Sparks and Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

For a measurement matrix in compressed sensing, its spark (or the smallest number of columns that are linearly dependent) is an important performance parameter. The matrix with spark greater than 2k guarantees the exact recovery of ksparse signals under an l0-optimization, and the one with large spark may perform well under approximate algorithms of the l0-optimization. Recently, Dimakis, Smarandache and Vontobel revealed the close relation between LDPC codes and compressed sensing and showed that good parity-check matrices for LDPC codes are also good measurement matrices for compressed sensing. By drawing methods and results from LDPC codes, we study the performance evaluation and constructions of binary measurement matrices in this paper. Two lower bounds of spark are obtained for general binary matrices, which improve the previously known results for real matrices in the binary case. Then, we propose two classes of deterministic binary measurement matrices based on finite geometry. Two further improved lower bounds of spark for the proposed matrices are given to show their relatively large sparks. Simulation results show that in many cases the proposed matrices perform better than Gaussian random matrices under the OMP algorithm.