References:

(1). Partial Differential Equations, by L.C.Evans (Graduate Studies in Mathematics, Vol.19, AMS)
(2). Mathematical Problems in Image Processing - PDE and the Calculus of Variations, Gilles Aubert and Pierre Kornprobst;
(3). Measure Theory and Fine Properties of Functions, L.C.Evans and R.F.Gariepy;
(4). Geometric Level Set Methods, Stanley Osher and Nikos Paragios.
(5). Paper reading.

Meeting Time and Room: MWF 5 at MAT.114

Office Hours: MWF 4 or by appointment

Course Outline:

The aim of the course is to brings a number of new mathematical concepts, theories, and methods into the field of biomedical imaging. This course will focus on the mathematical modeling, algorithm developing, and wellposedness study for the problems that arise from shape analysis, image restoration, reconstruction segmentation, and registration. Students will gain the ability in modeling and developing algorithms to solve real world problems.

Main Topics:

1: Functions in BV space and linear growth functionals
Minimizing linear growth functionals of measures
Existence, uniqueness and partial regularity for minimizes of certain linear growth functionals of measures
Relation between minimizing the total variation norm and the $L^p$ norm of the gradient

2. Medical image analysis:
Total variation based diffusion in image recovery and reconstruction
Edge or region based segmentation
Simultaneous segmentation and registration using priors
Shape modeling and analysis
Information theory and its applications in clustering and classification.
Diffusion weighted MRI, reconstruction of apparent diffusion coefficients, characterization of diffusion anisotropy, and fiber tracking

Grading:

Students will be required to present 2-3 papers including their own work and have one 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 the presentation and project.