Research On Deterministic Measurement Matrix For Compressed Sensing

Compressed Sensing (CS) is a novel signal sampling theory for sparse or compressible signals. It complements signal sampling and data compression at the same time. Measurement matrix plays a significant role in data sampling and signal reconstruction. Therefore, the research on measurement matrix for compressed sensing is of great importance. In this paper, we pay more attention to deterministic measurement matrix due to its advantages in hardware implementation. The main contributions on deterministic measurement matrix are summarized as follows.Based on the comparison and analysis of the common measurement matrixes, we mainly focused on Toeplitz measurement matrix. According to the distribution characteristics of one-dimensional signals and two-dimensional images in transformed domain, we proposed to adjust partial elements in Toeplitz measurement matrix through multiplying weighted coefficients in order to enhance the low frequency sampling. We called the new measurement matrix generalized rotation matrix. The simulation results show that the generalized rotation matrix performs better on reconstruction accuracy than other measurement matrixes.In order to further improve the performance of deterministic measurement matrix and satisfy the requirement of reconstruction algorithms, we proposed to introduce cycle direct product method into the construction of deterministic measurement matrix. The method is that starting from a small number of low-dimensional orthogonal "seed" vectors, we can obtain high-dimensional orthogonal matrix by cycle direct product and orthogonal normalization. And then the measurement matrix can be obtained by choosing some row vectors from the orthogonal matrix. We referred this kind of deterministic measurement matrix to the generalized Hadamard matrix. Simulation results show that the generalized Hadamard matrix is as efficient as Gaussian measurement matrix on reconstruction accuracy. Furthermore, it takes less time and needs less storage space to construct the generalized Hadamard matrix, which makes it more friendly to hardware design.Based on the above research on deterministic measurement matrix, we consider whether can improve the speed of constructing a measurement matrix through avoiding the redundant calculation. In this paper, with the further analysis of measurement matrix construction method, we proposed a new dynamic measurement matrix construction method based on an orthonormal basis. The basic idea is as follows. Firstly, an orthonormal basis is constructed and acts as the first M columns of measurement matrix. Then the remaining columns of measurement matrix are constructed by linear representation of the coefficient sequences generated by pseudo-random algorithms. Simulation results show that this method could increase the speed of constructing matrix and the obtained measurement matrix performs as well as Gaussian matrix on construction accuracy.