Research On Construction And Optimization Of Measurement Matrix For Compressed Sensing

Compressed Sensing (CS) theory is a new breakthrough to the the traditional Nyquist sampling theory. It is no longer respect to the bandwidth, but the sparse or compressibility of the signal. CS offers a joint compression and sensing processes. Measurement matrix is the basis of the CS, and runs through the whole process of sampling and reconstruction. Its performance plays a key role because it can directly affect the integrity of the data and the accuracy of the reconstructed signal. So the research on measurement matrix is of great significance. Based on the further study of the existing measurement matrix’s construction methods and optimization methods, this paper has conducted an exploratory and innovative research.The RIP criterion and the incoherent discrimination theory, which are the two key guiding theories for the measurement matrix’s construction, are first studied. And on this basis, we do an in-depth research and analysis on commonly-used measurement matrix’s construction method and optimization method. And then, we classify them into separate categories and summarize their respective advantages and disadvantages. These works provide a solid theoretical basis for further improvement and putting forward the innovative method.In the aspect of measurement matrix construction, we study the deterministic measurement matrix construction methods which have been the mainstream direction. However, deterministic measurement matrix performs not so well as random measurement matrix in signal reconstruction. To solve this problem, a new deterministic measurement matrix, called blocked ordered Vandermonde matrix, is proposed in this paper. Blocked ordered Vandermonde matrix is constructed on the basis of Vandermonde matrix, whose vectors are linearly independent. Then block operation is taken and its elements are sorted. This new measurement matrix realizes non-uniform sampling in time domain and is specifically suitable for natural images whose dimension is usually high. Simulation results show that this matrix is much superior to Gaussian matrix in image construction, and could be used in practice.In the aspect of measurement matrix optimization, we mainly care about the iterative optimization algorithm whose reconstruction precision is relative high. But this optimization algorithm usually requires hundreds of iterations to be convergence. So the algorithm computational complexity is generally on the high side. We proposed an optimization algorithm based on the power curve projection. This algorithm has a strong ability of projection. It could project all the sensing matrix’s correlation coefficients into smaller values and basically without iterative. So it greatly reduces the computational complexity, but without reducing the measurement matrix’s reconstruction precision. The experiment shows that, the optimization algorithm based on the power curve projection has the advantages of few iterations and high reconstruction precision.