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# COMPM081 - Geometry of Images

This database contains 2016-17 versions of the syllabuses. For current versions please see here.

Code COMPM081 (Also taught as: COMPGV11) 4 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. 2 Simon Arridge (50%) Lewis Griffin (50%) To introduce the generalisation of image processing to n-Dimensional data : volume data, scale space, time-series and vectorial data. 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