Research On The Primal And Dual Alternating Direction Methods Of Multipliers For Compressive Sensing Image Reconstruction

It is well known that,the compressive sensing theories provide the possibilities of accurately reconstructing images from highly undersampled data.The research on the problem of image reconstruction from undersampled data with Gauss noise have made abundant achievements.The alternating direction method of multipliers(ADMM)decom-poses the separable convex optimization problem into several solvable small sub-problems,and solves the each subproblem individually.In this thesis,we aim to employ the semi-proximal ADMMs with symmetric Gauss-Seidel(sGS)technique to solve the image re-construction problems corrupted by impulsive noises,analyze their convergence,and test their numerical effectiveness.In Chapter one,we briefly introduce the image reconstruction problem and some re-lated models,then summarize some typical algorithms subsequently.We summarize some preliminaries in optimization,and give the semi-proximal ADMM for two-block separable convex minimization problems.We also recall the sGS based ADMM for multiple-block separable minimization problems.Finally,we state the main motivation and contributions of this thesis,and list some symbols which will be used in the subsequent developments-In Chapter two,we construct the images reconstruction model at the case of impulsive noises,and then use a sGS based semi-proximal ADMM for its solution.Under certain conditions,we show that the proposed algorithm is equivalent to the two-block semi-proximal ADMM,which indicates that its convergence is guaranteed.In Chapter three,we derive the dual model of image reconstruction model proposed in Chapter two,and then implement the sGS based semi-proximal ADMM for its solution.We also show the equivalence between the proposed algorithm and the two-block semi-proximal ADMM with the special proximal term to ensure the algorithm's convergence theoretically.In Chapter four,we test the algorithms' numerical efficiency proposed in Chapter two and Chapter three using the simulated "Shepp-Logan" image and some magnetic reso-nance images.We test both algorithms under different sampling rates and do performance comparisons with the solver RecPF.In Chapter five,we conclude the thesis by listing some remarks and some further research topics.