- Lothar Reichel – “Iterative methods for Image Processing”
The restoration of images that have been contaminated by blur and noise gives rise to large systems of linear or nonlinear equations. The matrix of these (linearized) systems generally is numerically rank-deficient. Straightforward solution of these systems typically does not result in an accurate restoration due to severe propagation of the noise in the available contaminated image. The computation of an accurate restoration requires that the system to be solved be replaced by a nearby system whose solution is less sensitive to the noise. This replacement is known as regularization. A common approach to solving large scale problems is to first reduce them to problems of small size by a Krylov subspace method, and then solve the reduced problem by a factorization method. Regularization is achieved by only carrying out fairly few steps of the Krylov subspace method or by modifying the reduced problem. Tikhonov regularization is a commonly used regularization method. It allows the introduction of a regularization matrix whose choice may depends on the problem being solved. These lectures discuss the application of Krylov subspace and generalized Krylov subspace methods to reduce large-scale problems to small ones. Topics considered include the choice of Krylov subspace method and the choice of regularization matrix, and the use of the latter in Tikhonov and iterated Tikhonov methods.
- Jim Nagy: “Regularization methods in image processing”
In these lectures we provide some basic background on regularization methods, including filtering based on the singular (and spectral) value factorizations (SVD). Since problems in image processing are large scale, the filtering methods are not always applicable, so we describe situations where they can be used. We also discuss state-of-the-art iterative methods for cases where SVD methods cannot be used. However, since other lecturers will discuss mathematical and algorithmic details, the focus this particular set of lectures will be primarily on software, so that we can illustrate how various regularization methods perform on a variety of test problems. It would be beneficial for students to have a laptop with MATLAB so that they can get hands-on experience solving large scale inverse problems in imaging applications. We will provide MATLAB software that can be used to generate and solve a variety of test problems.
- Raymond Chan: “Variational Methods for Missing Data Recovery in Imaging”
In many practical problems in image processing, the observed data sets are often incomplete in the sense that features of interest in the images are missing partially or corrupted by noise. The recovery of missing data from incomplete data is an essential part of any image processing procedures whether the final image is utilized for visual interpretation or for automatic analysis.
The first part of the lectures will be on various variational methods for image recovery for missing data and the ways to solve these problems. The second part will touch on various applications in image processing such as, inpainting, impulse noise removal, segmentation, ground-based astronomy, hyper-spectral imaging, and super-resolution image reconstruction.
- Jean-Christophe Pesquet: “Proximal splitting methods in image processing”
Proximal methods are grounded on the use of the proximity operator which constitutes a natural extension of the projection onto a convex set. Although simple, this concept turns out to be fundamental to provide a unifying view of existing algorithms in convex optimization, but also to design new ones possibly applicable to the nonconvex case. The main advantage of proximal algorithms is their ability to solve large-scale optimization problems involving possibly non smooth functions such as sparsity measures or indicator functions of constraint sets. This course aims at introducing the proximity operator and the principles of proximal algorithms. Various kinds of minimization algorithms allowing to split an intricate objective function into a sum of easy-to-handle ones will be presented. Applications to image recovery and machine learning will be described.