Signal Sparse Representation Based On Relevant Dictionary

In recent years, sparse signal representation based on relevant dictionary has awide range of applications, such as radar, medical imaging, hyperspectral imaging,differential optical absorption spectroscopy and so on. But the existing algorithms areapplied in the model that the dictionary satisfies some unrelated conditions, such asthe matching pursuit algorithm, the basis pursuit algorithm, high resolution matchingpursuit algorithm, combinatorial optimization algorithm and greedy algorithm. In thisthesis, we discuss the case of the dictionary is relative, and find an optimizationalgorithm to solve the model and apply it to seismic data recover.When the column vector of random sampling observation matrix satisfies therelative conditions, the non-negative least squares method can get the sparsestsolution. As a general matrix satisfies the restricted isometry property,l1minimization often yields denser solutions. When the column vector of observationmatrix satisfies the uncorrelated conditions,l1minimization can’t get a densesolution any more. If we changel1tol1/l2orl1l2, in this way, we will obtainthe sparsest solution. This model can balance the influence of the coefficients to thesparse solution, in this thesis we also prove strictly the two models can make thesignal represent sparsely. We also improve the theoretical knowledge of signal in therelative dictionary sparse representation. In the sparse signal, the big coefficients havea great influence to the solution, the recovery effect of the signal is not accurateenough, we use the gradient projection approach and the alternating direction methodof multipliers. The two methods can improve the accuracy of the signal recovered andensure the solution is the best. In this thesis, we combine the model with compressedsensing theory and apply them to the seismic data reconstruction.