An impressive range of techniques have been developed in the fields of image processing, computer vision and computer graphics to manipulate and interpret image content for a variety of applications. So far, only a few of these techniques have been applied in the context of vision aid systems and even less have been carefully evaluated with patients. In the the team we have a particular expertise in PDEs and variational approaches for image processing.
PDE-IP library: A CImg plugin for image processing using partial differential equations and variational approaches
Back in the 90s, the domain of partial differential equations (PDE) appeared to be a promising mathematical framework to process images. In 2002, we wrote the first book in this area explaining in detail the techniques of functional analysis and the theory of partial differential equations to study several fundamental questions in image processing. The message we wanted to convey is that the intuition that leads to certain formulations and the underlying theoretical study are often complementary. Developing a theoretical justification of a problem is not simply “art for art’s sake.” In particular, a deep understanding of the theoretical difficulties may lead to the development of suitable numerical schemes or different models. To allow readers have a better feeling about numerical schemes and explore how state-of-the-art approaches are implemented, we developed the PDE-IP library which has been described in the second edition in 2006. It contains a selection of methods presented in the book (e.g., image enhancement, restoration, segmentation) implemented in C++ using the CImg library.
This book is concerned with the mathematical study of certain image processing problems. Thus we target two audiences:
- The first is the mathematical community and is achieved by showing the contribution of mathematics to this domain by studying classical and challenging problems which come from Computer Vision. It is also the occasion to highlight some difficult and unsolved theoretical questions.
- The second is the Computer Vision community: this is done by presenting a clear, self-contained and global overview of the mathematics involved for the problems of image restoration, image segmentation, sequence analysis and image classification.
We hope that this work will serve as a useful source of reference and inspiration for fellow researchers in Applied Mathematics and Computer Vision, as well as being a basis for advanced courses within these fields.
Reviews of the first edition
- “Mathematical Problems in Image Processing is a major, elegant, and unique contribution to the applied mathematics literature, oriented toward applications in image processing and computer vision. … Researchers and practitioners working in the field will benefit by adding this book to their personal collection. Students and instructors will benefit by using this book as a graduate course textbook.” (SIAM Review)
- “The Mathematician—and he doesn’t need to be a ‘die-hard’ applied mathematician—will love it because there are all these spectacular applications of nontrivial mathematical techniques and he can even find some open theoretical questions. The numerical analyst will discover many challenging problems and implementations. The image processor will be an eager reader because the book provides all the mathematical elements, including most of the proofs. …/… Both content and typography are a delight. I can recommend the book warmly for theoretical and applied researchers.” (Bulletin of the Belgian Mathematics)
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