Frame-based Approach for Sparse Representation of Signal Decomposition and Reconstruction

TL;DR

This paper explores the possibility of constructing dual frames in signal processing that optimize sparse representation by minimizing the l1-norm, extending classical results on l2-norm minimization.

Contribution

It introduces conditions under which dual frames can produce minimal l1-norm coefficients, and proposes methods to design such frames for sparse signal decomposition.

Findings

Dual frames cannot generally minimize l1-norm of coefficients.

Conditions are identified for dual frames to achieve minimal l1-norm.

New frame construction methods are proposed for sparse analysis and synthesis.

Abstract

Frame is the corner stone for designing decomposition and reconstruction operations in signal processing. Famous frames include wavelets, curvelets,and Gabor. A celebrated result indicates that if a synthesis frame is chosen for reconstruction, then its canonical dual frame is the analysis frame that performs decomposition, yielding coefficients that minimizes l2-norm of all coefficients of all dual frames. This paper tries to extend this result by inves-tigating whether a dual frame can be constructed so that the corresponding coefficients yield the minimum l1-norm. We show that this mission cannot be achieved for any over-complete frame. However, we present some conditions on a dual frame so that the minimizer of the l1-norm can be derived from the coefficients of the dual frame. We show that, based on frame structure, various applications that seek to optimize sparse analysis and synthesis co-efficients are logically equivalent. Finally, we present a general construction method of frames to demonstrate the proposed conditions are feasible. In addition, two design approaches are proposed to derive frames that meet the proposed conditions and yield small non-zero elements of frame coefficients.