Spark Deficient Gabor Frame Provides A Novel Analysis Operator For Compressed Sensing

Vicky Kouni, Holger Rauhut

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 frame, is a new (highly) redundant Gabor transform, which we use as a sparsifier 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.