Inverse Problems

The goal of inverse problems is to recover useful information about an unknown system or quantity from indirect and noisy measurements.

For example, a blurred and noisy image, such as one taken with an out-of-focus camera, contains only indirect information about the original scene. Recovering the sharp image from this measurement is a classic inverse problem.

Mathematically, image blur can be modelled as a convolution: the observed image is interpreted as a sharp image convolved with a kernel and corrupted by random noise. The corresponding inverse problem, recovering a sharp image from the blurred one, is called deconvolution. This inverse problem is ill-posed, meaning that the solution is highly sensitive to modelling errors and measurement noise. To obtain a stable and meaningful solution we can apply regularisation techniques.

Topics: Naïve inversion, (discrete) convolution, X-ray tomography, truncated singular value decomposition, Tikhonov and total variation regularisation, and short introduction to Bayesian approach to inverse problems.