Research On Optimized Projection Matrix In Compressed Sensing

Compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical Shannon-Nyquist Theorem. In recent years. CS has attracted widely attention of signal processing society. It has widely application in communication, image processing, blind source separation and pattern recognition. Practically, CS is one of the most popular research areas in signal processing society, due to its important theoretical value and widely application.In CS, projection matrix and dictionary influence the sparse recovery accuracy. Ran-dom projections were used since they present small coherence with almost any orthogonal dictionary. It was proved to be optimal in probabilistic sense. Recent researches show that optimizing the projection matrix toward decreasing the coherence is possible and can improve the reconstruction performance of CS. For a new sparse model, when the dictio-nary has special structure, the optimization of the projection matrix should be developed. For example, high-dimensional error correcting model and distributed compressed sens-ing (DCS) require the the optimization of the projection matrix should adapt the special structure of the dictionary. This thesis researches on the optimization of the-projection matrix in CS, the main contributions are as follows.1. To enhance the performance of CS-based high-dimensional sparse error correcting, PSO (Particle Swarm Optimization)-based algorithm to optimize the projection matrix is used to overcome the computational difficulty in high-dimensional cases. CS-based cross-and-bouquet (CAB) model was proposed by J. Wright et al. to reduce the com-plexity of sparse error correcting, where the random Gaussian projections are used. For the sake of leading to better performance of CS-based decoding for the CAB model, an algorithm is proposed in this thesis for constructing a well-designed projection matrix to minimize the average measures of mutual coherence. One was proposed by M. Elad, but it is not suitable for the high-dimensional cases. Another is proposed by this thesis for high dimensional cases. Using the equivalent dictionary, the dimensionality is reduced. Also, high-dimensional Singular Value Decomposition (SVD) is avoided in the procedure of constructing a well-designed projection matrix. The high-dimensional CAB model of sparse error correcting can be solved by the proposed algorithm without computational difficulty. At last, the validity of the proposed algorithm is illustrated by decoding ex-periments in high-dimensional cases.2. For the common CS model, this thesis gives a low-rank Gram matrix-based algorithm to optimize the projection matrix. Bring the multiplication of the projection matrix and the dictionary to be near an equiangular tight frame (ETF) was proposed as an idea in some previous works. Here, a low-rank Gram matrix model is introduced to realize it. Also, an algorithm is presented motivated from the computation method of the matrix nearness for low-rank cases. Simulations show that the proposed algorithm is better than some other algorithms to optimize the projection matrix in terms of image fusion and image denoising via sparse representation.3. The DCS theory rests on a new concept called multiple signals have joint sparse representation (JSR). These signals form a signal ensemble. Three joint sparsity models JSM-1, JSM-2and JSM-3were presented. In JSM-1model, all signals in one ensemble have a common sparse component, and each individual signal owns an innovation sparse component. The JSR offers lower computational complexity compared with the SR in dealing with multiple signals. This thesis proposed a novel dictionary learning method (MODJSR) whose dictionary updating procedure is derived employing the JSR structure with only once eigen value decomposition operation. Indeed, the MODJSR is the SR-based method of optimal directions (MOD) extended for the JSR. The MODJSR has lower complexity than the K-SVD algorithm which often used. To capture the image details more efficiently, this thesis extended the JSR to the GJSR (generalized joint sparse representation). The JSR models the common component and the innovation component by one dictionary while the GJSR depends on two dictionaries. The MODJSR is extended to MODGJSR in this case. For the JSR-based image fusion, this thesis gives a new fusion rule. MODJSR/MODGJSR can carry out dictionary learning, denoising and fusion of noisy source images, simultaneously. Some experiments on image fusion are given to demonstrate the validity of the proposed GJSR model, the MODJSR/MODGJSR and the new fusion rule.4. Based on the GJSR, the DCS is extended to generalized distributed compressed sensing (GDCS) in this letter. In this thesis, the minimization of the mutual coherence is summarized in the minimization of a non-convex function by the structure of the equiv-alent dictionary. Originally, this thesis proposed an algorithm to optimize the projection matrix for the GDCS using the structure of its equivalent dictionary. The algorithm belongs to the gradient method, its stepsize is chosen as Barzilai-Borwein stepsize. The algorithm is extended to block sparse model. The validity of the proposed algorithm is illustrated by some experiments for synthesized signals and real-world image fusion.Above all, the optimized projection matrix designing algorithms in CS are studied in this thesis. Firstly, an optimized projection matrix designing algorithm is given for CS-based high-dimensional sparse error correcting model. Secondly, a low-rank model based optimized projection matrix designing algorithm is proposed for the common CS model. Moreover, this thesis gives a dictionary learning method for the JSR. The JSR is extended to the GJSR. At last, an algorithm to optimize the projection matrix for the GDCS is proposed. Simultaneously, numerical experiments are given to illustrate the efficiency of the proposed algorithms in this thesis.