Section: New Results
Emerging activities on compressive sensing
Compressed sensing of Acoustic Wavefields (ECHANGE ANR project)
Main collaborations: Albert Cohen (Laboratoire Jacques-Louis Lions, Université Paris 6), Laurent Daudet, François Ollivier, Jacques Marchal (Institut Jean Le Rond d'Alembert, Université Paris 6)
Compressed sensing is a rapidly emerging field which proposes a new approach to sample data far below the Nyquist rate when the sampled data admits a sparse approximation in some appropriate dictionary. The approach is supported by many theoretical results on the identification of sparse representations in overcomplete dictionaries, but many challenges remain open to determine its range of effective applicability.
METISS has chosen to focus more specifically on the exploration of Compressed Sensing of Acoustic Wavefields. This research has began in the framework of the Ph.D. of Prasad Sudhakar (started in december 2007), and we have set up the ANR collaborative project ECHANGE (ECHantillonnage Acoustique Nouvelle GEnération) which is due to begin in January 2009. Rémi Gribonval is the coordinator of the project.
The main challenges are: a) to identify dictionaries of basic wavefield atoms making it possible to sparsely represent the wavefield in several acoustic scenarios of interest; b) to determine which types of (networks) of acoustic sensors maximise the identifiability of the sparse wavefield representation, depending on the acoustic scenario; c) to design scalable algorithms able to reconstruct the measured wavefields in a region of interest.
Compressed sensing of wideband signals
Participant : Rémi Gribonval.
Main collaborations: Laurent Jacques (EPFL & UCL Belgique), Pierre Vandergheynst (EPFL), Farid Nani Mohavedian
Compressed sensing is also the object of a collaboration with EPFL in the framework of the Equipe Associée SPARS 8.1.1 . In the framework of the summer internship of Mr Farid Naini Mohavedian, we studied the application of compressed sensing to ultra wide-band signals. More precisely, we studied a model where the considered signals are sparse linear superpositions of shifts of a known, potentially wide-band, pulse. This signal model is key for applications such as Ultra Wide Band (UWB) communications or neural signal processing. We compared several acquisition strategies and showed that the approximations recovered via 1 minimization are greatly enhanced if one uses Spread Spectrum analog modulation prior to applying random Fourier measurements. We complemented our experiments with a discussion of possible hardware implementation of our technique, and checked that a simplified hardware implementation did not degrade the performance of the compressed sensing system. The results have been published at the conference ICASSP 2009  .
Wavelets on graphs
Participant : Rémi Gribonval.
Main collaboration: Pierre Vandergheynst, David Hammond (EPFL)
Within the framework of the Equipe Associée SPARS 8.1.1 , we investigated the possibility of developping sparse representations of functions defined on graphs, by defining an extension to the traditional wavelet transform which is valid for data defined on a graph.
There are many problems where data is collected through a graph structure: scattered or non-uniform sampling, sensor networks, data on sampled manifolds or even social networks or databases. Motivated by the wealth of new potential applications of sparse representations to these problems, the partners set out a program to generalize wavelets on graphs. More precisely, we have introduced a new notion of wavelet transform for data defined on the vertices of an undirected graph. Our construction uses the spectral theory of the graph laplacian as a generalization of the classical Fourier transform. The basic ingredient of wavelets, multi-resolution, is defined in the spectral domain via operator-valued functions that can be naturally dilated. These in turn define wavelets by acting on impulses localized at any vertex. We have analyzed the localization of these wavelets in the vertex domain and showed that our multi-resolution produces functions that are indeed concentrated at will around a specified vertex. Our theory allowed us to construct an equivalent of the continuous wavelet transform but also discrete wavelet frames.
Computing the spectral decomposition can however be numerically expensive for large graphs. We have shown that, by approximating the spectrum of the wavelet generating operator with polynomial expansions, applying the forward wavelet transform and its transpose can be approximated through iterated applications of the graph Laplacian. Since in many cases the graph Laplacian is sparse, this results in a very fast algorithm. Our implementation also uses recurrence relations for computing polynomial expansions, which results in even faster algorithms. Finally, we have proved how numerical errors are precisely controlled by the properties of the desired spectral graph wavelets. Our algorithms have been implemented in a Matlab toolbox that will be released in parallel to the main theoretical article  . We also plan to include this toolbox in the SMALL project numerical platform.
We now foresee many applications. On one hand we will use non-local graph wavelets constructed from the set of patches in an image (or even an audio signal) to perform de-noising or in general restoration. An interesting aspect in this case, would be to understand how wavelets estimated from corrupted signals deviate from clean wavelets. In a totally different direction, we will also explore the applications of spectral graph wavelets constructed from brain connectivity graphs obtained from whole brain tractography. Our preliminary results show that graph wavelets yield a representation that is very well adapted to how the information flows in the brain along neuronal structures.