Section: New Results

Emerging activities on compressive sensing, learning and inverse problems

Compressive sensing, acoustic wavefields, audio inpainting,

Audio inpainting (SMALL FET-Open project)

Participants : Rémi Gribonval, Nancy Bertin, Corentin Guichaoua, Srdan Kitic.

Inpainting is a particular kind of inverse problems that has been extensively addressed in the recent years in the field of image processing. It consists in reconstructing a set of missing pixels in an image based on the observation of the remaining pixels. Sparse representations have proved to be particularly appropriate to address this problem. However, inpainting audio data has never been defined as such so far.

METISS has initiated a series of works about audio inpainting, from its definition to methods to address it. This research has begun in the framework of the EU Framework 7 FET-Open project FP7-ICT-225913-SMALL (Sparse Models, Algorithms and Learning for Large-Scale data) which began in January 2009. Rémi Gribonval was the coordinator of the project. The research on audio inpainting has been conducted by Valentin Emiya in 2010 and 2011.

The contributions consist of:

• defining audio inpainting as a general scheme where missing audio data must be estimated: it covers a number of existing audio processing tasks that have been addressed separately so far – click removal, declipping, packet loss concealment, unmasking in time-frequency;

• proposing algorithms based on sparse representations for audio inpainting (based on Matching Pursuit and on ${\ell }^1$ minimization);

• addressing the case of audio declipping (i.e. desaturation): thanks to the flexibility of our inpainting algorithms, they can be constrained so as to include the structure of signals due to clipping in the objective to optimize. The resulting performance are significantly improved. This work appeared as a journal paper [63] .

• addressing the case of audio declipping with the competitive cosparse approach, with promising result especially when the clipping level is low. A contribution was submitted to the iTwist'14 workshop [61] .

Current and future works deal with developping advanced sparse decomposition for audio inpainting, including several forms of structured sparsity (e.g. temporal and multichannel joint-sparsity), dictionary learning for inpainting, and several applicative scenarios (declipping, time-frequency inpainting).

Blind Calibration of Compressive Sensing systems

Participants : Rémi Gribonval, Cagdas Bilen.

Main collaborations: Gilles Chardon, Laurent Daudet (Institut Langevin), Gilles Puy (EPFL)

We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using ${\ell }^1$ minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly non-convex and riddled with local minima. In the considered context, when the gains are real valued and non-negative, we showed that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using off-the-shelf algorithms. Numerical simulations demonstrated the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We observed that the success/failure of the approach seems to obey sharp phase transitions  [84] . This year, we focused on extending the framework to phase-only decalibration, using techniques revolving around low-rank matrix recovery [27] , [26] , [34] , [52] , and to joint phase and gain decalibration [54] .

Compressive Gaussian Mixture estimation

Participants : Rémi Gribonval, Anthony Bourrier.

Main collaborations: Patrick Perez (Technicolor R&I France)

When fitting a probability model to voluminous data, memory and computational time can become prohibitive. In this paper, we propose a framework aimed at fitting a mixture of isotropic Gaussians to data vectors by computing a low-dimensional sketch of the data. The sketch represents empirical moments of the underlying probability distribution. Deriving a reconstruction algorithm by analogy with compressive sensing, we experimentally show that it is possible to precisely estimate the mixture parameters provided that the sketch is large enough. Our algorithm provides good reconstruction and scales to higher dimensions than previous probability mixture estimation algorithms, while consuming less memory in the case of numerous data. It also provides a privacy-preserving data analysis tool, since the sketch does not disclose information about individual datum it is based on [38] , [40] , [29] .