Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. 

Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

刘继军，东南大学二级教授，博士生导师，享受国务院政府特殊津贴专家。2009,12-2014,11任东南大学数学系主任、东南大学理学部副主任。现任东南大学丘成桐中心常务副主任，中国工业与应用数学学会常务理事，中国计算数学学会常务理事，全国大学生数学建模竞赛组委会委员，中国工业与应用数学学会数学建模竞赛专业委员会委员，江苏省工业与应用数学学会理事长，江苏省计算数学学会副理事长。国家精品资源共享课《数学建模与数学实验》主持人。长期从事数学物理反问题、大规模科学计算和介质成像的数学理论和方法的研究。主持完成国家自然科学基金重大研究计划培育项目、面上项目、国际合作项目、天元基金、中国博士后科学基金、江苏省自然科学基金等项目的研究。已在SIAM J. Appl. Math., SIAM J. Sci. Comput., Inverse Problems, Inverse Problems and Imaging， Science China Mathematics,  J. Comput. Maths., J. Sci. Comput.等发表学术论文100余篇，在科学出版社出版学术专著2本。曾受中国NSFC、德国DAAD、韩国21Brain Project等资助赴国外开展合作研究。2012-2017年任Inverse Problems in Sciences and Engineering编委，2018年起任J. Inverse and Ill-posed Problems编委。