Greedy Deconvolution of Point-like Objects

Abstract : The orthogonal matching pursuit (OMP) is an algorithm to solve sparse approximation problems. In [1] a sufficient condition for exact recovery is derived, in [2] the authors transfer it to noisy signals. We will use OMP for reconstruction of an inverse problem, namely the deconvolution problem. In sparse approximation problems one often has to deal with the problem of redundancy of a dictionary, i.e. the atoms are not linearly independent. However, one expects them to be approximatively orthogonal and this is quantified by incoherence. This idea cannot be transfered to ill-posed inverse problems since here the atoms are typically far from orthogonal: The illposedness of the (typically compact) operator causes that the correlation of two distinct atoms probably gets huge, i.e. that two atoms can look much alike. Therefore in [3], [4] the authors derive a recovery condition which uses the kind of structure one assumes on the signal and works without the concept of coherence. In this paper we will transfer these results to noisy signals. For our source we assume that it consists of a superposition of point-like objects with an a-priori known distance. We will apply it exemplarily to Dirac peaks convolved with Gaussian kernel as used in mass spectrometry.
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Communication dans un congrès
Rémi Gribonval. SPARS'09 - Signal Processing with Adaptive Sparse Structured Representations, Apr 2009, Saint Malo, France. 2009
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Dirk A. Lorenz, Dennis Trede. Greedy Deconvolution of Point-like Objects. Rémi Gribonval. SPARS'09 - Signal Processing with Adaptive Sparse Structured Representations, Apr 2009, Saint Malo, France. 2009. 〈inria-00369571〉

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