Finding sparse representation of quantized frame coefficients using 0-1 integer programming

Abstract

The use of overcomplete dictionaries, or frames, has received increased attention in low bit rate compression. Several vector selection algorithms, such as Matching Pursuit, Orthogonal Matching Pursuit and FOCUSS have been developed to get sparse representations of signals. In these algorithms, continuous.valued coeflcients are found and subsequently quantized. The latter part can cause unwanted effects on the quality of the reconstructed signal. We propose an algorithm that merges the selection and quantization procedures by using 0-1 integer programming. The object is to minimize the distortion measured by the 11 -norm, subject to a bound on the number of "ones" in a binary representation of the frame coefJicients. This bound is an indirect measure of the bit rate. Our new algorithm finds the globally optimal solution based on the above-mentioned criteria.

Sensitivity computation of the ℓ 1 minimization problem and its application to dictionary design of ill-posed problems

“…where R is a regularization functional, aimed for imposition of a priori information over the solution and α > 0 is a regularization parameter. A different approach for regularization is based on imposition of sparsity, see for example [1,2,10,49,54,59,66,75] and references therein. This approach has gained interest and popularity in the past few years.…”

“…A different approach for regularization is based on imposition of sparsity, see for example [1,2,10,49,54,59,66,75] and references therein. This approach has gained interest and popularity in the past few years.…”

Improved M-FOCUSS Algorithm With Overlapping Blocks for Locally Smooth Sparse Signals

“…T HE need for finding sparse solutions to systems of linear equations has been motivated by many applications such as those in signal and image processing [1]- [7], blind source separation [8]- [16], brain activity imaging [17]- [25], sparse component analysis [26], subband decomposition [27], [28], dictionary learning [29]- [32], sparse Bayesian learning [33], [34], sparse coding [35], sparse audio signal representation [28], [36], sparse regression [37]- [40], inverse synthetic aperture radar (ISAR) imaging [41], low-bit-rate compression [42], decision feedback equalization [43], or echo cancellation [44].…”