Stochastic Partial Differential Equations for Computer Vision with Uncertain Data

In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be-and more and more frequently are-taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate.

The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness.

This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.

Tobias Preusser studied mathematics at the University of Bonn and at New York University. He received his Ph.D. from the University of Duisburg with a thesis on anisotropic geometric diffusion in image processing and his habilitation in Mathematics from the University of Bremen with a thesis on image-based computing. He is a professor for the mathematical modeling of biomedical processes at Jacobs University Bremen, head of the modeling and simulation group, and member of the management board at the Fraunhofer Institute for Medical Image Computing MEVIS. On the one hand, his research interests include modeling and simulation of bio-medical processes with partial differential equations (PDEs), mathematical image processing, and scientific visualization based on PDEs. On the other hand, his research is driven by concrete and complex application problems in medicine. Robert M. (Mike) Kirby received an M.S. degree in applied mathematics, an M.S. degree in computer science, and a Ph.D. degree in applied mathematics from Brown University, Providence, Rhode Island, in 1999, 2001, and 2002, respectively. He is currently a Professor of Computing and Associate Director of the School of Computing, University of Utah, Salt Lake City, where he is also an Adjunct Professor in the Departments of Bioengineering and Mathematics and a member of the Scientific Computing and Imaging Institute. His current research interests include scientific computing and visualization. Torben Patz received his diploma degree in mathematics from the University of Bremen, Germany, in 2009 with a thesis focusing on the numerical simulation of radio-frequency ablation and his Ph.D. from Jacobs University Bremen, Germany, in 2009. In his Ph.D. thesis, he focused on the segmentation of stochastic images with stochastic PDEs, laying the basis for the book at hand. He was a postdoctoral fellow at Jacobs University Bremen while writing the book and is now a research scientist at the Fraunhofer Institute for Medical Image Computing MEVIS. His research interests include uncertainty modeling and propagation in medical applications as well as software support for interventional radiology. University of California, Berkeley