Construction of a circulant compressive measurement matrix based on chaotic sequence and RIPless theory

Abstract

Construction of a compressive measurement matrix is one of the key technologies of compressive sensing. A circulant matrix corresponds to the discrete convolutions with a high-speed algorithm, which has been widely used in compressive sensing. This paper combines the advantages of chaotic sequence with circulant matrix to propose a circulant compressive measurement matrix based on the chaotic sequence. The elements of a chaotic circulant measurement matrix are generated by taking advantage of the chaotic internal certainty, i.e. the independent identically distributed randomness sequence can be produced by the chaotic mapping formula using the initial value and a certain sampling distance. At the same time, the external randomness of chaotic sequence can satisfy the stochastic requirements of compressive measurement matrix. This paper presents the method of constructing chaotic circulant measurement matrix using a Cat chaotic map and the test method for RIPless property of the matrix. Measurement results of one-dimensional and two-dimensional signals using the chaotic circulant measurement matrix are studied and are compared with the results of conventional circulant measurement matrix. It can be shown that the chaotic circulant measurement matrix has good recovery results for both one-dimensional and two-dimensional signals. Moreover, it may get better results than the traditional matrix for the two-dimensional signal. From the point of view of phase diagram, the essential reason of chaotic circulant measurement matrix outperforms the conventional one is its integration of internal certainty with the external randomness of the chaotic sequence.