The Research Of The Inverse Problem Based On The Analysis Sparse Priori

The inverse problem is to obtain the system parameter or recovery the original input data from the observed data. The research on the inverse problem could be widely used in the fields of image recovery, blind speech separation and medical imaging, etc. Due to the illness of inverse problems, the solutions of them are hard to obtain compared with the forward problem. Generally, the sparse regularization term could be introduced to solve the inverse problem.There are two models for signal sparse representation, one is the synthesis sparse model and another is the analysis sparse model. The synthesis sparse model is researched for many years and got some achievements. Recently, the analysis sparse model has got more and more attention, because it has potential application value. To solve the inverse problem based on the analysis sparse priori, two problems are encountered, one is learning analysis dictionary and the other one is estimating the original signal. Most existing analysis dictionary learning algorithms need to pre-estimate the original signals for learning dictionary. This rends to a computationally slow optimization process and potentially unreliable estimation(if the noise level is high). With the learned analysis dictionary, the method to estimate original signals based on the synthesis sparse model can’t be applied to estimate the original signal based on the analysis sparse prior. So the method to estimate the original signal based on the analysis sparse model is need to be explored. The main contributions of this dissertation are summarized below:1. We proposed the orthogonality constrained analysis dictionary learning with iterative hard thresholding algorithm. In the proposed algorithm, the observed data is directly used to compute the analysis dictionary. This leads to a computationally very efficient algorithm. Moreover, the orthogonality constraint is enforced to avoid the trivial solution in the proposed algorithm.2. The weighted split Bregman algorithm is proposed to estimate the original signal based on the analysis sparse model. The 1l norm is replaced by the weighted 1l norm as the sparse regularization term to improve the performance of analysis sparse representation, and then the accuracy of the estimated original is improved too. At last, the convergence of the proposed algorithm is proved in theory.