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A first step to convolutive sparse representation | IEEE Conference Publication | IEEE Xplore

A first step to convolutive sparse representation


Abstract:

In this paper an extension of the sparse decomposition problem is considered and an algorithm for solving it is presented. In this extension, it is known that one of the ...Show More

Abstract:

In this paper an extension of the sparse decomposition problem is considered and an algorithm for solving it is presented. In this extension, it is known that one of the shifted versions of a signal s (not necessarily the original signal itself) has a sparse representation on an overcomplete dictionary, and we are looking for the sparsest representation among the representations of all the shifted versions of s. Then, the proposed algorithm finds simultaneously the amount of the required shift, and the sparse representation. Experimental results emphasize on the performance of our algorithm.
Date of Conference: 31 March 2008 - 04 April 2008
Date Added to IEEE Xplore: 12 May 2008
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Conference Location: Las Vegas, NV, USA

1. INTRODUCTION

In the classical atomic decomposition problem [1], we have a signal whose samples are collected in the signal vector and we would like to represent it as a linear combination of signal vectors . After [2], the vectors are called atoms and they collectively form a dictionary over which the signal is to be decomposed. We may write {\bf s}=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}=\Phi\alpha, \eqno{\hbox{(1)}} where is the dictionary (matrix) and is the vector of coefficients. A dictionary with is called overcomplete. Although, is sufficient to obtain such a decomposition (like what is done in Discrete Fourier Transform), using overcomplete dictionaries has a lot of advantages in many diverse applications (refer for example to [3] and the references in it). Note that for the overcomplete case, the representation is not unique, but all these applications need a sparse representation, that is, the signal s should be represented as a linear combination of as small as possible number of atoms of the dictionary. It has been shown [4], [5] that with some mild conditions on the dictionary matrix, if there is a sparse representation with at most non-zero coefficients, then this representation is unique. The main approaches for finding this sparse solution include Matching Pursuit (MP) [2], [6], FOCUSS [5], Basis Pursuit (BP) [1], and Smoothed (SLO) [7].

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