A fundamental pitfall in blind deconvolution with sparse and shift-invariant priors

Alexis Benichoux 1 Emmanuel Vincent 2, 3 Rémi Gribonval 1
1 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
2 METISS - Speech and sound data modeling and processing
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
3 PAROLE - Analysis, perception and recognition of speech
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : We consider the problem of blind sparse deconvolution, which is common in both image and signal processing. To counter-balance the ill-posedness of the problem, many approaches are based on the minimization of a cost function. A well-known issue is a tendency to converge to an undesirable trivial solution. Besides domain specific explanations (such as the nature of the spectrum of the blurring filter in image processing) a widespread intuition behind this phenomenon is related to scaling issues and the nonconvexity of the optimized cost function. We prove that a fundamental issue lies in fact in the intrinsic properties of the cost function itself: for a large family of shift-invariant cost functions promoting the sparsity of either the filter or the source, the only global minima are trivial. We complete the analysis with an empirical method to verify the existence of more useful local minima.
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Alexis Benichoux, Emmanuel Vincent, Rémi Gribonval. A fundamental pitfall in blind deconvolution with sparse and shift-invariant priors. ICASSP - 38th International Conference on Acoustics, Speech, and Signal Processing - 2013, May 2013, Vancouver, Canada. ⟨hal-00800770⟩

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