# COMPM078 - Inverse Problems in Imaging

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

Code | COMPM078 |
---|---|

Year | 4 |

Prerequisites | |

Term | 2 |

Taught By | Simon Arridge (100%) |

Aims | To introduce the concepts of optimisation, and appropriate mathematical and numerical tools. Applications in image processing and image reconstruction. |

Learning Outcomes | 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

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

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 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

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
- Obtain a minimum mark of 40% in each component worth ≥ 30% of the module as a whole.

# 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)