References:
(1). ``Geometric Partial Differential Equations and Image Analysis'' by G. Sapiro;
(2). ``Measure Theory and Fine Properties of Functions'' by L.C.Evans and R.F.Gariepy

Meeting Time and Room: MWF 3 at TUR 2346

Office Hours: MWF 4 or by appointment

Course Outline:
This course provides an introduction to the use of Partial Differential Equations in image processing. This is a relatively new research area that brings a number of new concepts into the field, providing a fundamental and formal approach to image processing. This course will focus on the problems of image denoising, restoraton, segmentation and registration. We will study the mathematical models to solve these problems, the mathematical theories for the well-posedness of the models, and the numerical solutions of the models. Students will gain practical experience by applying algorithms to real world problems.

Arrangement of the course:

Unit 1: Image de-noising and restoration
General linear filtering;
Curvature based image de-noising methods;
Existence, uniqueness and stability of the curvature based model -- viscosity solution;
Total variation based image de-noising methods;
solutions for total variation based regularizations in BV space;
Time-delay reglarization;

Unit 2: Image segmentation
Classical level set method and implicit representation;
Variational level set method;
Geodesic active contours and minimal surface;
Mumford-Shah's model;
Incorporating prior information into image segmentation;

Unit 3: Image registration
Intensity based image registration;
Feature based image registration;

Grading:

Students will be required to present a paper and do numerical and theoretical projects related to the course content. These projects may be related to problems of particular interest to the individual student. Grades will be assigned on the basis of these projects.