Patrick Marquié

Nonlinear Systems for Image Processing

I Introduction 79II Mechanical Analogy 83A Overdamped Case 84B Inertial Systems 90III Inertial Systems 95A Image Processing 95B Electronic Implementation 103IV Reaction-Diffusion Systems 108A One-Dimensional Lattice 108B Noise Filtering of a One-Dimensional Signal 111C Two-Dimensional Filtering: Image Processing 119V Conclusion 133VI Outlooks 134A Outlooks on Microelectronic Implementation 134B Future Processing Applications 135Acknowledgments 141Appendix A 142Appendix B 143Appendix C 144Appendix D 145References 146

Analog simulation of neural information propagation using an electrical FitzHugh–Nagumo lattice

A nonlinear electrical lattice modelling neural information propagation is presented. It is shown that our system is an analog simulator of the FitzHugh–Nagumo equations, and hence supports pulse propagation with the appropriate properties.

Nonlinear Schro¨dinger models and modulational instability in real electrical lattices

Abstract In nonlinear dispersive media, the propagation of modulated waves, such as envelope (bright) solitions or hole (dark) solitons, has been the subject of considerable interest for many years, as for example in nonlinear optics [A.C. Newell and J.V. Moloney, Nonlinear Optics (Addison-Wesley, 1991)]. On the other hand, discrete electrical transmission lines are very convenient tools to study the wave propagation in 1 D nonlinear dispersive media [A.C. Scott (Wiley-Interscience, 1970)]. In the present paper, we study the generation of nonlinear modulated waves in real electrical lattices. In the continuum limit, our theoretical analysis based on the Nonlinear Schrodinger equation (NLS) predicts three frequency regions with different behavior concerning the Modulational Instability of a plane wave. These predictions are confirmed by our experiments which show that between two modulationally stable frequency bands where hole solitons can be generated, there is a third band where spontaneous or induced modulational instability occurs and where envelope solitons exist. When lattice effects are considered the dynamics of modulated waves can be modeled by a discrete nonlinear Schrodinger equation which interpolates between the Ablowitz-Ladik and Discrete-self-trapping equations.

GENERATION OF NONLINEAR CURRENT–VOLTAGE CHARACTERISTICS: A GENERAL METHOD

A general method allowing to construct nonlinear resistors with arbitrary current-voltage (I-V) characteristics is proposed. The example of a cubic I-V characteristic is presented showing a perfect agreement between the theoretical desired resistor and its electronic realization based on analog multipliers.

NOISE-ENHANCED PROPAGATION IN A DISSIPATIVE CHAIN OF TRIGGERS

We study the influence of spatiotemporal noise on the propagation of square waves in an electrical dissipative chain of triggers. By numerical simulation, we show that noise plays an active role in improving signal transmission. Using the Signal to Noise Ratio at each cell, we estimate the propagation length. It appears that there is an optimum amount of noise that maximizes this length. This specific case of stochastic resonance shows that noise enhances propagation.

PROPAGATION FAILURE INDUCED BY COUPLING INHOMOGENEITIES IN A NONLINEAR DIFFUSIVE MEDIUM

Kink propagation failure induced by coupling inhomogeneities in a Nagumo lattice is investigated. Considering the case of weak couplings, we define analytically and numerically the coupling conditions leading to the pinning of the kink.

CONTOUR DETECTION BASED ON NONLINEAR DISCRETE DIFFUSION IN A CELLULAR NONLINEAR NETWORK

A contour detection based on a diffusive cellular nonlinear network is proposed. It is shown that there exists a particular nonlinear function for which, numerically, the obtained contour is satisfactory. Furthermore, this nonlinear function can be achieved using analog components.

Noise induced breather generation in a sine–Gordon chain

We consider a sine–Gordon chain driven sinusoidally at one end. In the absence of noise, there exists a well known critical value of the amplitude beyond which breather modes can be generated via the phenomenon of supratransmission. We consider values of the driving amplitude below the critical amplitude such that no breather propagates in the medium. We show that noise induces breather generation with a given probability depending on the noise intensity. We also propose a bifurcation diagram which extends the supratransmission effect to a more realistic signal, namely a noisy sinusoidal excitation. We finally discuss some promising signal processing applications that can be developed by taking into account the contribution of noise in the media sharing this supratransmission phenomenon.

EXPERIMENTS ON AN ELECTRICAL NONLINEAR OSCILLATORS NETWORK

We have recently proposed a Cellular Nonlinear Network (CNN) based on nonlinear oscillator properties to perform image processing tasks. We present here the electronic implementation of the elementary cell of this CNN. We experimentally verify the main property of the CNN, that is the possibility to enhance a weak difference of initial condition between two specific cells of the CNN at a given time. For this optimal time, a contrast enhancement of a weak contrasted gray scale is possible.

Bistability and Nonlinear Standing Waves in an Experimental Transmission-line

The bistability of the transfer function and the spatial dependence of the voltage envelope are investigated near the lower gap on a finite electrical transmission line. The nonlinear standing wave behaviour and the bistability in our experiments agree with our theoretical predictions, which take into account the damping of the line.