Smoothness of Nonlinear and Non-Separable Subdivision Schemes

We study in this paper nonlinear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in $L^p$ and Sobolev spaces

Nonlinear and Nonseparable Bidimensional Multiscale Representation Based on Cell-Average Representation

International audienceThe aim of this paper is to build a new nonlinear and nonseparable multiscale representations of piecewise continuous bidimensional functions. This representation is based on the definition of a linear projection and a nonlinear prediction operator which locally adapts to the function to be represented. This adaptivity of the prediction operator proves to be very interesting for image encoding in that it enables to considerably reduce the number of significant coefficients compared with other representations. Applications of these new nonlinear multiscale representation to image compression and super-resolution conclude the paper

A new optimization based approach to the empirical mode decomposition

International audienceIn this paper, an alternative optimization based approach to the empirical mode decomposition (EMD) is proposed. The principle is to build first an approximation of the signal mean envelope, which serves as initial guess for the optimization procedure. We develop several optimization strategies to approximate the mean envelope which compare favorably with the original EMD on AM/FM signals

A Novel Approach for Ridge Detection and Mode Retrieval of Multicomponent Signals Based on STFT

Time-frequency analysis is often used to study non stationary multicomponent signals, which can be viewed as the surperimposition of modes, associated with ridges in the TF plane. To understand such signals, it is essential to identify their constituent modes. This is often done by performing ridge detection in the time-frequency plane which is then followed by mode retrieval. Unfortunately, existing ridge detectors are often not enough robust to noise therefore hampering mode retrieval. In this paper, we therefore develop a novel approach to ridge detection and mode retrieval based on the analysis of the short-time Fourier transform of multicomponent signals in the presence of noise, which will prove to be much more robust than state-of-the-art methods based on the same time-frequency representation

A New Formalism for Nonlinear and Non-Separable Multi-scale Representation

In this paper, we present a new formalism for nonlinear and non-separable multi-scale representations. We first show that most of the one-dimensional nonlinear multi-scale representations described in the literature are based on prediction operators which are the sum of a linear prediction operator and a perturbation defined using finite differences. We then extend this point of view to the multi-dimensional case where the scaling factor is replaced by a non-diagonal dilation matrix $M$. The new formalism we propose brings about similarities between existing nonlinear multi-scale representations and also enables us to alleviate the classical hypotheses made to prove the convergence of the multi-scale representations

Interpolatory Nonlinear and Non-Separable Multi-scale Representation: Application to Image Compression

In this paper, we introduce the notion of nonlinear and non-separable multi-scale representation. We show how it can be derived from nonlinear and non-separable subdivision schemes associated to a non-diagonal dilation matrix. We focus on nonlinear multi-scale decomposition where the dilation matrix is either the quincunx or the hexagonal matrix. We then detail the encoding and decoding algorithm of the representation and, in particular, how the EZW (Embedded Zero-tree Wavelet) algorithm adapts in that context. Numerical experiments on image compression conclude the paper.

The Fourier-based Synchrosqueezing Transform

The short-time Fourier transform (STFT) and continuous wavelet transform (CWT) are intensively used to analyze and process multicomponent signals, ie superpositions of mod- ulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relations, inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the continuous wavelet transform, it provides a sharp, con- centrated representation, while remaining invertible. This technique received a renewed interest with the recent publi- cation of an approximation result, which provides guarantees for the decomposition of a multicomponent signal. This paper adapts the formulation of the synchrosqueezing to the STFT, and states a similar theoretical result. The emphasis is put on the differences with the CWT-based synchrosqueezing, and all the content is illustrated through numerical experiments

Smoothness Characterization and Stability of Nonlinear and Non-Separable Multi-scale Representations

International audienceThe aim of the paper is the construction and the analysis of nonlinear and non-separable multi-scale representations for multivariate functions. The described multi-scale representation is associated with an isotropic dilation matrix. We show that the smoothness of a function can be characterized by the rate of decay of its multi-scale coefficients. We also study the stability of these representations, a key issue in the designing of adaptive algorithms

On the Mode Synthesis in the Synchrosqueezing Method

Publication in the conference proceedings of EUSIPCO, Bucharest, Romania, 201

Time-Frequency Ridge Analysis Based on the Reassignment Vector

International audienceThis paper considers the problem of detecting and estimating AM/FM components in the time-frequency plane. It introduces a new algorithm to estimate the ridges corresponding to the instantaneous frequencies of the components, and to segment the time-frequency plane into different `basins of attraction', each basin corresponding to one mode. The technique is based on the structure of the reassignment vector, which is commonly used for sharpening time-frequency representations. Compared with previous approaches, this new method does not need extra parameters, exhibits less sensitivity to the choice of the window and shows better reconstruction performance. Its effectiveness is demonstrated on simulated and real datasets