This book is an introduction to the use of geometric partial differential equations (PDEs) in image processing and computer vision. This relatively new research area brings a number of new concepts into the field, providing, among other things, a very fundamental and formal approach to image processing. State-of-the-art practical results in problems such as image segmentation, stereo, image enhancement, distance computations, and object tracking have been obtained with algorithms based on PDE's formulations.
This book begins with an introduction to classical mathematical concepts needed for understanding both the subsequent chapters and the current published literature in the area. This introduction includes basic differential geometry, PDE theory, calculus of variations, and numerical analysis. Next we develop the PDE approach, starting with curves and surfaces deforming with intrinsic velocities, passing through surfaces moving with image-based velocities, and escalating all the way to the use of PDEs for families of images. A large number of applications are presented, including image segmentation, shape analysis, image enhancement, stereo, and tracking. The book also includes some basic connections among PDEs themselves as well as connections to other more classical approaches to image processing.
This book will be a useful resource for researchers and practitioners. It is intended to provide information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions. One of the main goals of this book is to provide a resource for a graduate course in the topic of PDEs in image processing. Exercises are provided in each chapter to help with this.