Convex Optimization for Image Reconstruction

Convex optimization is a particular class of mathematical tool in decision sciences and in analyzing physical systems for finding a point that maximizes/minimizes an objective function (subjected to equality constraints and/or inequality constraints) through iterative computations. In this regard, recent developments in the mathematics and computational fields have sparked and positioned convex optimization as one of the most powerful tools to formulate and solve image inverse problems. This chapter briefly introduces state-of-the-art convex optimization methods for such reconstruction problems and presents the most successful approaches and their interconnections. Section 3.1 summarizes the basic concepts of convex optimization, focusing on the convexity conditions; Section 3.2 introduces the principal mathematical optimization tools for image/signal processing, focusing on the role of convex optimization. Section 3.3 discusses a variety of conventional convex optimization-based reconstruction algorithms used in the image processing community. The scope of these mathematical concepts, algorithms, and their applications are extended as needed in the remainder of the book.