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Mathematical Problems in Image Processing

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It is surprising when we realize just how much we are surrounded by images. Images allow us not only to perform complex tasks on a daily basis, but also to communicate, transmit information, represent and understand the world around us. Just think, for instance about digital television, medical imagery, video-surveillance, etc. The tremendous development in information technology accounts for most of this. We are now able to handle more and more data. Many day to day tasks are now fully or partially accomplished with the help ofcomputers. Whenever images are involved we are entering the domains ofcomputer vision and image processing. The requirements for this are reliability and speed. Efficient algorithms have to be proposed to process these digital data. It is also important to rely on a well-established theory to justify the well-founded nature of the methodology.

Amongst the numerous approaches which have been suggested, we focus on Partial Differential Equations (PDEs), and Variational Approaches in this book. Traditionally applied in physics, these methods have been successfully and widely transferred in Computer Vision other the last decade. One ofthe main interests in using PDEs is that the theory behind the concept is well-established. Ofcourse, PDEs are written in a continuous setting refering to analog images, and once the existence and the uniqueness have been proven, we need to discretize them in order to find a numerical solution. It is our conviction that reasoning within a continuous framework makes the understanding ofph ysical realities easier and stimulates the intuition necessary to propose new models. We hope that this book will illustrate this idea effectively.

The message we wish to put over is that the intuition which leads to certain formulations and the underlying theoretical study are often complementary. Developing a theoretical justification ofa problem is not simply “art for art sake”. In particular, a deep understanding of the theoretical difficulties may lead to the development ofsuitable numerical schemes or different models.
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