A particle with an initial condition W

0

i

< m − α

√

2

evolves with an initial potential energy above the barrier .

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91

Neglecting the dissipative term, the particles of unitary mass arethen ruled

by the following nonlinear oscillator equations:

d

2

W

i

dt

2

= f (W

i

). (13)

1. Theoretical Analysis

We propose here to determine analytically the dynamics of the nonlinear

oscillators obeying Eq. (13) (Morfu and Comte, 2004; Morfu et al., 2006).

Setting x

i

= W

i

− m, Eq. (13) can be rewritten as

d

2

x

i

dt

2

=−ω

2

0

x

i

(x

i

− α)(x

i

+ α). (14)

Noting x

0

i

the initial position of the particle i and considering that all the

particles initially have a null velocity, the solutions of Eq. (14) can be

expressed with the Jacobian elliptic functions as

x

i

(t) = x

0

i

cn(ω

i

t, k

i

), (15)

where ω

i

and 0 ≤ k

i

≤ 1 represent, respectively, the pulsation and the

modulus of the cn function (see recall on the properties of Jacobian elliptic

function in Appendix B).

Deriving Eq. (15) twice and using the properties in Eq. (B3), yields

dx

i

dt

=−x

0

i

ω

i

sn(ω

i

t, k

i

)dn(ω

i

t, k

i

),

d

2

x

i

dt

2

=−x

0

i

ω

2

i

cn(ω

i

t, k

i

)

dn

2

(ω

i

t, k

i

) − k

i

sn

2

(ω

i

t, k

i

)

. (16)

Using the identities in Eq. (B4) and (B5), Eq. (16) can be rewritten as

d

2

x

i

dt

2

=−

2k

i

ω

2

i

x

0

2

i

x

x

2

−

2k

i

− 1

2k

i

x

0

2

i

. (17)

Identifying this last expression with Eq. (14), we derive the pulsation of

the Jacobian elliptic function

ω

i

= ω

0

x

0

2

i

− α

2

, (18)

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92

Saverio Morfu et al.

and its modulus

k

i

=

1

2

x

0

2

i

x

0

2

i

− α

2

. (19)

Finally, introducing the initial condition W

0

i

= x

0

i

+ m, the solution of

Eq. (13) can be straightforwardly deduced from Eqs. (15), (18), and (19):

W

i

(t) = m +

W

0

i

− m

cn(ω

i

t, k

i

), (20)

with

ω

i

W

0

i

= ω

0

W

0

i

− m

2

− α

2

and k

i

W

0

i

=

1

2

W

0

i

− m

2

W

0

i

− m

2

− α

2

.

(21)

Both the modulus and the pulsation are driven by the initial condition

W

0

i

. Moreover, the constraints to ensure the existence of the pulsation

ω

i

and of the modulus, respectively, are written as

W

0

i

− m

2

− α

2

≥ 0

and 0 ≤ k

i

≤ 1. These two conditions restrict the range of the allowed ini-

tial conditions W

0

i

to

−∞; m −α

√

2

m + α

√

2; +∞

, as shown in

Figure 6, where the pulsation and the modulus are represented versus the

initial condition W

0

i

. Note that this allowed range of initial conditions cor-

responds also to a particle with an initial potential energy exceeding the

barrier between the potential extrema (see Figure 5).

2. Nonlinear Oscillator Properties

To illustrate the properties of nonlinear oscillators, we consider a chain of

length N =2 particles with a weak difference of initial conditions and with

a null initial velocity. The dynamics of these two oscillators are ruled by

Eq. (20), where the pulsation and modulus of both oscillators are driven by

their respective initial condition. Moreover, we have restricted our study

to the case of the following nonlinearity parameters m = 2.58, α = 1.02,

ω

0

= 10

4

. We haveapplied the initial condition W

0

1

= 0 to theﬁrst oscillator,

while the initial condition of the second oscillator is set to W

0

2

= 0.2, which

corresponds to the situation of Figure 5.

Figure 7a shows that the oscillations of both particles take place in the

range [W

0

i

;2m − W

0

i

] as predicted by Eq. (20) [that is, [0; 5.16] for the

ﬁrst oscillator and [0; 4.96] for the second one]. Moreover, owing to their

difference of initial amplitude and to the nonlinear behavior of the sys-

tem, the two oscillators quickly attain a phase opposition for the ﬁrst

time at t = t

opt

= 1.64 × 10

−3

. This phase opposition corresponds to the

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93

0

0

0.5

1

1.5

2

2.5

0.5 1

Ⲑ

0

⌬Ⲑ

0

⌬Ⲑ

0

⌬W

i

0

⌬W

i

0

W

i

0

1.5 2

(a)

(b)

2.5 3 3.5 4 4.5 5

Forbidden range

of parameters

]m 2 ␣ŒW; m 1 ␣ŒW[

22

0

0

0.5

1

1.5

0.5 1

k

W

i

0

1.5 2 2.5 3 3.5 4 4.5 5

Forbidden range

of parameters

]m 2 ␣ŒW; m 1 ␣ŒW[

22

FIGURE 6 (a): Normalized pulsation ω/ω

0

versus the initial condition W

0

i

.

(b) Modulus parameter k versus W

0

i

. The parameters of the nonlinearity m = 2.58,

α = 1.02 impose the allowed amplitude range ]−∞; 1.137]

[4.023; +∞[.

t

opt

t

opt

w

1

(t)w

2

(t)

time

time

time

0

6

5

4

3

2

1

0

1 1.5 2

x(10

23

)

x (10

23

)

x (10

23

)

3

2.5

0.5

0

0

1

2

3

4

5

1 1.5 2 32.50.5

5

4

3

2

1

0

21

22

23

24

25

0.5 1.5132.520

(a) (b)

␦(t) 5 W

2

(t) 2 W

1

(t)

FIGURE 7 (a) Temporal evolution of the two oscillators. Top panel: evolution of

the first oscillator with initial condition W

0

1

= 0. Bottom panel: evolution of the

second oscillator with initial condition W

0

2

= 0.2. (b) Temporal evolution of the

displacement difference δ between the two oscillators. Parameters: m = 2.58, α = 1.02,

and ω

0

= 1.

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94

Saverio Morfu et al.

situation where the ﬁrst oscillator has reached its minimum W

1

(t

opt

) = 0,

whereas the second oscillator has attained its maximum W

2

(t

opt

) = 4.96.

As shown in Figure 7b, the displacement difference δ(t) = W

2

(t) − W

1

(t)

is then maximum for t = t

opt

and becomes δ(t

opt

) = 4.96. For this optimal

time, a “contrast enhancement” of the weak difference of initial conditions

is realized, since initially the displacement difference was δ(t = 0) = 0.2.

Note that in Figure 7b, the displacement difference between the two

oscillators also presents a periodic behavior with local minima and local

maxima. In particular, the difference δ(t) is null for t = 3.96 × 10

−5

, t =

1.81 × 10

−4

, t = 3.5 × 10

−4

, t = 5.21 × 10

−4

; minimum for t = 1.4 × 10

−4

,

t = 4.64 × 10

−4

, t = 1.47 × 10

−3

and maximum for t = 3 × 10

−4

, t = 6.29 ×

10

−4

, t = 1.64 × 10

−3

. These characteristic times will be of crucial interest

in image-processing context to deﬁne the ﬁltered tasks performed by the

nonlinear oscillators network.

Figure 6a reveals that the maximum variation of the pulsation com-

pared to the amplitude W

0

i

, that is, ω/ω

0

, is reached for W

0

i

= m − α

√

2,

that is, for a particle with an initial potential energy near the barrier .

Therefore, to quickly realize a great amplitude contrast between the two

oscillators, it could be interesting to launch them with an initial amplitude

near m − α

√

2, or to increase the potential barrier height . We chose to

investigate this latter solution by tuning the parameter of the nonlinear-

ity α, when the initial amplitude of both oscillators remains W

0

1

= 0 and

W

0

2

= 0.2. The results are reported in Figure 8, where we present the

evolution of the difference δ(t) for different values of α.

As expected, when the nonlinearity parameter α increases, the optimal

time is signiﬁcantly reduced. However, when α is adjusted near the critical

value (m − W

0

2

)/

√

2 as in Figure 8d, the optimum reached by the difference

δ(t) is reduced to 4.517 for α = 1.63 instead of 4.96 for α = 1.02. Even if it

is not the best contrast enhancement that can be performed by the sys-

tem, the weak difference of initial conditions between the two oscillators

is nevertheless strongly enhanced for α = 1.63.

To highlight the efﬁciency of nonlinear systems, let us consider the case

of a linear force f(W) =−ω

0

W in Eq. (13).

In the linear case, the displacement difference δ(t) between two har-

monic oscillators can be straightforwardly expressed as

δ(t) = cos(ω

0

t), (22)

where represents the slight difference of initial conditions between the

oscillators. This last expression shows that it is impossible to increase

the weak difference of initial conditions since the difference δ(t) always

remains in the range [−; ]. Therefore, nonlinearity is a convenient solu-

tion to overcome the limitation of a linear system and to enhance a weak

amplitude contrast.

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95

t

opt

01

25

23

21

1

3

5

23

␦(t) 5W

2

(t ) 2W

1

(t)

t 310

23

t

opt

01

25

23

21

1

3

5

23

␦(t) 5W

2

(t ) 2W

1

(t)

t 310

23

t

opt

01

25

23

21

1

3

5

23

␦(t) 5W

2

(t ) 2W

1

(t)

t 310

23

t

opt

01

25

23

21

1

3

5

23

␦(t) 5W

2

(t ) 2W

1

(t)

t 310

23

(a) (b)

(c) (d)

FIGURE 8 Influence of the nonlinearity parameter α on the displacement difference

δ between the two oscillators of respective initial conditions 0 and 0.2. Parameters

m = 2.58 and ω

0

= 1. (a): (t

opt

= 1.75 × 10

−3

; α = 0.4). (b): (t

opt

= 1.66 × 10

−3

;

α = 1.05). (c): (t

opt

= 1.25 × 10

−3

; α = 1.5). (d): (t

opt

= 0.95 × 10

−3

; α = 1.63).

III. INERTIAL SYSTEMS

This section presents different image-processing tasks inspired by the

properties of the nonlinear oscillators presented in Section II.B. Their

electronic implementation is also discussed.

A. Image Processing

By analogy with a particle experiencing a double-well potential, the pixel

number (i, j) is analog to a particle (oscillator) whose initial position

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96

Saverio Morfu et al.

corresponds to the initial gray level W

0

i, j

of this pixel. Therefore, if N × M

denotes the image size, we are led to consider a 2D network, or CNN,

consisting of uncoupled nonlinear oscillators. The node i, j of this CNN

relates to

d

2

W

i, j

dt

2

=−ω

2

0

(W

i, j

− m − α)(W

i, j

− m + α)(W

i, j

− m), (23)

with i = 1, 2 ...N and j = 1,2..,M.

Note that we take into account the range of oscillations [0; 2m − W

0

i, j

]

predicted in Section II.B.2 to deﬁne the gray scale of the images, namely,

0 for the black level and 2m = 5.16 for the white level.

The image to be processed is ﬁrst loaded as the initial condition at the

nodes of the CNN. Next, the ﬁltered image for a processing time t can be

deduced noting the position reached by all oscillators of the network at this

speciﬁc time t. More precisely, the state of the network at a processing time

t is obtained by solving numerically Eq. (23) with a fourth-order Runge–

Kutta algorithm with integrating time step dt = 10

−6

.

1. Contrast Enhancement and Image Inversion

The image to process with the nonlinear oscillator network is the weak

contrasted image of Figure 9a. Its histogram is restricted to the range

[0; 0.2], which means that the maximum gray level of the image (0.2) is the

initial condition of at least one oscillator of the network, while the mini-

mum gray level of the image (0) is also the initial condition of at least

one oscillator. Therefore, the pixels with initial gray level 0 and 0.2 oscil-

late with the phase difference δ(t) predicted by Figure 7b. In particular, as

explained in Section II.B.2, their phase difference δ(t) can be null for the

processing times t = 3.96 × 10

−4

, 1.81 × 10

−4

, 3.5 × 10

−4

, and 5.21 ×10

−4

;

minimum for t = 1.4 × 10

−4

, 4.64 ×10

−4

, and 1.47 × 10

−3

and maximum

for t = 3 ×10

−4

, 6.29 × 10

−3

, and 1.64 ×10

−3

. As shown in Figure 9b, 9d,

9f, and 9h, the image goes through local minima of contrast at the process-

ing times corresponding to the zeros of δ(t). Furthermore, the processing

times providing the local minima of δ(t) realize an image inversion with

a growing contrast enhancement (Figure 9c, 9g, and 9j). Indeed, since the

minima of δ(t) are negative, for these processing times the minimum of the

initial image becomes the maximum of the ﬁltered image and vice versa.

Finally, the local maxima of δ(t) achieve local maxima of contrast for the

corresponding processing times (Figure 9e, 9i, and 9k). Note that the best

enhancement ofcontrast is attainedat the processingtime t

opt

for whichδ(t)

is maximum. The histogram of each ﬁltered image in Figure 9 also reveals

the temporal dynamic of the network. Indeed, the width of the image his-

togram is periodically increased and decreased, which indicates that the

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Nonlinear Systems for Image Processing

97

(a)

(b) (c)

800

0 4.96

2500

0 4.96

500

0 4.96

2500

(d)

0 4.96

350

(e)

0 4.96

1200

(f)

0 4.96

(g)

(h)

(i)

300

0 4.96

1600

0 4.96

250

0 4.96

300

(j)

0 4.96

(k)

500

0 4.96

FIGURE 9 Filtered images and their corresponding histogram obtained with the

nonlinear oscillators network (23) for different processing times. (a) Initial image

(t = 0). (b) t = 3.96 × 10

−5

.(c)t = 1.4 × 10

−4

.(d)t = 1.81 × 10

−4

.(e)t = 3 × 10

−4

.

(f) t = 3.5 × 10

−4

. (g) t = 4.64 × 10

−4

. (h) t = 5.21 × 10

−4

. (i) t = 6.29 × 10

−4

.

(j) t = 1.47 × 10

−3

.(k)t = t

opt

= 1.64 × 10

−3

. Parameters: m = 2.58, α = 1.02, ω

0

= 1.

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98

Saverio Morfu et al.

contrast of the corresponding ﬁltered image is periodically enhanced or

reduced.

Another interesting feature of the realized contrast is determined by

the plot of the network response at the processing time t

opt

(Morfu, 2005).

This curve also represents the gray level of the pixels of the ﬁltered image

versus their initial gray level. Therefore, the horizontal axis corresponds to

the initial gray scale, namely, [0; 0.2], whereas the vertical axis represents

the gray scale of the processed image. Such curves are plotted in Figure 10

for different values of the nonlinearity parameter α, and at the optimal time

deﬁned by the maximum of δ(t). In fact, these times were established in

Section II.B.2 in Figure 8.

Moreover, to compare the nonlinear contrast enhancement to a uniform

one, we have superimposed (dotted line) the curve resulting from a simple

multiplication of the initial gray scale by a scale factor. In Figure 10a, since

the response of the system for the lowest value of α is most often above the

dotted line, the ﬁltered image at the processing time t

opt

= 1.75 × 10

−3

for

α = 0.4 will be brighter than the image obtained with a simple rescaling.

5

0

0 0.1

W

i

0.2

(a)

(c) (d)

(b)

2.5

5

0

0 0.1

W

i

0.2

2.5

5

0

0 0.1

W

i

0.2

2.5

5

0

0 0.1

W

i

0.2

2.5

W

0

i

W

0

i

W

0

i

W

0

i

FIGURE 10 Response of the nonlinear system for different nonlinearity parameters α

at the corresponding optimal time t

opt

(solid line) compared to a uniform rescaling

(dotted line). The curves are obtained with Eqs. (20) and (21) setting the time to

the optimum value defined by the maximum of δ(t) (see Figure 8). In addition, we

let the initial conditions W

0

i

vary in the range [0; 0.2] in Eqs. (20) and (21). (a): (t

opt

=

1.75 ×10

−3

; α = 0.4). (b): (t

opt

= 1.66 × 10

−3

; α = 1.05). (c): (t

opt

= 1.25 × 10

−3

;

α = 1.5). (d): (t

opt

= 0.95 × 10

−3

; α = 1.63), ω

0

= 1.

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Nonlinear Systems for Image Processing

99

As shown in Figure 10b, increasing the nonlinearity parameter α to 1.05

involves an optimum time 1.66 ×10

−3

and symmetrically enhances the

light and dark gray levels. When the nonlinearity parameter is adjusted

to provide the greatest potential barrier (Figure 10c and 10d), the contrast

of the medium gray level is unchanged compared to a simple rescaling.

Moreover, the dark and light grays are strongly enhanced with a greater

distortion when the potential barrier is maximum, that is, for the greatest

value of α (Figure 10d).

2. Gray-Level Extraction

Considering processing times exceeding the optimal time t

opt

, we propose

to perform a gray-level extraction of the continuous gray scale represented

in Figure 11a (Morfu, 2005). For the sake of clarity, it is convenient to

redeﬁne the white level by 0.2, whereas the black level remains 0.

For the nine speciﬁc times presented in Figure 11, the response of the

system displays a minimum that is successively reached for each level

of the initial gray scale. Therefore, with time acting as a discriminating

parameter, an appropriate threshold ﬁltering allow extraction of all pixels

with a gray level in a given range. Indeed, in Figure 11, the simplest case of

a constant threshold V

th

= 0.25 provides nine ranges of gray at nine closely

different processing times, which constitutes a gray-level extraction.

Moreover, owingtothe responseof thesystem,the width ofthe extracted

gray-level ranges is reduced in the light gray. Indeed, the range extracted

in the dark gray for the processing time t = 3.33 × 10

−3

(Figure 11c) is

approximatively twice greater than the range extracted in the light gray

for t = 3.51 ×10

−3

(Figure 11i). To perform a perfect gray-level extraction,

the threshold must match with a slight offset the temporal evolution of the

minimum attained by the response of the system. Under these conditions,

the width of the extracted gray range is set by the value of this offset.

Note that the response of the system after the optimal processing times

also allow consecutive enhancement of the fragment of the image with

different levels of brightness, which is also an important feature of image

processing. For instance, in Belousov–Zhabotinsky-type media this prop-

erty of the system enabled Rambidi et al. (2002) to restore individual

components of the picture when the components overlap. Therefore, we

trust that considering the temporal evolution of the image loaded in our

network could give rise to other interesting image-processing operations.

3. Image Encryption

Cryptography is another ﬁeld of application of nonlinear systems. In fact, AQ4

the chaotic behavior of nonlinear systems can sometimes produce chaotic

like waveforms that can be used to encrypt signals for secure commu-

nications (Cuomo and Oppenheim, 1993; Dedieu et al., 1993). Even if

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100

Saverio Morfu et al.

0.2

0

(a)

(b)

(e) (f) (g)

(h) (i) (j)

(c) (d)

W

i

0

W

i

0

W

i

0

W

i

0

W

i

0

W

i

0

66

0

0

0.2

6

0

W

i

W

i

0

W

i

0

W

i

0

0

0.2

6

0

W

i

0

0.2

6

0

w

i

0

0.2

6

0

W

i

W

i

W

i

W

i

W

i

W

i

0

0.2

6

0

0

0.2

6

0

0

0.2

0

0

0.2

0

0.2

0

6

FIGURE 11 Gray-level extraction. The response of the system is represented at the

top of each figure. At the bottom of each figure, a threshold filtering of the filtered

image is realized replacing the pixel gray level with 0.2 (white) if that gray level

exceeds the threshold V

th

= 0.25, otherwise with 0 (black). (a) Initial gray scale (t = 0).

(b) t = 3.3 ×10

−3

.(c)t = 3.33 × 10

−3

.(d)t = 3.36 × 10

−3

.(e)t = 3.39 × 10

−3

.

(f) t = 3.42 × 10

−3

. (g) t = 3.45 × 10

−3

. (h) t = 3.48 × 10

−3

. (i) t = 3.51 × 10

−3

.

(j) t = 3.54 ×10

−3

. Nonlinearity parameters: m = 2.58, α = 1.02, and ω

0

= 1

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Nonlinear Systems for Image Processing

101

many attempts to break the encryption key of these cryptosystems and to

retrieve the information have been reported (Short and Parker, 1998;

Udaltsov et al., 2003), cryptography based on the properties of chaotic

oscillators still attracts the attention of researchers because of the promis-

ing applications of chaos in the data transmission ﬁeld (Kwok and Tang,

2007).

Contrary to most studies, in which the dynamics of a single element

are usually considered, we propose here a strategy of encryption based on

the dynamics of a chain of nonlinear oscillators. More precisely, we con-

sider the case of a noisy image loaded as the initial condition in the inertia

network introduced in Section II.B. In addition, we add a uniform noise

over [−0.1; 0.1] to the weak-contrast picture of the Coliseum represented

in Figure 9a. Since the pixels of the noisy image assume a gray level in the

range [−0.1; 0.3], an appropriate change of scale is realized to reset the

dynamics of the gray levels to [0; 0.2]. The resulting image is then loaded

as the initial condition in the network. For the sake of clarity, the ﬁltered

images are presented at different processing times with the corresponding

system response in Figure 12.

Before the optimal time, we observe the behavior described in

Section III.A.1: the image goes through local minima and maxima of con-

trast until the optimum time t

opt

= 1.64 × 10

−3

, where the best contrast

enhancement is realized (Figure 12a).

Next, for processing times exceeding t

opt

, the noisy part of the image

seems to be ampliﬁed while the coherent part of the image begins to

be increasingly less perceptible (see Figure 12b and 12c obtained for

t = 3.28 × 10

−3

and t = 6.56 × 10

−3

). Finally, for longer processing times,

namely, t = 8.24 × 10

−3

and t = 9.84 × 10

−3

, the noise background has

completely hidden the Coliseum, which constitutes an image encryption.

Note that this behavior can be explained with the response of the sys-

tem, as represented below each ﬁltered image in Figure 12. Indeed, until

the response of the system versus the initial condition does not display

a “periodic-like” behavior, the coherent part of the image remains per-

ceptible (Figure 12a and 12b). By contrast, as soon as a “periodicity”

appears in the system response, the coherent image begins to disappear

(Figure 12c). Indeed, the response in Figure 12c shows that four pixels

of the initial image with four different gray levels take the same ﬁnal

value in the encrypted image (see the arrows). Therefore, the details of

the initial image, which corresponds to the quasi-uniform area of the

coherent image, are merged and thus disappear in the encrypted image.

Despite the previous merging of gray levels, since noise induces sudden

changes in the gray levels of the initial image, the noise conserves its ran-

dom feature in the encrypted image. Moreover, since the system tends to

enlarge the range of amplitude, the weak initial amount of noise is strongly

ampliﬁed whenever the processing time exceeds t

opt

. The periodicity of

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102

Saverio Morfu et al.

5

0

0 0.2

W

i

5

0

0 0.2

W

i

5

0

0 0.2

W

i

5

0

0 0.2

W

i

5

0

0 0.2

W

i

(a) (b) (c)

(d) (e)

W

0

i

W

0

i

W

0

i

W

0

i

W

0

i

FIGURE 12 Encrypted image and the corresponding response of the nonlinear

oscillators network for different times exceeding t

opt

. (a): Enhancement of contrast of

the initial image for t = t

opt

= 1.64 × 10

−3

. (b): t = 3.28 × 10

−3

. (c): t = 6.56 × 10

−3

.

(d): t = 8.24 × 10

−3

.(e):t = 9.84 ×10

−3

. Parameters: m = 2.58, α = 1.02, ω

0

= 1.

the system response can then be increased for longer processing times

until only the noisy part of the image is perceptible (Figure 12d and 12e).

A perfect image encryption is then realized.

To take advantage of this phenomenon for image encryption, the coher-

ent information (the enhanced image in Figure12a), must be restoredusing

the encrypted image of Figure 12e. Fortunately, owing to the absence of

dissipation, the nonlinear systems is conservative and reversible. It is thus

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Nonlinear Systems for Image Processing

103

possible to revert to the optimal time—when the information was the most

perceptible.

However, the knowledge of the encrypted image is not sufﬁcient to com-

pletely restore the coherent information, since at the time of encryption,

the velocity of the oscillators was not null. Consequently, it is neces-

sary to know both the position and the velocity of all particles of the

network at the time of encryption. The information then can be resto-

red solving numerically Eq. (23) with a negative integrating time step

dt =−10

−6

.

Under these conditions, the time of encryption constitutes the encryp-

tion key.

B. Electronic Implementation

The elementary cell of the purely inertial system can be developed accord-

ing to the principle of Figure 13 (Morfu et al., 2007). First, a polynomial

source is realized with analog AD633JNZ multipliers and classical invert-

ing ampliﬁer with gain −K. Taking into account the scale factor 1/10 V

−1

of the multipliers, the response of the nonlinear circuit to an input voltage

V

T

1W

0

W

i

m

AD633JN AD633JN

m 2 α

2K

2K

m 1 ␣

(W

i

2 m)(W

i

2 m 1 ␣)/10

P (W

i

) 5 (W

i

2 m 1 ␣)(W

i

2 m)(W

i

2 m 2 ␣)K

2

/100

1

R

2

C

2

2

ee

1

R

2

C

2

ee

W

i

5 2

P (W

i

)

i

FIGURE 13 Sketch of the elementary cell of the inertial system. m and α are adjusted

with external direct current sources, whereas −K is the inverting amplifier gain

obtained using TL081CN operational amplifier. The 1N4148 diode allows introduction

of the initial condition W

0

i

.

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104

Saverio Morfu et al.

W

i

is given by

P(W

i

) =

K

2

100

W

i

− m

W

i

− m − α

W

i

− m + α

, (24)

where the roots m, m − α, m + α of the polynomial circuit are set with

three different external direct current (DC) sources. As shown in Figure 14,

the experimental characteristic of the nonlinear source is then in perfect

agreement with its theoretical cubic law [Eq. (24)].

Next, a feedback between the input/output of the nonlinear circuits is

ensured by a double integrator with time constant RC such that

W =−

K

2

100R

2

C

2

W

i

− m + α

W

i

− m − α

W

i

− m

dt. (25)

Deriving Eq. (25) twice, the voltage W

i

at the input of the nonlinear circuit

is written as

d

2

W

i

dt

2

=−

K

2

100R

2

C

2

W

i

− m + α

W

i

− m − α

W

i

− m

, (26)

which corresponds exactly to the equation of the purely inertial system

AQ5

(13) with

ω

0

= K/(10RC). (27)

0.4

0.2

0

20.2

20.4

1.5 2 2.5

W

i

(Volt)

P (W

i

) (Volt)

3

3.5 4

FIGURE 14 Theoretical cubic law in Eq. (24) in solid line compared to the experi-

mental characteristic plotted with crosses. Parameters: m =2.58 V, α =1.02 V, K =10.

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Nonlinear Systems for Image Processing

105

Finally, the initial condition W

0

i

is applied to the elementary cell via

a1N4148 diode with threshold voltage V

T

= 0.7 V. We adjust the diode

anode potential to W

0

i

+ V

T

with an external DC source with the diode

cathode potential initially set to W

0

i

. Then, according to Section III, the

circuit begins to oscillate in the range [W

0

i

;2m − W

0

i

], while the potential

of the diode anode remains V

T

+ W

0

i

. Assuming that m > W

0

i

/2, which is

the case in our experiments, the diode is instantaneously blocked once the

initial condition is introduced. Note that using a diode to set the initial

condition presents the main advantage to “balance” the effect of dissi-

pation inherent in electronic devices. Indeed, the intrinsic dissipation of

the experiments tends to reduce the amplitude of the oscillations W

0

i

.As

soon as the potential of the diode cathode is below W

0

i

, the diode con-

ducts instantaneously, introducing periodically the same initial condition

in the elementary cell. Therefore, the switch between the two states of the

diode presents the advantage of refreshing the oscillation amplitude to

their natural value as in absence of dissipation.

In summary, the oscillations are available at the diode cathode and

are represented in Figure 15a for two different initial conditions, namely,

W

0

1

= 0 V (top panel) and W

0

2

= 0.2 V (bottom panel). As previously expla-

ined, the way to introduce the initial condition allows balancing the

dissipative effects since the oscillation remains with the same ampli-

tude, namely in the range [0 V; 5.34V] for the ﬁrst oscillator with ini-

AQ6

tial condition 0, and [0.2 V; 5.1 V] for the second one. Moreover, these

ranges match with fairly good agreement the theoretical predictions pre-

sented in Section II.B.2, that is [0 V; 5.16 V] for the ﬁrst oscillator and

[0.2 V; 4.96 V] for the second one. Figure 15a also reveals that the two

oscillators quickly achieve a phase opposition at the optimal time t

opt

=

1.46 ms instead of 1.64 ms as theoretically established in Section II.B.2. The

oscillations difference between the two oscillators in Figure 15b reaches

local minima and maxima in agreement with the theoretical behavior

observed in Section III. A maximum of 5.1 V is obtained correspond-

ing to the phase opposition W

1

(t

opt

) = 0 V and W

2

(t

opt

) = 5.1 V. There-

fore, the weak difference of initial conditions between the oscillators

is strongly increased at the optimal time t

opt

. Despite a slight discrep-

ancy of 11% for the optimal time, mainly imputable to the component

uncertainties, a purely inertial nonlinear system is then implemented

with the properties of Section III.

To perfectly characterize the experimental device, we now focus on

the response of the nonlinear system to different initial conditions in

the range [0 V; 0.2 V]. The plot of the voltage reached at the optimal

time t

opt

= 1.46 ms versus the initial condition is compared in Figure 16

to the theoretical curve obtained for the optimum time deﬁned in

Section II.B.2, namely, 1.64 ms. The experimental response of the system is

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106

Saverio Morfu et al.

t

opt

(a) (b)

0

Amplitude (Volt)

22

24

26

2

012

Time (ms)

345

4

6

t

opt

FIGURE 15 (a): Temporal evolution of two elementary cells of the chain with

respective initial conditions W

0

1

= 0 V (top panel) and W

0

2

= 0.2 V (bottom panel).

(b): Evolution of the voltage difference between the two oscillators. Parameters:

K = 10, R = 10 K, C = 10 nF, m = 2.58 V, α = 1.02V, t

opt

= 1.46 ms.

0

0 0.04 0.08 0.12 0.16 0.2

1

3

2

4

5

W

i

(t

opt

)

(Volt)

W

i

0

(Volt)

FIGURE 16 Response of the system to a set of initial conditions W

0

i

∈[0; 0.2] at the

optimal time. The solid line is obtained with Eqs. (20), (21), and (27) setting the time to

the theoretical optimal value 1.64 ms, the initial condition varying in [0; 0.2 V]. The

crosses are obtained experimentally for the corresponding optimal time 1.46 ms.

Parameters: R = 10 K, C = 10 nF, m = 2.58 V, α = 1.02 V, K = 10.

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Nonlinear Systems for Image Processing

107

then qualitatively conﬁrmed by the theoretical predictions, which allows

establishing the validityof theexperimental elementarycell forthe contrast

AQ7

enhancement presented in Section III.A.1.

Finally, we also propose to investigate the response of the system after

the optimum time, since it allows the extraction of gray levels. In order to

enhance the measures accuracy, we extend the range of initial conditions

to [0, 0.5 V] instead of [0, 0.2 V]. The corresponding experimental optimal

time becomes t

opt

= 564 μs, whereas the theoretical ones, deduced with

the methodology in Section II.B.2, is 610

μs. The resulting theoretical and

experimental responses are then plotted in Figure 17a, where a better

agreement is effectively observed compared to Figure 16.

0

0 0.1 0.2 0.3 0.4 0.5 0.6

21

20.1

1

3

2

4

5

6

W

i

( t

opt

)

(Volt)

W

i

0

(Volt)

W

i

( t

opt

)

(Volt)

0

0 0.1 0.2 0.3 0.4 0.5 0.6

20.5

20.1

1

0.5

2

1.5

3

2.5

4

3.5

4.5

W

i

0

(Volt)

W

i

( t

opt

)

(Volt)

0 0.1 0.2 0.3 0.4 0.5 0.620.1

0

1

0.5

2

1.5

3

2.5

4

3.5

4.5

5

W

i

0

(Volt)

0

0 0.1 0.2 0.3 0.4 0.5 0.6

21

20.1

1

3

2

4

5

6

W

i

( t

opt

)

(Volt)

W

i

0

(Volt)

(a) (b)

(c) (d)

FIGURE 17 Theoretical response of the purely inertial system (solid line) compared

to the experimental ones (crosses) for 4 different times and for a range of initial

conditions [0; 0.5 V]. Parameters: R = 10 K, C = 10 nF, m = 2.58 V, α = 1.02V,

K = 10. (a) Experimental time t = 564 μs corresponding to the theoretical time

t = 610 μs. (b) Experimental time t = 610 μs and theoretical time 713 μs.

(c) Experimental time t = 675 μs and theoretical time 789 μs. (d) Experimental

time t = 720 μs and theoretical time 841 μs.

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108

Saverio Morfu et al.

We have also reported the experimental device response for three

different times beyond the optimal time t

opt

= 564 μs in Figure 17b, c,

and d—namely, for the experimental times t = 610

μs, t = 675 μs, and

t = 720

μs. Since a time scale factor 610/564 = 1.1684 exists between the

experimental and the theoretical optimal time, we apply this scale factor

to the three previous experimental times. It provides the theoretical times

713

μs, 789 μs, and 841 μs. For each of these three times, we can then com-

pare the experimental response to the theoretical one deduced by letting

the initial condition vary in [0; 0.5 V] in Eqs. (20), (21), and (27). Despite

some slight discrepancies, the behavior of the experimental device is in

good agreement with the theoretical response of the system for the three

processing times exceeding the optimal time. Therefore, the extraction

of gray levels, presented in Section III.A.2, is electronically implemented

with this elementary cell.

IV. REACTION-DIFFUSION SYSTEMS

A. One-Dimensional Lattice

The motion Eq. (4) of the nonlinear mechanical chain can also describe the

evolution of the voltage at the nodes of a nonlinear electrical lattice. This

section is devoted to the presentation of this nonlinear electrical lattice.

The nonlinear lattice is realized by coupling elementary cells with lin-

ear resistors R according to the principle of Figure 18a. Each elementary

cell consists of a linear capacitor C in parallel with a nonlinear resistor

whose current-voltage characteristic obey the cubic law

I

NL

(u) = βu(u − V

a

)(u − V

b

)/(R

0

V

a

V

b

), (28)

where 0 < V

a

< V

b

are two voltages, β is a constant, and R

0

is the analog

to a weighting resistor.

RR RR

U

n21

R

NL

R

NL

D

0

D

1

R

1

R

2

R

4

R

4

R

3

D

2

R

NL

U

n

U

n11

C

CC

i

U

1Vcc

2Vcc

(a) (b)

FIGURE 18 (a) Nonlinear electrical lattice. (b) The nonlinear resistor R

NL

.

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Nonlinear Systems for Image Processing

109

0

260

250

240

230

220

210

10

20

30

40

0 0.2 0.4 0.6 0.8 1 1.2

Nonlinear Current (mA)

Voltage (V )

FIGURE 19 Current-voltage characteristics of the nonlinear resistor. The theoretical

law [Eq. (28)] in the solid line is compared to the experimental data plotted with

crosses. The dotted lines represent the asymptotic behavior of the nonlinear resistor.

Parameters: R

0

= 3.078 K, V

b

= 1.12 V, V

a

= 0.545 V, β = 1.

The nonlinear resistor can be developed using two different methods.

The ﬁrst method to obtain a cubic current is to consider the circuit of

Figure 18b with three branches (Binczak et al., 1998; Comte, 1996). A linear

resistor R

3

, a negative resistor, and another linear resistor R

1

are succes-

sively added in parallel thanks to 1N4148 diodes. Due to the switch of

the diodes, the experimental current-voltage characteristic of Figure 19

asymptotically displays a piecewise linear behavior with successively a

positive slope, a negative one, and ﬁnally a positive one.

This piecewise linear characteristics is compared to the cubic law

[Eq. (28)], which presents the same roots V

a

, V

b

, and 0 but also the same

area below the characteristic between 0 and V

a

. This last conditions leads

to β = 1 and R

0

= 3.078 K (Morfu, 2002c).

An alternative way to realize a perfect cubic nonlinear current is to

use a nonlinear voltage source that provides a nonlinear voltage P(u) =

βu(u − V

a

)(u − V

b

)/(V

a

V

b

) + u asshownin Figure20 (Comte andMarquié,

2003).

Thispolynomial voltageis realized with AD633JNZmultipliers andclas-

sical TL081CN operational ampliﬁers. A resistor R

0

ensures a feedback

between the input/output of the nonlinear source such that Ohm’s law

applied to R

0

corresponds to the cubic current in Eq. (28):

P(u) − u

R

0

= I

NL

(u). (29)

As shown in Figure 21, this second method gives a better agreement with

the theoretical cubic law [Eq. (28)].

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110

Saverio Morfu et al.

U

U

R

NL

I

NL

(U )

P(U)

I

NL

(U )

R

0

Polynomial

generation

circuits

FIGURE 20 Realization of a nonlinear resistor with a polynomial generation circuit.

β = 10V

a

V

b

.

22

22

21.5

21.5

21

21

20.5

20.5

0

0

0.5

0.5

1

1

1.5

1.5

2

2

I

NL

(mA)

U (Volts)

FIGURE 21 Current-voltage characteristics of the nonlinear resistor of Figure 20.

Parameters: β =−10 V

a

V

b

, V

a

=−2 V, V

b

= 2 V.

Applying Kirchhoff’s laws, the voltage U

n

at the n

th

node of the lattice

can be written as

C

dU

n

dτ

=

1

R

U

n+1

+ U

n−1

− 2U

n

− I

NL

(U

n

), (30)

where τ denotes the experimental time and n = 1 ...N represents the node

number of the lattice.

Moreover, we assume zero-ﬂux or Neumann boundary conditions,

which involves for n = 1 and n = N, respectively,

C

dU

1

dτ

=

1

R

U

2

− U

1

− I

NL

(U

1

), (31)

C

dU

N

dτ

=

1

R

U

N−1

− U

N

− I

NL

(U

N

). (32)

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111

Next, introducing the transformations

W

n

=

U

n

V

b

, D =

R

0

R

αβ, t =

τ

R

0

αCβ

, (33)

yields the discrete Nagumo equation in its normalized form,

dW

n

dt

= D

W

n+1

+ W

n−1

− 2W

n

+ f (W

n

). (34)

Therefore, an electronic implementation of the overdamped network

presented in Section II.A is realized.

B. Noise Filtering of a One-Dimensional Signal

One of the most important problems in signal or image processing is

removal of noise from coherent information. In this section, we develop

the principle of nonlinear noise ﬁltering inspired by the overdamped

systems (Marquié et al., 1998). In addition, using the electrical nonlinear

network introduced in Section IV.A, we also present an electronic imple-

mentation of the ﬁltering tasks.

1. Theoretical Analysis

To investigate the response of the overdamped network to a noisy sig-

nal loaded as an initial condition, we ﬁrst consider the simple case of a

constant signal with a sudden change of amplitude. Therefore, we study

the discrete normalized Nagumo equation

dW

n

dt

= D

W

n+1

+ W

n−1

− 2W

n

+ f (W

n

), (35)

with f (W

n

) =−W

n

(W

n

− α)(W

n

− 1) in the speciﬁc case α = 1/2. Further-

more, the initial condition applied to the cell n is assumed to be uniform

for all cells, except for the cell N/2, where a constant perturbation b

0

is added; namely:

W

n

(t = 0) = V

0

∀n =

N

2

W

N/2

(t = 0) = V

0

+ b

0

. (36)

The solution of Eq. (35) to the initial condition in Eq. (36) can be expressed

with the following form

W

n

(t) = V

n

(t) + b

n

(t). (37)

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112

Saverio Morfu et al.

Inserting Eq. (37) in Eq. (35), we collect the terms of order 0 and 1 in

with the reductive perturbation methods to obtain the set of differential

equations (Taniuti and Wei, 1968; Taniuti and Yajima, 1969):

dV

n

dt

= D(V

n+1

+ V

n−1

− 2V

n

) + f (V

n

) (38)

db

n

dt

= D(b

n+1

+ b

n−1

− 2b

n

) − (3V

2

n

− 2V

n

(1 + α) + α)b

n

(39)

Assuming that V

n

is a slow variable, Eq. (38) reduces to

dV

n

dt

= f (V

n

), (40)

which provides the response of the system to a uniform initial condition

V

0

(see details in Appendix A):

V(t) =

1

2

⎛

⎜

⎝

1 +

V

0

−

1

2

(V

0

−

1

2

)

2

− V

0

(V

0

− 1)e

−

t

2

⎞

⎟

⎠

. (41)

Next, to determine the evolution of the additive perturbation, it is

convenient to consider a perturbation under the following form:

b

n

(t) = I

n

(2Dt)g(t), (42)

where I

n

is the modiﬁed Bessel function of order n (Abramowitz and

Stegun, 1970). Substituting Eq. (42) in Eq. (39), and using the property

of the modiﬁed Bessel function,

dI

n

(2Dt)

dt

= D(I

n+1

+ I

n−1

), (43)

we obtain straightforwardly

dg

dt

=−2Dg −

3V

2

n

− 2V

n

(1 − α) + α

g, (44)

that is,

dg

g

=−2Ddt −

3V

2

n

− 2V

n

(1 − α) + α

dt. (45)

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113

Noting that

df (V

n

)

dt

=−

3V

2

n

− 2V

n

(1 − α) + α

dV

n

dt

, (46)

and deriving Eq. (40) versus time, we derive

V

n

V

n

=−

3V

2

n

− 2V

n

(1 − α) + α

, (47)

where V

n

and V

n

denote the ﬁrst and second derivative versus time.

Combining Eq. (47) and Eq. (45) allows expression of g(t) as:

g(t) = Ke

−2Dt

dV

n

dt

, (48)

where K is an integrating constant.

Deriv