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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. |