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# COMP0115 Geometry of Images

This database contains the 2018-19 versions of syllabuses.

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

 Academic session 2018-19 Module Geometry of Images Code COMP0115 Module delivery 1819/A7P/T2/COMP0115 Postgraduate Related deliveries 1819/A7U/T2/COMP0115 Masters (MEng) Prior deliveries COMPGV11 Level Postgraduate FHEQ Level L7 FHEQ credits 15 Term/s Term 2 Module leader Arridge, Simon Contributors Arridge, SimonGriffin, Lewis Module administrator Horslen, Caroline

## Aims

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

## Learning outcomes

On successful completion of the module, a student will be able to:

1. understand the principles of image processing in n-dimensions, time-series analysis and scale space.
2. understand the relations between geometric objects and sampled images.

## Availability and prerequisites

This module delivery is available for selection on the below-listed programmes. The relevant programme structure will specify whether the module is core, optional, or elective.

In order to be eligible to select this module as optional or elective, where available, students must meet all prerequisite conditions to the satisfaction of the module leader. Places for students taking the module as optional or elective are limited and will be allocated according to the department’s module selection policy.

 Programmes on which available: MRes Virtual Reality MSc Computer Graphics, Vision and Imaging MSc Scientific Computing Prerequisites: Students must have taken the term 1 module Image Processing (COMP0026). Students must also have a strong competency in mathematical and programming skills, including: Fourier Theory (discrete and continuous; sampling; convolution); Calculus (functions of multiple variables; calculus of variationm, integration by parts); Partial Differential Equations (Green’s function methods, finite differencing methods); Matlab programming (multidimensional arrays; image visualisation; anonymous functions) Students should take the self-test available here http://www0.cs.ucl.ac.uk/staff/S.Arridge/teaching/ndsp/GV11test.pdf to assess their mathematical ability for this course

## Content

• Basic Image Operations
• Fourier Transforms
• Convolution and Differentiation in Fourier Domain, Recursive Filters
• Marching Square/Cubes
• Level Set Methods
• Introduction to Differential Geometry
• 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
• Curvature
• Contour curvature
• Image curvature
• 3D curvature: the Weingarten mapping, Gaussian and mean curvatures
• Scale Space
• Linear Scale Space
• Introduction and background
• Formal properties
• Gaussian kernels and their derivitives
• 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
• Multispectral Images and Statistical Classification
• Feature Space
• Introduction
• Definitions of feature space
• Clustering
• Statistical Methods
• Linear and non-linear discriminant functions
• Supervised learning
• Unsupervised learning
• Bayesian and Information Theoretic Approaches
• Bayesian Image Restoration
• Markov Random Fields
• Definitions of Entropy and Mutual information
• Deconvolution with image priors (statistical and structural)