## COMP 790-098: Optimal Estimation in Image Analysis

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

**Instructor: Marc Niethammer****Email:** (mn -at- cs.unc.edu)**Phone:** 919-843-7449**Office:**219 Sitterson Hall

## Overview

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 http://www.nr.com/) | 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 http://www.stanford.edu/~boyd/cvxbook/) | 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)

Date |
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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 |
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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 |
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Optional Lecture | Dynamic filtering: Kalman filter, unscented Kalman filter, etc. | |

Optional Lecture | Advanced registration methods: higher order models, regression models, ... |