Aims:

To introduce the concepts of inverse problems, optimisation, and appropriate mathematical and numerical tools applications in image processing and image reconstruction.Learning outcomes:

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

Understand the principles of forward and inverse problems, illposedness, and regularisation.
	Demonstrate acquired skills in mathematical methods and programming techniques for solving inverse problems using optimisation and other methods.
	Demonstrate gained experience in practical problems in Imaging Science, including image enhancement, reconstruction and tomography.
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-Quadratic Regularisation.
	Non-Convex Regularisation.
	Methods for selection of regularisation parameters.
Numerical Tools:

Descent Methods:
	Steepest Descent.
		Conjugate Gradients.
		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.
	Non-Quadratic Problems.
	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.
		Direct and Adjoint Differentiation.
	
Other Approaches:

Learning Based Methods.
Requisites:

To be eligible to select this module as an option or elective, a student must: (1) be registered on a programme and year of study for which it is a formally available; (2) have taken Machine Vision (COMP0137) in term 1 (if not, contact the Module leader); and (3) have a strong competency in mathematical and programming skills, including Fourier Theory (discrete and continuous, sampling, convolution), Linear Algebra (Eigenvalues and Eigenvectors, Matrix Algebra), Calculus (functions of multiple variables, calculus of variation), Probability (Gaussian and Poisson probabilities, Bayes Theorem), and MATLAB programming (multidimensional arrays, image visualisation, anonymous functions).

Self-Assessment Test:

Students should take the self-test available here, to assess their mathematical ability for this module.