# COMPGV08 - Inverse Problems in Imaging

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

Code COMPGV08 MSc N/A 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;
Variational and Iterative Concepts.

Mathematical Tools
Linear Algebra: Solving Systems of Linear Equations; Over and Under Determined Problems; Eigen-Analysis and SVD; Preconditioning
Variational Methods: Calculus of Variation; Multivariate Derivatives; Frechet and Gateaux Derivatives
Regulariation: Tikhonov and Generalised Tikhonov Non-Quadratic Regularisation, Non-Convex Regularisation.

Numerical Tools
Non-Gradient Methods: Simplex Method; Powell's Method
Descent Methods: Steepest Descent; Conjugate Gradients; Line Search
Newton Methods: Gauss Newton and Full Newton; TrustRegion and Globalisation; Quasi-Newton; Inexact Newton

Unconstrained Optimisation
Least-Squares Problems: Linear Least Squares; Non-linear Lesat Squares
Non-Quadratic Problems: Poisson Likelihood; Kullback-Leibler Divergence

Regularisation Parameter Selection
Discrepancy Principles; The L-Curve Method; Generalised Cross-Validation

Constrained Optimisation
Equality Constraints: Lagranian Penalties
Inequality Constraints: Positivity Constraints; Upper and Lower Bounds; Active Sets
Primal-Dual Methods: Primal-Dual Interior Point Methods

Bayesian Approach
Bayesian Priors and Penalty Functions, Maximum Likelihood and Maximum A Posterior: Best Linear Unbiased Estimation; Expectation-Minimisation
Posterior Sampling: Confidence-Limits; Monte Carlo Markov Chain; Applications Image Deblurring: Deconvolution; Anisotropic Denoising
Linear Image Reconstruction: Tomographic Reconstruction; Reconstruction from Incomplete Data
Non-Linear Parameter Estimation: General Concepts; Direct and Adjoint Differentiation

Other Approaches
Simulated Annealing; Genetic Algorithms

# 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.

# Resources:

C.R. Vogel, Computational Methods for Inverse Problems (SIAM 2002)

J.E. Dennis and R.B Schnabel, Numerical Methods for Unconstrained Optimisation and Nonlinear Equations (SIAM 1996)

J. Nocedal and S.J Wright, Numercal Optimisation (Springer 1999)

S.Boyd and L.Vandenberghe, Convex Optimisation, (Cambridge University Press, 2004)