Research And Design Of Measurement Matrix For Compressed Sensing

Compressed Sensing is an emerging methodology for data acquisition and signal recovery, which states that a sparse or compressible signal that only contains a few nonzero elements can be reconstructed from a small number of random linear measurements. Different from traditional sampling and compressing schemes, CS captures and compresses the signal at the same time and can reconstruct signal using optimal reconstruction algorithms at a sampling rate below the Nyquist rate. The structure of measurement matrix has a significant influence on the computational complexity and recovery performance. Therefore it is meaningful to study the properties of current CS measurement matrices and design a new measurement matrix which is more applicable to CS. This paper studies the measurement matrix of Compressed Sensing from the principle, development and applications, and proposes novel structured measurement matrices to solve the problems of existing matrices.1. In order to solve the problems of high computational complexity and difficulty in hardware realization of current measurement matrices, a novel structured measurement matrix referred to as block-based sparse strip-type circulant(BSSC) matrix is proposed. It supports independent block processing, and each block is a highly-sparse circulant matrix of strip-type that generated by horizontally cyclic shifting a sparse Toeplitz matrix with random shift amount. It takes entries from {0, 1,-1}, which is easy in implementation. The nature of circulation and sparsity significantly reduces sensing complexity along with storage requirement. Theoretical analysis proves it satisfies the RIP with overwhelming probability which guarantees exact recovery from compressed measurements. The simulation results validate that the BSSC matrix achieves higher recovery accuracy than Gaussian random matrix with significantly lower sensing complexity.2. To reduce the storage and computational complexity of Compressed Sensing in imaging application, especially in medical imaging application, a fast and efficient sensing operator is proposed with a block-based sparse circulant matrix. Based on the block processing operator, each block image can be compressed and reconstructed independently in a real time. Compared with the most widely used Block Hadamard matrix in 2D image processing, using the proposed measurement matrix can highly reduce the storage and computational complexity as well as the processing time in terms of the same reconstruction performance. Besides, with the proposed sensing operator, it doesn’t need to acquire the whole image before sampling, therefore it is more applicable to CS image processing.