Researches On Nonlinear Compressed Sensing Theory Based Sparse Hidden Space

With the rapid development of science and technology, the amount of information is increasing rapidly. In particular, the rapid growth of people's needs makes the signals needed to be deal with more and more complex, dimension of signals increasing, and also the signal bandwidth increasing, the sampling rate is also higher and higher in signal processing, which proposes a very serious challenge to the traditional sampling system based on the principle of Shannon- Nyquist Sampling Law. In 2006, the compressed sensing theory is put forward to effectively solve the problem of sampling rate, the theory can accurately recovery the original signals from fewer observations at the rate that is far lower Nyquist sampling rate. After ten years of development, the Compressed Sensing Theory has become mature.The current methods of Compressed Sensing are based on the explicit linear sparse representation model. Linear sparse representation model is simple and intuitive, easy to understand, easy to operate and so on. However, explicit linear models assume that various transform of all the signals is linear, and this is not true with the real situation of the natural environment. Real scenarios are more complex, and simple linear transformation cannot fully express the actual signal. And because linear sparse representation model obtain the sparse coefficient with Higher sparse degree, more measured values are required to reconstruct the original signal. At the same time, in signal processing with Compressed Sensing, the majority of cases are based on random observations. This kind of random observation matrix cannot be correlated with most of the sparse dictionary, and can get good results, but it still belongs to the non-adaptive observation matrix. The advantage of the non-adaptive observation matrix is universal, but for different signals, it does not have to be targeted, and therefore it is likely to lack a certain kind of information which is unique to the signal. In the application of nonlinear compressed sensing model, the existing nonlinear dictionary learning method is less, and the quality of the dictionary learning algorithm is very important for the results of the compressed sensing. In view of the above problems, this paper studies the nonlinear compressive sensing theory based on Sparse Hidden Space. Under the basic nonlinear framework, with the kernel method, deep the basic module of Compressed Sensing, and put forward a new method, to further enhance the reconstruction results of signals. Specific work is as follow:Firstly, the random observation matrix is universal and cannot be related to most of the orthogonal dictionaries. However, the random observation matrix cannot guarantee that it is not related to the sparse dictionary which is acquired by Dictionary Learning. In view of the above problems, with the method of coupling observation in linear space, we propose a nonlinear compression imaging algorithm based on Gram Matrix observation optimization. According to different sparse dictionaries, this algorithm can construct a more optimized observation matrix. At the same time, it is ensured that the observation matrix is not related to sparse dictionary, and it meets the Restricted Isometry Property. We apply the Nonlinear Gram Matrix observation optimization algorithm to the Nonlinear Compressed Sensing model to deal with the multidimensional data, and have some experiments on three groups of high spectral data. The experimental results show, compared with the situation of the fixed Gauss random matrix as the observation matrix, the reconstruction effect of the proposed algorithm is better, its PSNR is increased by 1~2d B, and MSE is greatly reduced.Secondly, KPCA is a very important method for dictionary learning under the nonlinear sparse representation model. But the KPCA method only uses two order statistical information of the image, and the KICA(kernel independent component analysis) method can not only extract the image irrelevant features, the higher-order statistical information of the image are also used. In view of the above problems, we propose a nonlinear compression imaging algorithm based on KICA. After signal processing with the regular KICA dictionary learning method, the final data components obtained not only removes the correlation, each other is statistically independent, and is not Gauss distribution. The proposed method is superior to KPCA. We have some experiments on three groups of high spectral data with this algorithm. The experimental results show, the algorithm can obtain better reconstruction effect, and its PSNR is increased in the range of 1~4d B, compared with different methods(KPCA, KKSVD and KMOD).Thirdly, at present, the dictionary learning methods based on the nonlinear sparse representation model, are kernelization of the dictionary learning method in the linear model. However, not all of the methods can be applied to the higher dimensional feature space by simple "kernelization". On the other hand, the size of the kernel matrix constructed by the kernel function depends on the number of samples, which is often large. But it needs to be kept in the operation process, and it will bring higher time complexity and storage space complexity. In view of the above problems, we propose a nonlinear compression imaging algorithm based on Linear Kernel Dictionary Learning. Linear Kernel Dictionary Learning method, not only can reduce the complexity of some of the existing methods of nonlinear dictionary learning, but also can be popularized and used, and it is suitable for all the dictionary learning methods under the linear model. Experimental results show, the proposed algorithm can improve the reconstruction effect in a certain extent, and the experimental running time has shrunk by about half.