Augmented Lagrangian Based Dictionary Learning Algorithm And Its Application To Medical Imaging And Image Processing

Sparse representation of image patches with adaptive dictionary learning is originally from issue of "effective coding hypothesis". Since the model has been first published in Nature in 2001, it attacted attentions of many researchers. The mainstream research direction is to update the dictionary learning and sparse coding stage alternatively. This thesis proposes a series of novel dictionary learning methods based on augmented Lagrangian scheme. Compared to conventional algorithms they alleviate two drawbacks: sensitivity to initial values and heavy computation load. These proposed methods have been successfully used in medical imaging and image processing with the "ill-posed" feature, particularly in image denoising, image deconvolution and MRI reconstruction. The main work and innovations are as follows:Speaking from the theoretical framework, to overcome the problems in the traditional dictionary learning approaches that are sensitive to initial values and bring heavy computation load, we try to propose fast and efficient augmented Lagrangian-based dictionary learning algorithms. The main idea of these methods is to let the dictionary updated after each inner iteration of augmented Lagrangian. According to the equivalence between augmented Lagrangian and the Bregman iterative method proposed recently, assuming that each step as a scale, then the main advantage of this update satretage is that per iteration can be viewed as a dictionary refinement operation. The iterative refinement procedure makes the dictionary updated from low scale to high. On one hand, the optimization pathway makes the algorithm largely avoid the potential of falling into local optimum. On the other hand, from the point of iteration convergence view, the objective function value and PSNR values change very quickly in the initial steps, the dictionary also changes intensely during the first few steps. The algorithms have good convergence properties.Within this theoretical framework, these methods are applied to a variety of models: 1) In image denoising, the constrained and unconstrained optimization models are considered. Particularly for the non-constrained model, we extend it to a general mode, and use it for Gaussian and salt pepper noise callenation, respectively. Compared to other existing methods, this method has made significant improvements in the reconstruction result and calculation time. 2) In image deconvolution, for the unconstrained optimization model, the algorithm is finally decomposed to alternative minimization by derivation, that is, it alternatively updates the image patches, image solutions itself, and the dictionary. Numerical experiments show that our algorithm is comparable or even superior to the state-of-the-art methods, saving six times of compuatation load. 3) In MRI reconstruction, we consider the constrained optimization model. In particular, the priori information of solution constraint is considered, by taking advantage of the excellent expansion capability of the augmented Lagrangian, to get better reconstruction result. Compared with other algorithms, the parameter of the extened method is very robust due to the nature of Bregman iteration method, which is very beneficial in applications such as reconstruction. Numerical experiments show the superiority of the method to total variation (TV) and other wavelet-based algorithms, the gap of the peak signal to noise ratio (PSNR) between them is even as high as 14dB sometimes.