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From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images (2007)
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@MISC{Bruckstein07fromsparse,
author = {Alfred M. Bruckstein and David L. Donoho and Michael Elad},
title = {From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images },
year = {2007}
}
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Abstract
A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Keyphrases
sparse solution sparse modeling image processing problem theoretical result underdetermined system bold perspective linear system image processing nonzero entry undetermined system surprising phenomenon easily-verifiable condition efficient method many solution effective computational method full-rank matrix ir optimally-sparse solution linear equation ax recent year considerable evidence wide variety several well-known signal important practical problem
