Current students

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)
Year 4
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.


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


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