In many computer vision problems the sought-after object (e.g., an image) is defined as the minimizer of an objective function, called an energy. This minimizer is required to extract information from the recorded data via a data-fidelity term and also to satisfy some specific requirements via a regularization term. These terms are often constructed using calculus of variations or statistics. Independently of this, the features of a minimizer are implicitly determined by shape of the energy. This tutorial will present a systematic approach to the problem of the choice of an energy so that its (local) minimizers exhibit specific properties. It is based on a series of analytical results providing a rigorous knowledge of the main features of a minimizer according to the shape of the energy function. Thus an intrinsic relationship between image modelling and optimization is provided. Our tutorials will explain some of these relationships and show how to use them in practice.
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