ABSTRACT

Recently, convex optimization has become one of most powerful tools to formulate the introduced problems of many different applications, e.g. image processing, computer vision and machine learning, etc., and develop efficient algorithms, especially the first-order optimization algorithms, for the related large-scale optimization problems. It provides not only an elegant mathematical theory for analysis but also a tractable numerical framework for designing tractable algorithmic schemes. Particularly for medical image analysis, convex optimization has delivered most advanced image processing techniques to discover valuable information for physicians, image guided therapy and surgery, and monitoring of the treated organ during therapy. As the key study, segmentations of organs and lesions from the acquired images are used to quantify volumes and shapes used in diagnosis and monitoring treatment. The challenges from clinical-motivated image segmentation applications introduce novel and sophisticated mathematical problems which inspire developments of advanced optimization and computing methods, especially convex optimization attaining optimum in a global sense, and hence, bring an enormous spread of research topics for recent computational medical image analysis. In addition, distinct from the usual image segmentation, most medical images have a big volume of acquired data, often in 3D or 4D (3D + t), along with great noise or incomplete image information, which pose hard large-scale optimization problems; how to process such poor ‘big data’ of medical images efficiently and solve the corresponding optimization problems robustly are the key factors of modern medical image segmentation.

Keywords: Convex Optimization Medical Image Segmentation Primal-Dual Methods Image Processing.