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CHAPTER3

Nonlinear Systems for Image

Processing

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

and Dominique Ginhac*

Contents

I Introduction

II Mechanical Analogy

A Overdamped Case

B Inertial Systems

III Inertial Systems

A Image Processing

B Electronic Implementation

IV Reaction-Diffusion Systems

A One-Dimensional Lattice

B Noise Filtering of a One-Dimensional Signal

C Two-Dimensional Filtering: Image Processing

V Conclusion

VI Outlooks

A Outlooks on Microelectronic Implementation

B Future Processing Applications

Acknowledgments

Appendix A

Appendix B

Appendix C

Appendix D

References

79

83

84

90

95

95

103

108

108

111

119

133

134

134

135

141

142

143

144

145

146

I. INTRODUCTION

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

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

AQ1

*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).

Intherecentfieldofengineeringscience(Agrawal1,2002;Zakharovand

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)wasintroducedbyChuaandYangasanovelclassof

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,informationscience,nonconventionalmethodsofcom-

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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81

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.Noiseremovalwithanonlineardissipativelattice(Comteetal.,1998;

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

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

noise(Hongleretal.,2003),optimizationbynoiseofnonoptimumproblems

orsignaldetectionaidedbynoiseviathestochasticresonancephenomenon

(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,undercertainconditions(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)latticeallowssignalfiltering,whileextendingtheconcept

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).Inparticular,sincethepioneeringworksbyHirotaandSuzuki

(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.

Especiallyconsideringthefeaturesofpurelyinertialsystems,weshowhow

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.

Thethirdpartofthischapterisdevotedexclusivelytothefilteringappli-

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

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.

AQ:2

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. If Wnrepresents the

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

is written as

Md2Wn

dt2

+ λdWn

dt

= −d?

dWn

+ Rn,(1)

whereMd2W

force. Furthermore, the resulting elastic force Rnapplied to the nthparticle

by its neighbors can be defined by:

?

dt2representstheinertiatermandλdW

dt

correspondstoafriction

Rn= D

?

j∈Nr

Wj− Wn

?

, (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

Md2W

dt2

λdW

dt.

>> λdW

dt, and the overdamped one deduced when Md2W

dt2

<<

A. Overdamped Case

Inthissection,anoverdampedsystemispresentedbyneglectingtheinertia

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),

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

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

tial,namely,thewellbottoms,whereastherootαrepresentsthepositionof

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,

(3)

0f(u)du as repre-

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

0.04

20.2

0.02

0

00.20.40.60.811.2

0.06

?50.8

?50.4

?50.2

?50.3

?50.7

?50.6

W

(b)

0

20.2

20.02

20.04

00.2 0.40.60.811.2

0.02

W

(a)

F(W)

F(W)

FIGURE 1

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

barrier is given by ? =?α

energy, and the potential barrier is ? =?α

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

0f(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

1f(u)du = φ(α) − φ(1).

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Nonlinear Systems for Image Processing

85

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

thefirstderivativeandwhenλ = 1,Eq.(1)reducestothediscreteversionof

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

(Fisher, 1937):

dWn

dt

= D(Wn+1+ Wn−1− 2Wn) + f(Wn).(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 W0obeys

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

steady states W∗∈ {0,1,α}. If fW(W∗) denotes the derivative versus W of

the nonlinear force for W = W∗, we are led to solve:

dW

dt

AQ:3

= fW(W∗)(W − W∗) + f(W∗).(6)

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

W(t) = W∗+ CefW(W∗)t−

f(W∗)

fW(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 fW(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, fW(W∗) is

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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 W0can be

expressed by

W(t) =1

2

⎛

⎜

⎝1 +

W0−1

2)2− W0(W0− 1)e−1

2

?

(W0−1

2t

⎞

⎟

⎠. (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 W0is below the unstable state α = 1/2, the particle

evolves toward the steady state 0. Otherwise, if the initial condition W0

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)

displacement W

F(W )(1023)

0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

012345678910 1680

Stable

state

Stable

state

Unstable

state

FIGURE 2

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 W0∈ [0; 1]. The

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

as a reference.

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

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Nonlinear Systems for Image Processing

87

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) = −α4

(see Figure 3b).

12+α3

6

(Morfu, 2003)

D

D

D

W2

D

D

D

W4

W3

W2

W1

?(Wn)

Δ(?)

Wn

(t50)

(a)

0

?

1

(b)

W4

W3

D

?

W1

?(Wn)

Δ(?)

W2

(t.0)

Wn

0

?

1

FIGURE 3

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.

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

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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

particleofthechainsuccessivelyundergoesatransitionfromtheexcited

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

thusstayspinnedatapositionw in[0; α[:itisthewell-knownpropaga-

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. Thevelocityofthekinkanditsprofilecanbe

theoreticallyobtainedinthelimitofcontinuousmedia—whenthecoupling

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∂2W

∂z2+ 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-

tantmeaninginunderstandingtransportmechanisminbiologicalsystems

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

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

propagative kink solution only if?1

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

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

0f(u)du ?= 0, which reduces to α ?= 1/2

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

W(ξ) =1

2

?

1 ± tanh

?

1

2√2D(ξ − ξ0)

??

,(10)

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Nonlinear Systems for Image Processing

89

where ξ0is the initial position of the kink for t = 0 and where the kink

velocity is defined by c = ±?D/2(1 − 2α).

reststate1spreadsinthechain,whichsetthesignofthevelocityaccording

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

Whenα < 1/2,theexcitedstateis0,andthereststateis1.Therefore,the

1. If the profile is given by W(ξ) =1

propagates from left to right with a positive velocity c =?D/2(1 − 2α)

2. Otherwise, if the profile is set by W(ξ) =1

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:

?

left with a negative velocity c =?D/2(1 − 2α) (Figure 4b, left).

2

?

1 − tanh

?

1

2√2D(ξ − ξ0)

??

, a kink

(Figure 4a, left).

2

?

1 + tanh

?

1

2√2D(ξ − ξ0)

??

,

1. If W(ξ) =1

2

1 − tanh

?

1

2√2D(ξ − ξ0)

??

, a kink propagates from right to

215

250

Z

515

0

0.2

1

0.6

215

250

Z

515

W

W(z)

0

0.2

1

0.6

W(z)

1

0.5

0

0.5 21.5 23.5

F(W) 102

(a)

215

250

Z

5 15

0

0.2

1

0.6

215

250

Z

5 15

W

W(z)

0

0.2

1

0.6

W(z)

1

0.5

0

420

F(W) 102

(b)

FIGURE 4

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.

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

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Saverio Morfu et al.

2. Else if W(ξ) =1

to right with a positive velocity c = −?D/2(1 − 2α) (Figure 4b, right).

B. Inertial Systems

2

?

1 + tanh

?

1

2√2D(ξ − ξ0)

??

, a kink propagates from left

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) = −ω2

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 ? = ω2

0α4/4 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: W1

0

second particle: W2

0

0

21

22

23

24

25

26

27

280

W2

0.511.522.533.545 4.5

0

m2?

m 1?

W (Arb.Unit)

2m2W2

0

Potential energy

!W

2

!W

2

FIGURE 5

for m=2.58, α=1.02, and ω0=1. A particle with an initial condition W0

evolves with an initial potential energy above the barrier ?.

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

i< m − α√2

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Neglectingthedissipativeterm,theparticlesofunitarymassarethenruled

by the following nonlinear oscillator equations:

d2Wi

dt2

= f(Wi).(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 xi= Wi− m, Eq. (13) can be rewritten as

d2xi

dt2= −ω2

0xi(xi− α)(xi+ α). (14)

Noting x0

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

expressed with the Jacobian elliptic functions as

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

xi(t) = x0

icn(ωit,ki), (15)

where ωi and 0 ≤ ki≤ 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

dxi

dt

d2xi

dt2= −x0

= −x0

iωisn(ωit,ki)dn(ωit,ki),

iω2

icn(ωit,ki)

?

dn2(ωit,ki) − kisn2(ωit,ki)

?

. (16)

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

d2xi

dt2= −2kiω2

i

x02

i

x

?

x2−2ki− 1

2ki

x02

i

?

. (17)

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

the Jacobian elliptic function

ωi= ω0

?

x02

i− α2,(18)

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92

Saverio Morfu et al.

and its modulus

ki=1

2

x02

i

x02

i− α2.(19)

Finally, introducing the initial condition W0

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

Wi(t) = m +?W0

with

??W0

i= x0

i+ m, the solution of

i− m?cn(ωit,ki), (20)

ωi

?W0

i

?= ω0

i− m?2− α2

and

ki

?W0

i

?=1

2

?W0

i− m?2

?W0

i− m?2− α2.

(21)

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

W0

ωiand of the modulus, respectively, are written as?W0

tial conditions W0

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

initial condition W0

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

barrier ? between the potential extrema (see Figure 5).

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

i− m?2− α2≥ 0

?

and 0 ≤ ki≤ 1. These two conditions restrict the range of the allowed ini-

ito

?

−∞; m − α√2

???

m + α√2; +∞

, as shown in

i. Note that this allowed range of initial conditions cor-

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),wherethepulsationandmodulusofbothoscillatorsaredrivenby

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

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

ω0= 104.WehaveappliedtheinitialconditionW0

while the initial condition of the second oscillator is set to W0

corresponds to the situation of Figure 5.

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

range [W0

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 = topt= 1.64 × 10−3. This phase opposition corresponds to the

1= 0tothefirstoscillator,

2= 0.2, which

i; 2m − W0

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

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Nonlinear Systems for Image Processing

93

0

0

0.5

1

1.5

2

2.5

0.51

???0

????0

????0

?Wi

0

?Wi

0

Wi0

(a)

1.52

(b)

2.533.54 4.55

Forbidden range

of parameters

]m2?ŒW; m1?ŒW[

22

0

0

0.5

1

1.5

0.51

k

Wi0

1.522.533.54 4.55

Forbidden range

of parameters

]m2?ŒW; m1?ŒW[

22

FIGURE 6

(b) Modulus parameter k versus W0

α = 1.02 impose the allowed amplitude range ] − ∞; 1.137]?[4.023; +∞[.

(a): Normalized pulsation ω/ω0versus the initial condition W0

i. The parameters of the nonlinearity m = 2.58,

i.

topt

topt

w1(t)

w2(t)

time

time

(b)

time

0

6

5

4

3

2

1

0

11.52

x(1023)

x(1023)

x(1023)

3

2.5

0.5

0

0

1

2

3

4

5

11.5232.50.5

5

4

3

2

1

0

21

22

23

24

25

0.51.5132.520

(a)

?(t)5W2(t)2W1(t)

FIGURE 7

the first oscillator with initial condition W0

second oscillator with initial condition W0

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

and ω0= 1.

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

1= 0. Bottom panel: evolution of the

2= 0.2. (b) Temporal evolution of the