Semi-Proximal Alternating Direction Method Of Multipliers With Applications In Image Processing

In 1980s, with the rapid development of magnetic resonance imaging technology based on nuclear magnetic resonance theory, imaging and biochemistry research has made revolutionary progress. Transform invariant low rank textures extractor is an efficient method for extracting geometric information and texture from the given image. The semi-proximal alternating direction method of multipliers makes multi-block convex composite quadratic programming easy to be solved relying on its unique cycle structure. In this thesis, we focus on the semi-proximal alternating direction method of multipliers for the magnetic resonance imaging problem and transform invariant low rank textures problem,analyze their convergence properties, and test their numerical performances in some image problems.In Chapter one, we review magnetic resonance imaging model and transform invariant low rank textures model, summarize some well-known algorithms for solving their solu-tions; and simply introduce iterative form of alternating direction method of multipliers for solving 2-block separable convex optimization problems, then give the iterative form and convergence theorem of the semi-proximal alternating direction method of multipli-ers with symmetric Gauss-Seidel technique for multi-block separable convex optimization problems. Finally, we state the main contributions of this thesis, and list some symbols and concepts which used in the sequent chapters.In Chapter two, we present the dual of magnetic resonance imaging problem and pro-pose our algorithm based on the semi-proximal alternating direction method of multipliers with symmetric Gauss-Seidel technique. Under some mild conditions, we show that our proposed algorithm is equivalent to 2-block semi-proximal alternating direction method,and then the convergence property of the proposed algorithm is analyzed. Finally, we illustrate the superiority of the proposed algorithm by numerical experiments.In Chapter three, we propose our algorithm for transform invariant low rank textures problem based on the semi-proximal alternating direction method of multipliers with sym-metric Gauss-Seidel technique. Under some certain conditions, the convergence property of the proposed algorithm is analyzed and experimental results are reported to explain the effectiveness of our proposed algorithm.In Chapter four, we conclude the thesis by listing some remarks and some further research topics.