## Introduction to Mathematical Image Analysis

Yunmei Chen

·  References:
(1). Partial Differential Equations by L.C.Evans;
(2). Measure Theory and Fine Properties of Functions by L.C.Evans and R.F.Gariepy
(3). Mathematical Problems in Image Processing - PDE and the Calculus of Variations, by Gilles Aubert and Pierre Kornprobst;
(4). The Handbook of Mathematical Models in Computer Vision, Nikos Paragios, Yunmei Chen, and Olivier Faugeras;

·  Meeting Time and Room: MWF 5 at LIT233

·  Office Hours: MWF 4 or by appointment

·  Course Outline:

This course is an introduction to mathematical imaging. We will study basic mathematical methods for modeling and solving certain fundamental problems in image analysis, such as image segmentation, registration, restoration, and reconstruction. We will focus on  the use of variational methods, statistical dependence measures and analysis for modeling. We will also learn several currently developed efficient numerical methods for solving large scale and ill-conditioned linear inverse problems with total variation regularization, and discuss their applications in solving image analysis problems. Students will gain knowledge on mathematical theories, methods, and practical experience in solving real world problems in image analysis.

·  Arrangement of the course:

Unit 1:  1. Image Segmentation

(1). Edge based active contour method:  Snake model, Geodesic active contour model, Level set formulation;

(2). Region based active contour method:  Mumford-Shah's model and CV model, Region based active contour with parametric density estimation, Region based active contour     with non-parametric density estimation; Maximizing likelihood function (MLE), Maximizing a posterior estimate (MAP), Non-parametric density estimation.

(3). Hybrid methods: Combination of edge and regional intensity information, or incorporating prior information into image segmentation;

(4). Fuzzy/ soft segmentation methods: Relaxing models and global minimum.

·  Unit 2: Image Registration

(1). Mono-modal image registration: Rigid and deformable segmentation, MLE and MAP for intensity difference, Correlation coefficient (CC), Smoothness of deformation field;
(2). Multi- modal image registration: Joint entropy, Mutual information, Kullback-Leibler (KL) distance
, Renyi’s statistical dependence measure, Models based on these measures and their local versions;

(3). Joint segmentation and registration.

·  Unit 3: Image Restoration and Reconstruction

(1).  Subdifferential, Shrinkage operator, Fenchel transform;

(2). First-order gradient method, Iterative shrinkage-thresholding algorithm (ISTA), and fast iterative shrinkage-thresholding algorithm (FISTA); Operator splitting;
(3).  Bregman iterative algorithm, Split Bregman, Augmented Lagranging, Alternating minimization algorithm (AMA), Alternating direction method of multipliers (ADMM), Linearized Bregman, Bregman operator splitting (BOS);

(4). First-order primal-dual algorithms, and Chambolle's method for solving dual problem;

(5). Primal-dual hybrid gradient (PDHG) algorithm;
(6). Applications in image denoising, deblurring, and multi-channel MR image reconstruction with arbitrary under-sampling pattern.