Inverse Problems in Imaging

This page is for information related to the 4th year/VIVE course GV08 : Inverse Problems in Imaging.

Notes

I have put some lecture notes from last year here (these are liable to change before this years course)

First Examples

Example of model fitting (over determined) modelfit.m
Example of model fitting (under determined) modelfit_under.m
Example of ill conditioned matrix inversion ip1.m
Second example inverts this matrix with a Gaussian prior with covariance C ip3.m.
Example call for this : ip3(0.2,0.05,[1 1; 1 -1]);

One dimensional blur of function in interval [0,1] linblur.m
Regularised inversion of linblur using Truncated SVD linsvd_truncsvd.m
Regularised inversion of linblur using Zero-Order Tikhonov linsvd_tk0.m

Regularisation Parameter Selection

Here are some functions that calculate various regularisation parameter selection procedures for the 1 dimensional linear blurring example

Discrepency Principle DP.m

Miller Criterion Miller.m

Predictive Risk predrisk.m

Unbiased Predictive Risk Estimator UPRE.m

Generalised Cross-Validation GCV.m

Here's an example how to use these compare_regselect.m
Further example, that compares zero-order and first order Tikhonov compare_TK0TK1.m This example requires a first order finite difference derivative operator, such as the one produced by this function lindf.m

Nonlinear Optimisation

Here are some examples for nonlinear optimisation of the Rosenbrock function. These make use of the following Line Search Function

Generic Steepest Descents and Rosenbrock Example

Generic Conjugate Gradients and Rosenbrock Example

Generic Damped Gauss Newton and Rosenbrock Example

Generic Levenburg-Marquardt and Rosenbrock Example

Constrained Optimisation

Function for inequality constrained optimisation using active sets
Example applied to a quadratic matrix function quadratic matrix function

Equality constrained Lp-norm minimisation. Function returning Lp-norm of a vector.
2D example with single equality consraint : Lpmin.m

Poisson noise

Example of adding Poisson noise to an image

Example of solving 1D deblurring from Poisson noise using Richardson-Lucy (MLEM in 1D).

Stochastic Optimisation

The basic tools are the Metropolis-Hasting Sampling Method and the Gibbs Sampling Method

Here are some examples using each method

Simple unimodal Gaussian sampling using Metropolis-Hasting and Gibbs

Mixture of Gaussian sampling using Metropolis-Hasting and Gibbs

Highly non-linear function sampling using Metropolis-Hasting and Gibbs

Posterior sampling of regularised inverse problem using Metropolis-Hasting and Gibbs

The Metropolis Hastings algorithm is easily turned into a Simulated Annealing method. Here is one way, and an example using the above non-linear function