Tutorials

Tutorial title: What energy to minimize?

Lecturer: Mila Nikolova
Affiliation: Centre de Mathématiques et de Leurs Applications (CMLA), ENS de Cachan, France

Tutorial outline:
Many applications in signal and image processing are solved by minimizing an energy function composed of a data-fidelity term and a regularization term. Such energies are classically defined either from a PDE standpoint or in a Bayesian MAP estimation framework. In both approaches, the control on the solutions remains limited. Finer knowledge of these energies and of their minimizers is the key to obtain good solutions. This tutorial will present a systematic approach to the problem of the choice of pertinent energies for signal and image reconstruction.

The last decade an important effort was done in order to gain an insight into these energies and the solutions they yield. Behind the wide variety of this research, several important orientations can be drawn. The interplay between energy minimization and constrained minimization gave rise to interpretations of the parameters and opened connections with the models based on frames (e.g. splines, wavelets or packets). Studying the spaces of functions (e.g. signals and images) underlying different energies helped to better understand sampling and modeling, and gave rise to new tools for reconstruction and decomposition. It also launched an examination and a clarification of the goals of signal and image reconstruction. Challenging theories established bridges between disparate methods based on energy minimization, diffusion and shrinkage estimation of frame representations. Even though holding in restricted conditions, these results suggest various practical ramifications for reconstruction and optimization.

We focus more closely on the energy minimizers since a solution, defined as the minimizer of an energy, is an implicit function of both the data and the shape of the energy. This point of view raises the question of how the features of the reconstructed signals and images are determined by the shape of the energy. It hence provides a framework to unify the theory on energy minimization methods and to address rigorously the problem of the choice of energies for signal and image reconstruction. The objectives of this tutorial are the following:

• to understand the practical issues governing the proper choice of an energy for signal and image reconstruction;
• to show how to conceive energies in such a way that their minimizers exhibit some desired properties;
• to provide a systematic way to compare existing energies for signal and image reconstruction.

This talk is based on a series of analytical results which characterize some essential features exhibited by the minimizers of regularized energies, in connection with the shape of the energy. Points of interest are for instance the recovery of homogeneous regions, textures and edges, and the processing of signals and images containing outliers or spikes. These are shown to be determined by some attributes of the energy function relevant to its (non)smoothness or its (non)convexity. Numerical examples are used to illustrate the theory and stability results are provided. Afterwards we present several applications where specific energies are conceived using the mathematical results on minimizers. Indications on implementation issues are also given.

The minimizer approach and results invoke a critical analysis of the ways to construct energies and a new understanding of modeling. By way of conclusion, open questions ranging from concepts to practical signal and image reconstruction are discussed.


Information about the lecturer:



Mila Nikolova received the Ph.D. degree in signal processing from the Université Paris-Sud, Paris, France, in 1995. Currently, she is senior research fellow with the National Center for Scientific Research (CNRS), France and performs her research at the Centre de Mathématiques et de Leurs Applications (CMLA), ENS de Cachan, France. Her research interests are in Image and signal reconstruction, Regularization and variational methods, Scientific computing. The last few years she published a series of papers analyzing the properties of the minimizers of regularized energy functions.