Convex Variational Methods and Optimization Techniques for Image Processing

Abstract

[eng] Image processing problems have emerged as essential in our society. Indeed, in a world where technology and computers play a central role, images are always present in our everyday life. Therefore, the necessity of improving the quality of certain images, such as medical images or remote sensing images, has called the attention of contemporary mathematics. Several mathematical techniques can be used in order to process an image. We analyze the basics on variational methods for image processing problems, from the Calculus of Variations to Convex Analysis and Continuous Optimization. These items are preceded by a general overview on Functional Analysis topics. Lastly, we consider the problems of denoising and conditioned interpolation of digital images. We will propose several variational models to solve them and perform several experimental results to compare their efficiency.