This tutorial introduces three fundamental image processing problems, i.e. image denoising, image segmentation and image registration, and shows that all of them can be phrased in a variational manner, i.e. as a minimization problem. As such, these seemingly different processing problems have a lot common structure. We will touch on general questions that arise when considering minimizers, i.e. their existence, uniqueness and characterization. The characterization of minimizers is a key to algorithms that can efficiently compute minimizers, and thus solve the corresponding processing problems. In particular, we will see how the non-convex smooth optimization problems arising in image registration can be solved in practice. Moreover, we will investigate how non-smooth but convex optimization problems can be solved using the proximal map and operator splitting. Such problems, for instance, arise when using the total variation as regularizer. Example applications will illustrate the usefulness of these approaches for electron microscopy.
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