References:
(1). Mathematical Problems in Image Processing - PDE and the Calculus of Variations, by Gilles Aubert and Pierre Kornprobst;
(2). Measure Theory and Fine Properties of Functions by L.C.Evans and R.F.Gariepy
(3). Geometric Level Set Methods, Stanley Osher and Nikos Paragios.
(4). The Handbook of Mathematical Models in Computer Vision, Nikos Paragios, Yunmei Chen, and Olivier Faugeras;
(5). Paper reading.

Meeting Time and Room: MWF 4 at LIT305

Office Hours: MWF 5 or by appointment

Course Outline:
This course is an introduction to the use of Partial Differential Equations and variational methods in image processing. It will provides standard methods, fundamental theories, and new developments in the field of mathematical imaging and image analysis. We will focus on the problems of image denoising, restoration, 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: Some basic mathematical concepts
1. Strong and weak convergences
2. Sobolev space and BV space
3. Convexity, low-semicontinuty and direct method in calculus of variation
4. Convolution and smoothing

Unit 2: Image de-noising and restoration
1. P-energy minimization based filtering
Isotropic smoothing, Total variation based smoothing
P-energy minimization based smoothing, Differences and connections between these methods
2. Geometric PDEs in image restoration
Curvature based image de-noising methods
3. Existence and uniqueness discussion
BV solution and viscosity solution

Unit 3: Image segmentation
1. Classical level set method and implicit representation
2. Variational level set method
3. Active contour based segmentation methods
Edge based method - geodesic active contour
Region based method - Mumford-Shah's model based approach
4. Fuzzy/ soft segmentation methods
5. Incorporating prior information into image segmentation

Unit 4: Image registration
1. Intensity based image registration:
Rigid and deformable segmentation
Monomodality and multimodality image registration
Mutual information, joint entropy
2. Feature based image registration
joint segmentation and 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.