The Study Of Sparse Representation Of Signals And Its Applications

One of the key points in signal processing is to represent signal in an effective way. Especially in the age of big data, concise representation of the data has become a hot issue in the field of signal processing. Sparse representation of signals is a novel signal representation method, which represents signal in the way of sparse approximation instead of original signal to reduce costs of signal processing. The proposed method improves the efficiency of signal processing. Sparse representation of signals is developed in the last century. As one of the hot topics in the field of signal processing, sparse representation has been utilized to deal with various tasks such as signal coding, hyperspectural unmixing, and blind source separation. The sparse representation model is powerful because of its solid theory foundation and its additivity to various data sources. In brief, sparse representation has a bright future in the field of signal processing.This paper is mainly on the dictionary learning and the application in signal cryptosystem of the sparse representation theory,which are based on non-smooth non-negative matrix factorization algorithm and the compressed sensing theory. In this paper, we first introduce researching status in domestic and overseas, then the theory frame of sparse representation is proposed; Moreover, we present a fast nonsmooth nonnegative matrix factorization algorithm and analyze the security of compressed sensing-based signal cryptosystem under the extend Wyner-sense and Shannon-sense perfect secrecy criteria.In this paper, the main contents are as follows:A fast nonsmooth nonnegative matrix factorization algorithm is proposed, in which Nesterov’s optimal gradient method is utilized to optimize each nonnegative least square (NLS) subproblem. In the proposed algorithm, we optimally minimize one matrix with another fixed. In particular, we update two sequences recursively for optimizing each factor, one sequence stores the approximation solutions and another sequence stores the search points. Based on this optimization scheme,our algorithm achieves optimal convergence. Numerical experiments both on synthetic and real-world datasets show the effectiveness of the proposed method, which is an adaptive method of dictionary learning in sparse representation.The security of the CS-based signal cryptosystem is also analyzed. It is conducted in the information theory frame, where the entropy,the conditional entropy, the mutual information (MI) of the ciphertext, key. and plaintext are involved. Moreover, the perfect secrecy criteria (Shannon-sense and Wyner-sense) are extended to measure the security. It is further proved that under some conditions, the key and the plaintext could be cracked partly by using modern information processing technology based on the blind source separation (knowing only the ciphertexts).