TIES6820 COM1: Variational models and fast numerical schemes in image processing and computer vision (JSS28) (2–3 op)

Content

This course will introduce a number of problems in image processing and computer vision, and describe how they can be tackled by modern techniques based on calculus of variations and partial differential equations. We will focus especially (but not exclusively) on image reconstruction (denoising, deblurring, inpainting, as well as some inverse problems) and image segmentation. These procedures are fundamental in many applications, such as medical imaging and target recognition. Numerical solution of the models, which involve minimizing appropriate energies (often by solving associated partial differential equations) will be a major concern of the course: A variety of numerical techniques for this purpose, including level set and diffuse interface methods for evolving curves and surfaces, will be introduced and covered in detail. In addition, important theoretical questions about the various models and how they have been answered will be presented. We will try to cover some of the newest developments for these problems which have not been covered in any other standard textbook. Tentative outline:

(1) Mathematical preliminaries.
• Some elementary partial differential equations.
• Basic about minimization and calculus of variations
• Functions of bounded variation.

(2) Image restoration, inpainting and deblurring.
• The total variation model of Rudin, Osher, and Fatemi.
• Mumford-Shah model.
• Euler’s Elastica model.
• Higher order methods of PDE nonlinear filters.
• Other geometrical models for image filters.

(3) Fast numerical schemes
• Gradient descent method.
• Operator splitting and AOS schemes.
• Dual approaches.
• Split-Bregman.
• Augmented Lagrangian approach.
• Other fast minimization approaches for image processing.

(4) Image segmentation and geometrical PDEs
• Geodesic active contours; implementation using level-sets.
• Mumford-Shah and Chan-Vese models using level-sets.
• Piecewise Constant Level set method.
• Graph cut approach for interface problems and image segmentation.
• Recent fast numerical schemes and global minimizations.
• Discussion of a few other vision problems.

Completion methods

Lectures and assignments.

Assessment details

Obligatory attendance at lectures and completing the given exercises.