Spark Deficient Gabor Frame Provides a Novel Analysis Operator for Compressed Sensing

TL;DR

This paper introduces a novel spark deficient Gabor frame as an analysis operator for compressed sensing, demonstrating its superior performance over existing Gabor transforms in reconstructing signals from fewer measurements.

Contribution

The paper proposes a new highly redundant Gabor analysis operator based on a spark deficient frame, enhancing analysis sparsity models in compressed sensing.

Findings

The proposed Gabor transform outperforms existing Gabor transforms in all tested cases.

Computational experiments confirm improved signal reconstruction with fewer measurements.

The new analysis operator is effective on both synthetic and real-world data.

Abstract

The analysis sparsity model is a very effective approach in modern Compressed Sensing applications. Specifically, redundant analysis operators can lead to fewer measurements needed for reconstruction when employing the analysis $l_1$-minimization in Compressed Sensing. In this paper, we pick an eigenvector of the Zauner unitary matrix and -- under certain assumptions on the ambient dimension -- we build a spark deficient Gabor frame. The analysis operator associated with such a spark deficient Gabor frame, is a new (highly) redundant Gabor transform, which we use as a sparsifying transform in Compressed Sensing. We conduct computational experiments -- on both synthetic and real-world data -- solving the analysis $l_1$-minimization problem of Compressed Sensing, with four different choices of analysis operators, including our Gabor analysis operator. The results show that our proposed redundant Gabor transform outperforms -- in all cases -- Gabor transforms generated by state-of-the-art window vectors of time-frequency analysis.