Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes

This paper provides central limit theorems for the wavelet packet decomposition of stationary band-limited random processes. The asymptotic analysis is performed for the sequences of the wavelet packet coefficients returned at the nodes of any given path of the $M$-band wavelet packet decomposition tree. It is shown that if the input process is centred and strictly stationary, these sequences converge in distribution to white Gaussian processes when the resolution level increases, provided that the decomposition filters satisfy a suitable property of regularity. For any given path, the variance of the limit white Gaussian process directly relates to the value of the input process power spectral density at a specific frequency.Comment: Submitted to the IEEE Transactions on Signal Processing, October 200

Signal norm testing in additive and independant standard Gaussian noise

This paper addresses signal norm testing (SNT), that is, the problem of deciding whether a random signal norm exceeds some specified value τ > 0 or not, when the signal has unknown probability distribution and is observed in additive and independent standard Gaussian noise. The theoretical framework proposed for SNT extends usual notions in statistical inference and introduces a new optimality criterion. This one takes the invariance of both the problem and the noise distribution into account, via conditional notions of power and size and, more specifically, the introduction of the spherically-conditioned power function. The theoretical results established with respect to this criterion extend those deriving fromstandard statistical inference theory in the case of an unknown deterministic signal. Thinkable applications are problems where signal amplitude deviations from some nominal reference must be detected above a certain tolerance τ, possibly chosen by the user on the basis of his experience and know-how. In this respect, the theoretical results of this paper are applied to an SNT formulation for the problem of detecting random signals in noise,with a specific focus on the case where the noise standard deviation is unknown

Graph reconstruction from the observation of diffused signals

Signal processing on graphs has received a lot of attention in the recent years. A lot of techniques have arised, inspired by classical signal processing ones, to allow studying signals on any kind of graph. A common aspect of these technique is that they require a graph correctly modeling the studied support to explain the signals that are observed on it. However, in many cases, such a graph is unavailable or has no real physical existence. An example of this latter case is a set of sensors randomly thrown in a field which obviously observe related information. To study such signals, there is no intuitive choice for a support graph. In this document, we address the problem of inferring a graph structure from the observation of signals, under the assumption that they were issued of the diffusion of initially i.i.d. signals. To validate our approach, we design an experimental protocol, in which we diffuse signals on a known graph. Then, we forget the graph, and show that we are able to retrieve it very precisely from the only knowledge of the diffused signals.Comment: Allerton 2015 : 53th Annual Allerton Conference on Communication, Control and Computing, 30 september - 02 october 2015, Allerton, United States, 201

Characterization and Inference of Graph Diffusion Processes from Observations of Stationary Signals

Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its nodes. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures.Comment: Submitted to IEEE Transactions on Signal and Information Processing over Network

Algorithms based on sparsity hypotheses for robust estimation of the noise standard deviation in presence of signals with unknown distributions and concurrences

Inmany applications, d-dimensional observations result fromthe randompresenceor absence of randomsignals in independent and additivewhite Gaussiannoise. An estimate of the noise standard deviation can then be very useful todetect or to estimate these signals, especially when standard likelihood theory cannot apply because of too little prior knowledge about the signal probability distributions. Recent results and algorithms have then emphasized the interest of sparsity hypotheses to design robust estimators of the noise standard deviation when signals have unknown distributions. As a continuation, the present paper introduces a new robust estimator for signals with probabilities of presence less than or equal to one half. In contrast to the standard MAD estimator, it applies whatever the value of d. This algorithm is applied to image denoising by wavelet shrinkage as well as to non-cooperative detection of radiocommunications.In both cases, the estimator proposed in the present paper outperforms the standard solutions used in such applications and based on the MAD estimator. The Matlab code corresponding to the proposed estimator is available at http://perso.telecom-bretagne.eu/pasto

On the Statistical Decorrelation of the Wavelet Packet Coefficients of a Band-Limited Wide-Sense Stationary Random Process

International audienceThis paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients resulting from the decomposition of a random process, stationary in the wide-sense, whose power spectral density is bounded with support in [-\pi,\pi]. Consider two quadrature mirror filters (QMF) that depend on a parameter r, such that these filters tend almost everywhere to the Shannon QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemarié filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function). Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packet associated with a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet. Another consequence of our derivation is that, when the coefficients associated with a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag $0$ is close to the value taken by the power spectral density of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration. Some simulations highlight the good quality of the ''whitening'' effect that can be obtained in practical cases