Sparse Decomposition Over Non-Full-Rank Dictionaries

Massoud Babaie-Zadeh (1) , Vincent Vigneron (2) , Christian Jutten (3)

1 EE department of SUT [Tehran] - Department of Electrical Engineering [Tehran]
2 IBISC - Informatique, Biologie Intégrative et Systèmes Complexes
3 GIPSA-SIGMAPHY - GIPSA - Signal Images Physique

Sparse Decomposition (SD) of a signal on an overcomplete dictionary has recently attracted a lot of interest in signal processing and statistics, because of its potential application in many different areas including Compressive Sensing (CS). However, in the current literature, the dictionary matrix has generally been assumed to be of full-rank. In this paper, we consider non-full-rank dictionaries (which are not even necessarily overcomplete), and extend the definition of SD over these dictionaries. Moreover, we present an approach which enables to use previously developed SD algorithms for this non-full-rank case. Besides this general approach, for the special case of the Smoothed L0-norm (SL0) algorithm, we show that a slight modification of it covers automatically non-full-rank dictionaries.

Mots clés: Overcomplete Signal Representation; Sparse Component Analysis (SCA); Sparse Decomposition; Atomic Decomposition; Compressive Sensing (CS)

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Massoud Babaie-Zadeh, Vincent Vigneron, Christian Jutten. Sparse Decomposition Over Non-Full-Rank Dictionaries. ICASSP 2009 - IEEE International Conference on Acoustics, Speech and Signal Processing, Apr 2009, Taipei, Taiwan. pp.2953-2956. ⟨hal-00400474⟩