Variational PDE Models in Image Processing, Volume 50, Number 1

14 NOTICES OF THE AMS VOLUME 50, NUMBER 1 I mage processing, traditionally an engineering field, has attracted the attention of many mathematicians during the past two decades. From the point of view of vision and cognitive science, image processing is a basic tool used to reconstruct the relative order, geometry, topology, patterns, and dynamics of the three-dimensional (3-D) world from two-dimensional (2-D) images. Therefore, it cannot be merely a historical coincidence that mathematics must meet image processing in this era of digital technology. The role of mathematics is determined also by the broad range of applications of image processing in contemporary science and technology. These applications include astronomy and aerospace exploration, medical imaging, molecular imaging, computer graphics, human and machine vision, telecommunication, autopiloting, surveillance video, and biometric security identification (such as fingerprints and face identification). All these highly diversified disciplines have made it necessary to develop common mathematical foundations and frameworks for image analysis and processing. Mathematics at all levels must be introduced to address the crucial criteria 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. This article gives a broad picture of mathematical image processing through one of the most recent and very successful approaches—the variational PDE (partial differential equation) 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 that we personally have worked on: inpainting and segmentation. By no means, however, do we intend to give a comprehensive review of the entire field of image processing. Many of the authors’ articles and preprints related to the subject of this paper can be found online at our group homepage [11], where an extended bibliography is also available. Image Processing as an Input-Output System Directly connected to image processing are two dual fields in contemporary computer science: computer vision and computer graphics. Vision (whether machine or human) tries to reconstruct the 3-D world from observed 2-D images, while This paper is based on a plenary presentation given by Tony F. Chan at the 2002 Joint Mathematics Meetings in San Diego. The work was supported in part by NSF grants DMS-9973341 (Chan), DMS-0202565 (Shen), and ITR0113439 (Vese); by ONR grant N00014-02-1-0015 (Chan); and by NIH grant NIH-P20MH65166 (Chan and Vese).