Jean-Jacques Fuchs

Multipath time-delay detection and estimation

A transmitted and known signal is observed at the receiver through more than one path in additive noise. The problem is to estimate the number of paths and, for each of them, the associated attenuation and delay. We propose a deconvolution approach with an additive regularization term built around an l/sub 1/ norm. The underlying optimization problem is transformed into a quadratic program and is, thus, easily and quickly solved by standard programs. The procedure is able to handle more severe conditions than previous methods.

Recovery of exact sparse representations in the presence of noise

The paper extends some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. The type of question addressed so far is: given a (n,m)-matrix A with m>n and a vector b=Ax, find a sufficient condition for b to have a unique sparsest representation as a linear combination of the columns of A. The answer is a bound on the number of nonzero entries of, say, x/sub o/, that guarantees that x/sub o/ is the unique and sparsest solution of Ax=b with b=Ax/sub o/. We consider the case b=Ax/sub o/+e where x/sub o/ satisfies the sparsity conditions requested in the noise-free case and seek conditions on e, a vector of additive noise or modeling errors, under which x/sub o/ can be recovered from b in a sense to be defined.

An inverse problem approach to robust regression

When recording data, large errors may occur occasionally. The corresponding abnormal data points, called outliers, can have drastic effects on the estimates. There are several ways to cope with outliers-detect and delete or adjust the erroneous data,-use a modified cost function. We propose a new approach that allows, by introducing additional variables, to model the outliers and to detect their presence. In the standard linear regression model this leads to a linear inverse problem that, associated with a criterion that ensures sparseness, is solved by a quadratic programming algorithm. The new approach (model+criterion) allows for extensions that cannot be handled by the usual robust regression methods.

Linear programming in spectral estimation. Application to array processing

We address the narrow-band source localization problem for arbitrary arrays with known geometry in the presence of arbitrary noise of unknown spatial spectral density. Very few methods are able to handle this problem. We present a very unsophisticated approach whose algorithmic part relies on a standard linear programming algorithm (such as the simplex algorithm available in any scientific program library). The computational complexity of the method is reasonable, the performance appear to be remarkable on simulations. The justification of the procedure and the asymptotic analysis is more complex and much work remains to be done.

Detection and estimation of superimposed signals

The problem of fitting a model composed of a number of superimposed signals to noisy observations is addressed. An approach allowing us to evaluate both the number of signals and their characteristics is presented. The idea is to search for a parsimonious representation of the data. The parsimony is insured by adding to the maximum likelihood criterion a regularization term built upon the l/sub 1/-norm of the weights. Different equivalent formulations of the criterion are presented. They lead to appealing physical interpretations. Due to limited space, we only sketch an analysis of the performance of the algorithm that has been successfully applied to different classes of problems.

Extension of the Pisarenko method to sparse linear arrays

When applied to array processing, the Pisarenko harmonic decomposition (PHD) method is limited to linear equispaced arrays. We present an approach that allows us to extend it to general arrays, although for the ease of exposition, we consider only sparse linear arrays. We exploit the fact that the PHD can be seen as a deconvolution or model-fitting approach that minimizes an t/sub 1/ norm and can thus be implemented as a standard linear program. Looking at the PHD from this point of view has two advantages: it allows us to extend its applicability to arbitrary arrays, and by diverging slightly from the basic philosophy, it allows us to improve its performance, which is often quite poor in its original version.

Sparsity and uniqueness for some specific under-determined linear systems

The paper extends some results on sparse representations of signals in redundant bases developed for arbitrary bases to two frequently encountered bases. The general problem is the following: given an n/spl times/m matrix, A, with m>n, and a vector, b=Ax/sub 0/, with x/sub 0/ having q<n nonzero components, find sufficient conditions for x/sub 0/ to be the unique sparsest solution to Ax=Ax/sub 0/. The answer gives an upper-bound on q depending upon A. We consider the cases where A is a Vandermonde matrix or a real Fourier matrix and the components of x/sub 0/ are known to be greater than or equal to zero. The sufficient conditions we get are much weaker than those valid for arbitrary matrices and guarantee further that x/sub 0/ can be recovered by solving a linear program.

Multipath time-delay estimation

A transmitted and known signal is observed at the receiver through more than one path in additive noise. The problem is to estimate the number of paths and for each of them the associated attenuation and delay. It is a frequent problem in sonar, radar and geophysics. We propose an algorithm that is easy to implement, that has a reasonable computational load and seems to be able to solve the problem under more severe conditions (lower SNR) than previous methods.

A deconvolution approach to source localization

Due to its low resolution the conventional beamformer (CBF) has generally been neglected in the recent literature on array processing and considerable attention devoted to the high-resolution (HR) techniques. Applying a deconvolution approach to the output of the CBF, the authors obtain a source localization procedure whose performances are comparable to those of the most efficient HR techniques. The scheme, moreover, also furnishes estimates of the source powers and comprises a test that detects the number of sources present. It can indeed be seen as a means to obtain maximum likelihood estimates (MLE) when the CBF-output is taken as the observed data. >

Atomic decomposition dedicated to AVC and spatial SVC prediction

This work, we propose the use of sparse signal representation techniques to solve the problem of closed-loop spatial image prediction. The reconstruction of signal in the block to predict is based on basis functions selected with the matching pursuit (MP) iterative algorithm, to best match a causal neighborhood. We evaluate this new method in terms of PSNR and bitrate in a H.264/AVC encoder. Experimental results indicate an improvement of rate-distortion performance. In this paper, we also present results concerning the use of this technique for intra-inter layer prediction refinement, in a scalable video coding (SVC) like scheme.