# COMPM081 - Geometry of Images

**This database contains the 2017-18 versions of syllabuses.** Syllabuses from the 2016-17 session are available here.

**Note:** Whilst every effort is made to keep the syllabus and assessment records correct, the precise details must be checked with the lecturer(s).

Code | COMPM081 (Also taught as: COMPGV11) |
---|---|

Year | 4 (Masters) |

Prerequisites | Successful completion of years 1 and 2 of the Computer Science programme, including the mathematics course/option, or core courses in computer science and mathematics. Plus Image Processing in Year 3. |

Term | 2 |

Taught By | Simon Arridge (50%) Lewis Griffin (50%) |

Aims | To introduce the generalisation of image processing to n-Dimensional data : volume data, scale space, time-series and vectorial data. |

Learning Outcomes | To understand the principles of image processing in n-dimensions, time-series analysis and scale space, and to understand the relations between geometric objects and sampled images. |

# Content

**0. Basic Image Operations**

Fourier Transforms

Convolution and Differentiation in Fourier Domain Recursive Filters

Marching Square/ Cubes

Level Set Methods

1. Introduction to Differential Geometry

1.1 Images as functions

- Definitions

- Taylor Series expansion and the Koenderick jet

- Properties of the local Hessian

- Definition of extrema and saddle points

- Ridges in n-dimensions

- Image invarients up to fourth order

1.2 Curvature

- Contour curvature

- Image curvature

- 3D curvature: the Weingarten mapping, Gaussian and mean curvatures

2. Scale Space

2.1 Linear Scale Space

- Introduction and background

- Formal properties

- Gaussian kernels and their derivitives

2.2 Non-linear Scale Space

- Motivation

- Edge-effected diffusion (Perona-Malik)

- Classification of Alvarez and Morel

- Euclidian and Affine shortening flow

- Numerical methods for computing scale spaces

3. Multispectral Images and Statistical Classification

3.1 Feature Space

- Introduction

- Definitions of feature space

- Clustering

3.2 Statistical Methods

- Linear and non-linear discriminant functions

- Supervised learning

- Unsupervised learning **4. Bayesian and Information Theoretic Approaches**

Bayesian Image Restoration

Markov Random Fields

Definitions of Entropy and Mutual information

Deconvolution with image priors (statistical and structural

# Method of Instruction

Lecture presentations with associated class coursework and laboratory sessions

# Assessment

The course has the following assessment components:

- Written Examination (2.5 hours, 75%)
- Coursework Section (2 pieces, 25%)

To pass this course, students must:

- Obtain an overall pass mark of 50% for all sections combined
- Obtain a minimum mark of 40% in each component worth ≥ 30% of the module as a whole.

The examination rubric is:

Choice of 3 questions from 5. All questions carry equal marks.

# Resources

Reading list available via the UCL Library catalogue.