Nonlinear Systems for Image Processing

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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
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. 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
jNr
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
n1
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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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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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
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
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
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
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
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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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
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
i
t, k
i
)dn
i
t, k
i
),
d
2
x
i
dt
2
=−x
0
i
ω
2
i
cn
i
t, k
i
)
dn
2
i
t, k
i
) k
i
sn
2
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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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
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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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
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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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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(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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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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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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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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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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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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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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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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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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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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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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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
n1
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
N1
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
n1
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
n1
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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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
n1
2V
n
) + f (V
n
) (38)
db
n
dt
= D(b
n+1
+ b
n1
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
n1
), (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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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