Algorithms for the Construction of Incoherent Frames Under Various Design Constraints

TL;DR

This paper presents algorithms for designing various types of low coherence frames, which are crucial in compressed sensing for accurate sparse signal recovery, by solving convex optimization problems.

Contribution

It introduces novel algorithms for constructing diverse low coherence frames under multiple constraints using convex optimization techniques.

Findings

Algorithms effectively reduce frame coherence compared to the Welch bound.

Numerical experiments demonstrate improved performance in compressed sensing applications.

Proposed methods outperform existing algorithms in various frame design scenarios.

Abstract

Unit norm finite frames are generalizations of orthonormal bases with many applications in signal processing. An important property of a frame is its coherence, a measure of how close any two vectors of the frame are to each other. Low coherence frames are useful in compressed sensing applications. When used as measurement matrices, they successfully recover highly sparse solutions to linear inverse problems. This paper describes algorithms for the design of various low coherence frame types: real, complex, unital (constant magnitude) complex, sparse real and complex, nonnegative real and complex, and harmonic (selection of rows from Fourier matrices). The proposed methods are based on solving a sequence of convex optimization problems that update each vector of the frame. This update reduces the coherence with the other frame vectors, while other constraints on its entries are also imposed. Numerical experiments show the effectiveness of the methods compared to the Welch bound, as well as other competing algorithms, in compressed sensing applications.