Quantized overcomplete expansions in IR/sup N/: analysis, synthesis, and algorithms

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Sparse Approximate Solutions to Linear Systems

B. K. Natarajan (1995)

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Vector Quantization and Signal Compression

Allen Gersho, Robert M. Gray (1992)

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Image restoration by the method of convex projections: part 1 theory.

D Youla, Lynsey H Webb (1981)

A projection operator onto a closed convex set in Hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms. Moreover, it minimizes distance and is nonexpansive, and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds. In this paper, we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically. Their common conceptual basis is as follows. Every known property of an original image f is envisaged as restricting it to lie in a well-defined closed convex set. Thus, m such properties place f in the intersection E(0) = E(i) of the corresponding closed convex sets E(1),E(2),...EE(m). Given only the projection operators PE(i) onto the individual E(i)'s, i = 1 --> m, we restore f by recursive means. Clearly, in this approach, the realization of the P(i)'s in a Hilbert space setting is one of the major synthesis problems. Section I describes the geometrical significance of the three main theorems in considerable detail, and most of the underlying ideas are illustrated with the aid of simple diagrams. Section II presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal-processing applications, and the Appendix contains proofs of all the major results.

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Title: IEEE Transactions on Information Theory

Abbreviated Title: IEEE Trans. Inform. Theory

Publisher: Institute of Electrical and Electronics Engineers (IEEE)

ISSN (Print): 00189448

Publication date (Print): Jan. 1998

Volume: 44

Issue: 1

Pages: 16-31

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DOI: 10.1109/18.650985

SO-VID: 6ec6bdee-af72-46d4-8c41-703cd5556c01

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Quantized overcomplete expansions in IR/sup N/: analysis, synthesis, and algorithms

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Abstract

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Core Readings in Statistical Mediation Analysis

Most cited references 17

Sparse Approximate Solutions to Linear Systems

Vector Quantization and Signal Compression

Image restoration by the method of convex projections: part 1 theory.

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