Constructing circulant measurement matrix through alternating optimizing amplitudes together with chaotic stochastic phases of the matrix generating elements

Abstract

Circulant measurement matrix has been widely used in compressive sensing because of its high-speed discrete convolution algorithm. The typical work of optimizing circulant measurement matrix was introduced by Wotao Yin in Reference [16]. Motivated by his work, the construction of circulant measurement matrix in this paper is explored from the view point of generating elements' amplitudes and phases; and the optimal construction procedures are proposed based on alternately optimizing amplitudes in conjunction with chaotic stochastic phases of the matrix generating elements. The main idea of this paper is based on two innovations: The first one is to reduce the mutual coherence between column vectors of equivalent dictionary by alternately optimizing the generating elements' amplitudes, thus improving the recovery performance of the circulant measurement matrix. From the different expressions of the circulant matrixes of one-dimensional and two-dimensional signal, by setting the Welch bound for the coefficient of mutual coherence between the column vectors of equivalent dictionary as the approximation objective, two novel unified mathematical models are derived from the optimizing function for generating elements' amplitudes of the two different matrixes. Optimal solutions for generating elements' amplitudes are gained by alternately optimizing method. The second innovation is to construct the generating elements' phases of circulant measurement matrix by utilization of a chaotic sequence with independent property. The chaotic stochastic phase of the circulant measurement matrix generating elements are generated by taking advantage of the chaotic internal certainty, which means an independent identically-distributed randomness sequence can be produced by the chaotic map with the initial value at certain sampling distance. At the same time, the external randomness of chaotic sequence can satisfy the stochastic requirement of circulant measurement matrix. This paper presents the method of constructing chaotic stochastic phase using Cat chaotic map. Experimental results of one-dimensional and two-dimensional signals in the optimized circulant measurement matrix are studied in this paper, which has a better performance as compared with the results of conventional circulant measurement matrixes, such as Gaussian circulant matrix and optimized circulant matrix proposed by Wotao Yin. The column vectors of equivalent dictionary in the optimized circulant measurement matrix have lower mutual coherence, this is the essence of the superiority of the optimized circulant matrix.