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In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems,… (More)

- Alfred M. Bruckstein, David L. Donoho, Michael Elad
- SIAM Review
- 2009

A full-rank matrix A ∈ IRn×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… (More)

In recent years there is a growing interest in the study of sparse representation for signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described by sparse linear combinations of these atoms. Recent activity in this field concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a… (More)

- Nahum Kiryati, Yuval Eldar, Alfred M. Bruckstein
- Pattern Recognition
- 1991

- Michael Elad, Alfred M. Bruckstein
- IEEE Trans. Information Theory
- 2002

An elementary proof of a basic uncertainty principle concerning pairs of representations of vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main… (More)

- S. D. Yanowitz, Alfred M. Bruckstein
- 2009 Asia-Pacific Conference on Computational…
- 1989

On the basis of analyzing the blur images with noise, this paper presents a new segmentation method which is based on the morphology method, fuzzy K-means algorithm and some parts operator of the Canny algorithm. Because of the Canny's good performance on good detection, good localization and only one response to a single edge, we introduce the course of… (More)

- Ron Rubinstein, Alfred M. Bruckstein, Michael Elad
- Proceedings of the IEEE
- 2010

| Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a pre-specified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: i)… (More)

- Alfred M. Bruckstein, Michael Elad, Michael Zibulevsky
- IEEE Transactions on Information Theory
- 2008

An underdetermined linear system of equations <i>Ax</i> = <i>b</i> with nonnegativity constraint <i>x</i> ges 0 is considered. It is shown that for matrices <i>A</i> with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence… (More)

In recent years there is a growing interest in the study of sparse representation for signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described as sparse linear combinations of these atoms. Recent activity in this field concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a… (More)

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ‖x‖0. Such solutions find applications in signal and image processing, where the topic is typically… (More)