Research On The Measurement Matrix Of Compressed Sensing Based On Chaos Theory

The compressed sensing theory has broken the limit of the traditional Nyquist sampling theorem,which sampled the signal with a much lower sampling rate than the Nyquist frequency.And,the signal is accurately reconstructed with a small number of observations that contain most of the information of the original signal.The construction of measurement matrix is the key of compression sensing,and its performance directly affects the performance of signal sampling and signal reconstruction.Therefore,it is of great theoretical significance and application value to design a measurement matrix with good performance.Because the sequence generated by the chaotic system is deterministic and pseudorandom,the measurement matrix constructed with this sequence can overcome the instability of the random measurement matrix.Therefore,this paper applies the chaotic system to the construction of the measurement matrix.The main contents of this paper are as follows:Firstly,this paper introduced the compressed sensing theory and chaotic systems theory,and analyzed a new chaotic system-Hybrid chaotic system.It is proved that the measurement matrix generated by the Hybrid chaotic system can be the measurement matrix of compressed sensing,and compared with the Gaussian random measurement matrix and the measurement matrix constructed by Logistic chaotic sequence.Secondly,a image encryption scheme is designed.Based on the properties of Hybrid chaotic system which are deterministic,pseudo randomness,sensitivity and so on,and the advantages of compressed sensing,which can reduce data processing and save storage space,this paper proposed a new image encryption scheme.Experimental simulation analysis shows that the encryption scheme has the ability of low data transmission,sensitive to key response,and resistance to many attacks.Thirdly,a Hybrid chaos-Bernoulli block circulant measurement matrix is designed to avoid the disadvantages of the Hybrid chaotic measurement matrix,such as the large number of independent elements and large storage space.Based on the fact that the Hybrid chaotic sequence is processed by the symbolization function to save storage space,and the merits of the fast algorithm of cyclic matrix and the principle of circular convoluting in two directions when dealing with two-dimensional signals in the cyclic matrix,this paper construct the chaos-Bernoulli block circulant measurement matrix.The measurement matrix is compared with the Gauss random measurement matrix,the Bernoulli random measurement matrix,the chaotic measurement matrix and the Gauss cyclic measurement matrix in one dimension signal and two dimensional signal.The results show that the measurement matrix is feasible.Finally,in order to improve the signal reconstruction quality of the Hybrid chaotic measurement matrix,a measurement matrix optimization algorithm is designed to optimize the Hybrid chaotic measurement matrix and improve its performance.According to that the correlation between measurement matrix and sparse matrix and column independence of sensing matrix affect the signal sampling and recovery,this paper proposed a combination of the variable step size gradient descent method which is used to reduce the correlation between the measurement matrix and the sparse matrix,and the QR decomposition method which is used to improve the column independence of the sensing matrix,to optimize the measurement matrix.The Hybrid chaotic measurement matrix optimized by this optimization algorithm is compared with the unoptimized Hybrid chaotic measurement matrix,the Hybrid chaotic measurement matrix optimized by Elad algorithm and the Hybrid chaotic measurement matrix optimized by Xu algorithm in the two-dimensional signal simulation.The result shows that the optimization algorithm proposed in this paper is obviously better than the other algorithms,so it has certain value.