Frame-based Sparse Analysis and Synthesis Signal Representations and Parseval K-SVD

TL;DR

This paper explores frame theory in sparse signal representation, demonstrating limitations of dual frames for over-complete frames and introducing Parseval K-SVD for learning tight-frame dictionaries to improve image recovery.

Contribution

The paper introduces a novel Parseval K-SVD algorithm for learning tight-frame dictionaries and analyzes the limitations of dual frames in over-complete settings for sparse representations.

Findings

Dual frames cannot be constructed for over-complete frames.

Analysis coefficients from the canonical dual frame approximate sparse synthesis solutions.

Image recovery quality correlates with frame bounds of learned dictionaries.

Abstract

Frames are the foundation of the linear operators used in the decomposition and reconstruction of signals, such as the discrete Fourier transform, Gabor, wavelets, and curvelet transforms. The emergence of sparse representation models has shifted of the emphasis in frame theory toward sparse l1-minimization problems. In this paper, we apply frame theory to the sparse representation of signals in which a synthesis dictionary is used for a frame and an analysis dictionary is used for a dual frame. We sought to formulate a novel dual frame design in which the sparse vector obtained through the decomposition of any signal is also the sparse solution representing signals based on a reconstruction frame. Our findings demonstrate that this type of dual frame cannot be constructed for over-complete frames, thereby precluding the use of any linear analysis operator in driving the sparse synthesis coefficient for signal representation. Nonetheless, the best approximation to the sparse synthesis solution can be derived from the analysis coefficient using the canonical dual frame. In this study, we developed a novel dictionary learning algorithm (called Parseval K-SVD) to learn a tight-frame dictionary. We then leveraged the analysis and synthesis perspectives of signal representation with frames to derive optimization formulations for problems pertaining to image recovery. Our preliminary, results demonstrate that the images recovered using this approach are correlated to the frame bounds of dictionaries, thereby demonstrating the importance of using different dictionaries for different applications.