a spectral approach for sifting process in empirical mode decomposition in this paper we propose an alternative to the algorithmic definition of the sifting process used in the original huang s empirical mode decomposition emd method although it has been proven to be particularly effective in many applications emd method has several drawbacks the major problem with emd is the lack of theoretical framework which leads to difficulties for the characterization and evaluation this approach on top of the mathematical model there are other concerns with mode mixing and transient phenomena such as intermittency or pure tones separation this paper follows a previous published nonlinear diffusion based filtering to solve the mean envelope estimation in sifting process the major improvements made in this present work are a non iterative resolution scheme for the previously proposed partial differential equation pde a new definition of the stopping function used in the pde framework and finally an automatic regularization process based on inverse problem theory to deal with mode mixing or transient detection problem obtained results confirm good properties of the new version of the pde based sifting process and its usefulness for decomposition of various kinds of data the efficiency of the method is illustrated on some examples using informative and pathological signals for which standard emd algorithm fails

multivariate empirical mode decomposition despite empirical mode decomposition emd becoming a de facto standard for time frequency analysis of nonlinear and non stationary signals its multivariate extensions are only emerging yet they are a prerequisite for direct multichannel data analysis an important step in this direction is the computation of the local mean as the concept of local extrema is not well defined for multivariate signals to this end we propose to use real valued projections along multiple directions on hyperspheres n spheres in order to calculate the envelopes and the local mean of multivariate signals leading to multivariate extension of emd to generate a suitable set of direction vectors unit hyperspheres n spheres are sampled based on both uniform angular sampling methods and quasi monte carlo based low discrepancy sequences the potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real world human motion signals

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

a priori guaranteed evolution within the neural network approximation set and robustness expansion via prescribed performance control a neuroadaptive control scheme for strict feedback systems is designed which is capable of achieving prescribed performance guarantees for the output error while keeping all closed loop signals bounded despite the presence of unknown system nonlinearities and external disturbances the aforementioned properties are induced without resorting to a special initialization procedure or a tricky control gains selection but addressing through a constructive methodology the longstanding problem in neural network control of a priori guaranteeing that the system states evolve strictly within the compact region in which the approximation capabilities of neural networks hold moreover it is proven that robustness against external disturbances is significantly expanded with the only practical constraint being the magnitude of the required control effort a comparative simulation study clarifies and verifies the approach

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

partial differential equation based approach for empirical mode decomposition application on image analysis the major problem with the empirical mode decomposition emd algorithm is its lack of a theoretical framework so it is difficult to characterize and evaluate this approach in this paper we propose in the 2 d case the use of an alternative implementation to the algorithmic definition of the so called sifting process used in the original huang s emd method this approach especially based on partial differential equations pdes was presented by niang in previous works in 2005 and 2007 and relies on a nonlinear diffusion based filtering process to solve the mean envelope estimation problem in the 1 d case the efficiency of the pde based method compared to the original emd algorithmic version was also illustrated in a recent paper recently several 2 d extensions of the emd method have been proposed despite some effort 2 d versions for emd appear poorly performing and are very time consuming so in this paper an extension to the 2 d space of the pde based approach is extensively described this approach has been applied in cases of both signal and image decomposition the obtained results confirm the usefulness of the new pde based sifting process for the decomposition of various kinds of data some results have been provided in the case of image decomposition the effectiveness of the approach encourages its use in a number of signal and image applications such as denoising detrending or texture analysis

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

optimal gabor filters for texture segmentation texture segmentation involves subdividing an image into differently textured regions many texture segmentation schemes are based on a filter bank model where the filters called gabor filters are derived from gabor elementary functions the goal is to transform texture differences into detectable filter output discontinuities at texture boundaries by locating these discontinuities one can segment the image into differently textured regions distinct discontinuities occur however only if the gabor filter parameters are suitably chosen some previous analysis has shown how to design filters for discriminating simple textures designing filters for more general natural textures though has largely been done ad hoc we have devised a more rigorously based method for designing gabor filters it assumes that an image contains two different textures and that prototype samples of the textures are given a priori we argue that gabor filter outputs can be modeled as rician random variables often approximated well as gaussian rv s and develop a decision theoretic algorithm for selecting optimal filter parameters to improve segmentations for difficult texture pairs we also propose a multiple filter segmentation scheme motivated by the rician model experimental results indicate that our method is superior to previous methods in providing useful gabor filters for a wide range of texture pairs

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

two dimensional empirical mode decomposition by finite elements empirical mode decomposition emd is a powerful tool for analysis of non stationary and nonlinear signals and has drawn significant attention in various engineering application areas this paper presents a finite element based emd method for two dimensional data analysis specifically we represent the local mean surface of the data a key step in emd as a linear combination of a set of two dimensional linear basis functions smoothed with bi cubic spline interpolation the coefficients of the basis functions in the linear combination are obtained from the local extrema of the data using a generalized low pass filter by taking advantage of the principle of finite element analysis we develop a fast algorithm for implementation of the emd the proposed method provides an effective approach to overcome several challenging difficulties in extending the original one dimensional emd to the two dimensional emd numerical experiments using both simulated and practical texture images show that the proposed method works well

prescribed performance adaptive control for multi input multi output affine in the control nonlinear systems we consider the tracking problem of unknown robustly stabilizable multi input multi output mimo affine in the control nonlinear systems with guaranteed prescribed performance by prescribed performance we mean that the tracking error converges to a predefined arbitrarily small residual set with convergence rate no less than a prespecified value exhibiting maximum overshoot as well as undershoot less than some sufficiently small preassigned constants utilizing an output error transformation we obtain a transformed system whose robust stabilization is proven necessary and sufficient to achieve prescribed performance guarantees for the output tracking error of the original system provided that initially the transformed system is well defined consequently a switching robust control lyapunov function rclf based adaptive state feedback controller is designed to solve the stated problem the proposed controller is continuous and successfully overcomes the problem of computing the control law when the approximation model becomes uncontrollable simulations illustrate the approach

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

multiscale image fusion using complex extensions of emd empirical mode decomposition emd is a fully data driven technique for decomposing signals into their natural scale components however the problem of uniqueness caused by the empirical nature of the algorithm and its sensitivity to changes in parameters makes it difficult to perform fusion of data from multiple and heterogeneous sources a solution to this problem is proposed using recent complex extensions of emd which guarantees the same number of decomposition levels that is the uniqueness of the scales the methodology is used to address multifocus image fusion whereby two or more partially defocused images are combined in automatic fashion so as to create an all in focus image

adaptive neural control for output feedback nonlinear systems using a barrier lyapunov function in this brief adaptive neural control is presented for a class of output feedback nonlinear systems in the presence of unknown functions the unknown functions are handled via on line neural network nn control using only output measurements a barrier lyapunov function blf is introduced to address two open and challenging problems in the neuro control area 1 for any initial compact set how to determine a priori the compact superset on which nn approximation is valid and 2 how to ensure that the arguments of the unknown functions remain within the specified compact superset by ensuring boundedness of the blf we actively constrain the argument of the unknown functions to remain within a compact superset such that the nn approximation conditions hold the semiglobal boundedness of all closed loop signals is ensured and the tracking error converges to a neighborhood of zero simulation results demonstrate the effectiveness of the proposed approach

mixed mode oscillations and chaotic solutions of jerk newtonian equations we analyze jerk equations third order odes resulting from an underlying prototypical model of mixed mode oscillations and propose their circuit realizations in this paper the scalar odes and their corresponding circuit realizations are obtained from a system of first order odes with one nonlinearity third degree polynomial one of the jerk equations is newtonian as it is obtained by computing the time derivative of the second newton s law x f m 0 for a constant mass m and specially designed nonlinear force function f x x t the second jerk equation is non newtonian the two circuits are op amp rc circuits with interesting dynamical properties including the mixed mode and chaotic oscillations the mixed mode oscillations follow the rules of farey arithmetic and the circuits dynamics is of a fractal nature

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

turning tangent empirical mode decomposition a framework for mono and multivariate signals a novel empirical mode decomposition emd algorithm called 2t emd for both mono and multivariate signals is proposed in this correspondence it differs from the other approaches by its computational lightness and its algorithmic simplicity the method is essentially based on a redefinition of the signal mean envelope computed thanks to new characteristic points which offers the possibility to decompose multivariate signals without any projection the scope of application of the novel algorithm is specified and a comparison of the 2t emd technique with classical methods is performed on various simulated mono and multivariate signals the monovariate behaviour of the proposed method on noisy signals is then validated by decomposing a fractional gaussian noise and an application to real life eeg data is finally presented

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

adaptive neural control of nonlinear mimo systems with time varying output constraints in this paper adaptive neural control is investigated for a class of unknown multiple input multiple output nonlinear systems with time varying asymmetric output constraints to ensure constraint satisfaction we employ a system transformation technique to transform the original constrained in the sense of the output restrictions system into an equivalent unconstrained one whose stability is sufficient to solve the output constraint problem it is shown that output tracking is achieved without violation of the output constraint more specifically we can shape the system performance arbitrarily on transient and steady state stages with the output evolving in predefined time varying boundaries all the time a single neural network whose weights are tuned online is used in our design to approximate the unknown functions in the system dynamics while the singularity problem of the control coefficient matrix is avoided without assumption on the prior knowledge of control input s bound all the signals in the closed loop system are proved to be semiglobally uniformly ultimately bounded via lyapunov synthesis finally the merits of the proposed controller are verified in the simulation environment

a new iterated two band diffusion equation theory and its application we propose an iterated two band filtering method to solve the selective image smoothing problem we prove that a discrete computation step in an iterated nonlinear diffusion based filtering algorithm is equivalent to a sequence of operations including decomposition regularization and then reconstruction in the proposed two band filtering scheme to correctly separate the high frequency components from the low frequency ones in the decomposition process we adopt a dyadic wavelet based approximation scheme in the regularization process we use a diffusivity function as a guide to retain useful data and suppress noises finally the signal of the next stage which is a smoother version of the signal at the previous stage can be computed by reconstructing the decomposed low frequency component and the regularized high frequency component based on the proposed scheme the smoothing operation can be applied to the correct targets experimental results show that our new approach is really efficient in noise removing

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

crenelated fast oscillatory outputs of a two delay electro optic oscillator an electro optic oscillator subject to two distinct delayed feedbacks has been designed to develop pronounced broadband chaotic output its route to chaos starts with regular pulsating gigahertz oscillations that we investigate both experimentally and theoretically of particular physical interest are the transitions to various crenelated fast time periodic oscillations prior to the onset of chaotic regimes the two delay problem is described mathematically by two coupled delay differential equations which we analyze by using multiple time scale methods we show that the interplay of a large delay and a relatively small delay is responsible for the onset of fast oscillations modulated by a slowly varying square wave envelope as the bifurcation parameter progressively increases this envelope undergoes a sequence of bifurcations that corresponds to successive fixed points of a sine map

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

a hierarchy of transitions to mixed mode oscillation in an electrochemical system abstract the dynamic behavior of the electrocatalytic oxidation of hydrogen on a platinum anode in the presence of cu 2 and cl ions under galvanostatic conditions has been studied experimentally in two control parameters current density and copper ion concentration this parameter plane can roughly be divided into three broad regions a steady state is found at low values of the current density at medium values the system shows small amplitude oscillations whereas at high current densities typical mixed mode oscillations mmos are found depending on the cu 2 concentration the transition from the small amplitude to the mixed mode regime takes place from a simple periodic period doubled or chaotic attractor arising from a feigenbaum route in this paper we focus on the characteristics of these different transitions to mmos and try to construct a bifurcation diagram the bifurcation from the small chaotic attractor to mmos very likely constitutes an interior crisis a bifurcation at which the small chaotic attractor is destroyed by colliding with a stable manifold of a saddle type limit set at lower copper ion concentrations the transition to mmos moved through the period double cascade we compile different properties of the bifurcation diagram and discuss a scenario which fits the observations in a consistent way

noise removal using smoothed normals and surface fitting in this work we use partial differential equation techniques to remove noise from digital images the removal is done in two steps we first use a total variation filter to smooth the normal vectors of the level curves of a noise image after this we try to find a surface to fit the smoothed normal vectors for each of these two stages the problem is reduced to a nonlinear partial differential equation finite difference schemes are used to solve these equations a broad range of numerical examples are given in the paper

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding