Deterministic Construction of Bipolar Matrices for Compressed Sensing

Ranjan, S and Vidyasagar, Mathukumalli (2019) Deterministic Construction of Bipolar Matrices for Compressed Sensing. In: 58th IEEE Conference on Decision and Control, CDC, 11-13 December 2019, France.

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Abstract

Compressed sensing refers to the recovery of highdimensional but sparse (or nearly sparse) vectors from a small number of linear measurements. Until now most of the attention has been focused on measurement matrices that consist of real number, or are binary. Relatively less attention has been paid to the design of bipolar matrices, where every element is plus or minus one. Such matrices are preferred in applications such as the design of touchpads for cell phones. Previously the design of bipolar matrices was based on algebraic codes such as the BCH codes, and the methodology was based on satisfying the restricted isometry property (RIP). In the present paper, we adopt a different approach, namely, to start with binary measurement matrices that have uniform column weight (the same number of ones in each column), and show that a simple modification leads to bipolar matrices that satisfy the robust null space property (RNSP). Since RIP implies the RNSP, as shown by the authors in another paper, our approach leads to a far smaller number of bipolar measurements compared to existing methods for the same.

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