COMP 790: Optimal Estimation in Image Analysis (Spring 2012)

COMP 790-098: Optimal Estimation in Image Analysis

Spring 2012, T Th 12:30pm-1:45pm, SN 115

Instructor: Marc Niethammer
Email: (mn -at-
Phone: 919-843-7449
Office:219 Sitterson Hall


The amount of image data obtained from standard cameras, microscopes, ultrasound or magnetic resonance scanners is ever increasing. This makes image analysis (the automatic extraction of quantitative information from such data) essential. While a huge number of algorithms exist to tackle problems ranging from image segmentation, image registration, to 3D scene reconstruction, many of these algorithms require the formulation and solution of similar optimization problems.

The focus of this course is therefore to introduce students to the basic approaches in formulating and solving such optimization problems. The course will be split in three parts: (i) background material in numerical optimization, (ii) continuous formulations, and (iii) discrete formulations of optimization problems in image analysis. Special emphasis will be given to numerical solution methods for example problems.

Prerequisites: The material covered will be diverse. Prior exposure to all the relevant topics is therefore unrealistic. Consequentially, the course lecture will be supplemented with background material as needed. Focus will be on the understanding of the overall concepts in the context of image analysis. Exposure to standard probability theory, basics of differential equations, and an overall interest in mathematical problem formulations will be a sufficient background.

Format: The course will combine standard lectures as well as paper and project presentations.

Resources: There is no standard textbook covering the material of this course and a textbook is therefore not required. Related conference and journal articles will be provided through blackboard. Further, I will make use of material from the following books (some of them available freely or electronically at UNC -- as indicated):

1. Geometric Partial Differential Equations and Image Analysis by Guillermo Sapiro
2. Level Set Methods and Dynamic Implicit Surfaces by Stanley Osher and Ronald Fedkiw
3. Finite Difference Schemes and Partial Differential Equations by John Strikwerda
4. Numerical Recipes in C (available at by William H. Press et al.
5. Markov Random Field Modeling in Image Analysis (available as an electronic resource from the UNC library) by Stan Z. Li
6. Numerical Optimization (available as an electronic resource from the UNC library) by Nocedal/Wright
7. Convex Optimization (available at by Boyd/Vandenberghe

Lecture materials will be posted on this web page as well as on Sakai

Grading: The course grade will be based on: (i) a set of homework assignment (theory + implementation); (ii) presentation of a research paper or related background material; (iii) a final project related to the course subject.

This is a tentative outline of the lectures. They will change, subject to student interest and student background.

Schedule (in flux)

Tue Jan. 10 Introduction General course outline. Examples of optimization problems in image analysis.
Thu Jan. 12 Lecture 1 Overview of the formulation of common optimization problems in image analysis. Low-dimensional/high-dimensional/infinite-dimensional formulations.
Tue Jan. 17 Lecture 2 Model-based formulations: Probabilistic modeling, energy interpretations, simple noise models.
Thu Jan. 19 Lecture 3 Basics of robust estimation: mean, median, M-estimators.
    Basics of optimization
Tue Jan. 24 Lecture 4 Unconstrained optimization
Thu Jan. 26 Lecture 5 Constrained Optimization
Tue Jan. 31 Lecture 6 Constrained Optimization
Thu Feb. 2 Lecture 7 Constrained Optimizaton
Tue Feb. 7 Lecture 8 Convexity
Thu Feb. 9 Lecture 9 Convexity
    Continuous formulations
Tue Feb. 14 Lecture 10 Bayesian interpretation
Thu Feb. 16 Lecture 11 Review: Calculus of variations
Tue Feb. 21 Lecture 12 Spatial and temporal constraint
Thu Feb. 23 Lecture 13 Examples from segmentation and registration
Tue Feb. 28 Lecture 14 Examples cont.
Thu Mar. 1 Lecture 15 Basics of partial differential equations: types and stability
Tue Mar. 6 Lecture 16 Basics of partial differential equations, cont.
Thu Mar. 8    
    Discrete formulations
Tue Mar. 13 Lecture 17 Graph Cuts
Thu Mar. 15 Lecture 18 Graph Cuts
Tue Mar. 20 Lecture 19 Belief Propagation
Thu Mar. 22 Lecture 20 Dynamic Programming
Tue Mar. 27 Lecture 21 Convex relaxations
Thu Mar. 29 Lecture 22  
Tue Apr. 3 Lecture 23  
Thu Apr. 5 Lecture 24  
Tue Apr. 10 Paper presentations  
Thu Apr. 12 Paper presentations  
Tue Apr. 17 Paper presentations  
Thu Apr. 19 Paper presentations  
Tue Apr. 24 Paper presentations  
Thu May 3 Exam day Project presentations
  Optional Lecture Dynamic filtering: Kalman filter, unscented Kalman filter, etc.
  Optional Lecture Advanced registration methods: higher order models, regression models, ...