Bruno Torrésani

Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed. >

Characterization of signals by the ridges of their wavelet transforms

The characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.

Practical Time-Frequency Analysis

The purpose of the book is to give a self-contained presentation of the techniques of time-frequency/time-scale analysis of 1-D signals and provide a set of useful tools to perform the analyses. Such a package should be especially attractive to the part of the scientific community interested in mathematical and practical issues, especially if they involve random or noisy signals with possibly nonstationary features. The use of the S language is a reflection of the intent to reach the statistical community that, despite the traditional interest in the spectral analysis of time series and the pervasive use of orthonormal wavelet bases, has seen very few attempts to understand the benefits of the continuous transforms. The first part of the book is intended to be a hands-on crash course on some of the major components of the time-frequency analysis of signals. A special emphasis is put on the analyses of noisy signals, and great care is taken to address the stationarity issue and describe the statistical significance of the spectral analyses and the denoizing procedures. The second part of the book is a reference manual for the library of S functions, which is written to perform all the computations relative to the examples described in the first part of the monograph.

Multiridge detection and time-frequency reconstruction

The ridges of the wavelet transform, the Gabor transform, or any time-frequency representation of a signal contain crucial information on the characteristics of the signal. Indeed, they mark the regions of the time-frequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges. The procedure is based on an original form of Markov chain Monte Carlo algorithm especially adapted to the present situation. We show that this detection algorithm is especially useful for noisy signals with multiridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e., the restriction of the transform to the ridges). After reviewing several known procedures, we introduce a new reconstruction algorithm, and we illustrate its efficiency on speech signals and its robustness and stability on chirps perturbed by synthetic noise at different SNRs.

Hybrid representations for audiophonic signal encoding

In this paper, we discuss a new approach for signal models in the context of audio signal encoding. The method is based upon hybrid models featuring simultaneously transient, tonal and stochastic components in the signal. Contrary to several existing approaches, our method does not rely on any prior segmentation of the signal. The three components are estimated and encoded using a strategy very much in the spirit of transform coding. While the details of the method described here are tailored to audio signals, the general strategy should also apply to other types of signals exhibiting significantly different features, for example images.

The Linear Time Frequency Analysis Toolbox

The Linear Time Frequency Analysis Toolbox is a MATLAB/Octave toolbox for computational time-frequency analysis. It is intended both as an educational and computational tool. The toolbox provides the basic Gabor, Wilson and MDCT transform along with routines for constructing windows (filter prototypes) and routines for manipulating coefficients. It also provides a bunch of demo scripts devoted either to demonstrating the main functions of the toolbox, or to exemplify their use in specific signal processing applications. In this paper we describe the used algorithms, their mathematical background as well as some signal processing applications.

Sparse Linear Regression With Structured Priors and Application to Denoising of Musical Audio

We describe in this paper an audio denoising technique based on sparse linear regression with structured priors. The noisy signal is decomposed as a linear combination of atoms belonging to two modified discrete cosine transform (MDCT) bases, plus a residual part containing the noise. One MDCT basis has a long time resolution, and thus high frequency resolution, and is aimed at modeling tonal parts of the signal, while the other MDCT basis has short time resolution and is aimed at modeling transient parts (such as attacks of notes). The problem is formulated within a Bayesian setting. Conditional upon an indicator variable which is either 0 or 1, one expansion coefficient is set to zero or given a hierarchical prior. Structured priors are employed for the indicator variables; using two types of Markov chains, persistency along the time axis is favored for expansion coefficients of the tonal layer, while persistency along the frequency axis is favored for the expansion coefficients of the transient layer. Inference about the denoised signal and model parameters is performed using a Gibbs sampler, a standard Markov chain Monte Carlo (MCMC) sampling technique. We present results for denoising of a short glockenspiel excerpt and a long polyphonic music excerpt. Our approach is compared with unstructured sparse regression and with structured sparse regression in a single resolution MDCT basis (no transient layer). The results show that better denoising is obtained, both from signal-to-noise ratio measurements and from subjective criteria, when both a transient and tonal layer are used, in conjunction with our proposed structured prior framework.

Classification and construction of Quasisimple Lie algebras

Abstract We study a class of (possibly infinite-dimensional) Lie algebras, called the Quasisimple Lie algebras (QSLAs), and generalizing semisimple and affine Kac-Moody Lie algebras. They are characterized by the existence of a finite-dimensional Cartan subalgebra, a non-degenerate symmetric ad-invariant Killing form, and nilpotent rootspaces attached to non-isotropic roots. We are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems; moreover, we construct explicit realizations of some of them as (untwisted and twisted) current algebras, generalizing the affine loop algebras.

Continuous wavelet decompositions, multiresolution, and contrast analysis

A continuous version of multiresolution analysis is described, starting from usual continuous wavelet decompositions. Scale discretization leads to decomposition into functions of arbitrary bandwidth, satisfying QMF-like conditions. Finally, a nonlinear multiresolution scheme is described, providing multiplicative reconstruction formulas.

Decoding the nucleoid organisation of Bacillus subtilis and Escherichia coli through gene expression data

BackgroundAlthough the organisation of the bacterial chromosome is an area of active research, little is known yet on that subject. The difficulty lies in the fact that the system is dynamic and difficult to observe directly. The advent of massive hybridisation techniques opens the way to further studies of the chromosomal structure because the genes that are co-expressed, as identified by microarray experiments, probably share some spatial relationship. The use of several independent sets of gene expression data should make it possible to obtain an exhaustive view of the genes co-expression and thus a more accurate image of the structure of the chromosome.ResultsFor both Bacillus subtilis and Escherichia coli the co-expression of genes varies as a function of the distance between the genes along the chromosome. The long-range correlations are surprising: the changes in the level of expression of any gene are correlated (positively or negatively) to the changes in the expression level of other genes located at well-defined long-range distances. This property is true for all the genes, regardless of their localisation on the chromosome.We also found short-range correlations, which suggest that the location of these co-expressed genes corresponds to DNA turns on the nucleoid surface (14–16 genes).ConclusionThe long-range correlations do not correspond to the domains so far identified in the nucleoid. We explain our results by a model of the nucleoid solenoid structure based on two types of spirals (short and long). The long spirals are uncoiled expressed DNA while the short ones correspond to coiled unexpressed DNA.