Compressed Sensing and Redundant Dictionaries

Details Compressed Sensing and Redundant Dictionaries Compressed Sensing and Redundant Dictionaries

2007-01-04

arxiv.org/abs/math/0701131v2

IEEE Trans. Inform. Theory, 54(5):2210-2219, 2008

Mathematics

Probability

error in a constant corrected

Scientific paper

10.1109/TIT.2008.920190

This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.

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