Deterministic Construction of Compressed Sensing Measurement Matrix with Arbitrary Sizes via QC-LDPC and Arithmetic Sequence Sets

Abstract

It is of great significance to construct deterministic measurement matrices with good practical characteristics in Compressed Sensing (CS), including good reconstruction performance, low memory cost and low computing resources. Low-density-parity check (LDPC) codes and CS codes can be closely related. This paper presents a method of constructing compressed sensing measurement matrices based on quasi-cyclic (QC) LDPC codes and arithmetic sequence sets. The cyclic shift factor in each submatrix of QC-LDPC is determined by arithmetic sequence sets. Compared with random matrices, the proposed method has great advantages because it is generated based on a cyclic shift matrix, which requires less storage memory and lower computing resources. Because the restricted isometric property (RIP) is difficult to verify, mutual coherence and girth are used as computationally tractable indicators to evaluate the measurement matrix reconstruction performance. Compared with several typical matrices, the proposed measurement matrix has the minimum mutual coherence and superior reconstruction capability of CS signal according to one-dimensional (1D) signals and two-dimensional (2D) image simulation results. When the sampling rate is 0.2, the maximum (minimum) gain of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) is up to 2.89 dB (0.33 dB) and 0.031 (0.016) while using 10 test images. Meanwhile, the reconstruction time is reduced by nearly half.