Variational PDE Models in Image Processing

Abstract

Abstract : Image processing, a traditionally engineering field, has attracted the attention of many mathematicians during the past two decades. From the vision and cognitive science point of view, image processing is a basic tool used to reconstruct the relative order, geometry, topology, patterns, and dynamics of the 3-D world from 2-D images. Therefore, it cannot be merely a historic coincidence that mathematics must meet image processing in this digital technology era. The role of mathematics is also determined by the broad range of applications of image processing in contemporary science and technology. These include astronomy and aerospace exploration, medical imaging, molecular imaging, computer graphics, human and machine vision, telecommunication, auto-piloting, surveillance video, and biometric security identification (such as fingerprints and face identification), etc. All these highly diversified disciplines have made it necessary to develop the common mathematical foundation and frameworks for image analysis and processing. Mathematics at all levels must be introduced to meet the crucial qualities demanded by this new era-genericity, well-posedness, accuracy, and computational efficiency, just to name a few. In return, image processing has created tremendous opportunities for mathematical modeling, analysis, and computation. In this article, we intend to give a broad picture of mathematical image processing through one of the most recent and very successful approaches - the variational PDE method. We first discuss two crucial ingredients for image processing: image modeling or representation, and processor modeling. We then focus on the variational PDE method. The backbone of the article consists of two major problems in image processing - inpainting and segmentation, which we have personally worked on, but by no means do we intend to have a comprehensive review of the entire field of image processing.