# COMPGV08 - Inverse Problems in Imaging

This database contains 2017-18 versions of the syllabuses. For current versions please see here.

Code COMPGV08 (Also taught as COMPM078) MSc This course requires good mathematical and programming skills. In particular students should be familiar with : Fourier Theorydiscrete and continuoussamplingconvolution Linear AlgebraEigenvalues and EigenvectorsMatrix Algebra Calculusfunctions of multiple variablescalculus of variation ProbabilityGaussian and Poisson probabilitiesBayes Theorem Matlab programmingmultidimensional arraysimage visualisation anonymous functions If in doubt about your suitability for the course, please review the notes and examples on the course web page, and/or consult the instructor. 2 Simon Arridge (100%) To introduce the concepts of optimisation, and appropriate mathematical and numerical tools applications in image processing and image reconstruction. To understand the principles of optimisation and to acquire skills in mathematical methods and programming techniques.

# Content

### Introduction

• Example problems
• Data Fitting Concepts
• Existence
• Uniqueness
• Stability
• Bayesian interpretation

### Mathematical Tools

• Linear Algebra
• Solving Systems of Linear Equations
• Over and Under Determined Problems
• Eigen-Analysis and SVD.
• Variational Methods
• Calculus of Variation
• Multivariate Derivatives
• Frechet and Gateaux Derivatives
• Regulariation
• Tikhonov and Generalised Tikhonov
• Non-Convex Regularisation
• Methods for selection of regularisation parameters

### Numerical Tools

• Descent Methods
• Steepest Descent
• Line Search

Newton Methods

• Gauss Newton and Full Newton
• Trust-Region and Globalisation
• Quasi-Newton
• Inexact Newton

### Optimisation Methods

• Least-Squares Problems
• Linear Least Squares
• Non-linear Least Squares
• Poisson Likelihood
• Kullback-Leibler Divergence
• Lagrangian penalties and constrained optimisation
• Proximal methods

### Concepts of sparsity

• L1 and total variation
• wavelet compression
• dictionary methods.

### Bayesian Approach

• Maximum Likelihood and Maximum A Posteriori estimates
• Expectation-Minimisation
• Posterior Sampling
• Confidence-Limits
• Monte Carlo Markov Chain

### Applications

• Image Denoising
• Image Deblurring
• Inpainting
• Linear Image Reconstruction
• Tomographic Reconstruction
• Reconstruction from Incomplete Data
• Non-Linear Parameter Estimation
• General Concepts
• Direct and Adjoint Differentiation

### Other Approaches

• Learning Based Methods

# 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 (2 pieces, 25%)

To pass this course, students must:

• Obtain an overall pass mark of 50% for all sections combined.

# Resources

Reading list available via the UCL Library catalogue.