We develop new Besov prior methods to overcome the non-smoothing problem for inverse problems.
Inverse problems are the procedures of determining causal relations from observations. In science inverse problems are the mathematical theory and the numerical analysis of problems, where an unknown quantity is estimated from indirect observations. One important application of inverse problems is medical imaging where examples of inverse problems are X-ray Computed Tomography, Electrical Impedance Tomography, and Magnetic Resonance Imaging. In X-ray Computed Tomography the body of a patient is emitted by X-rays, and then by solving an inverse problem a picture of the patient’s interior is computed. This method is often used to detect cancer, strokes, and other diseases.
When solving inverse problems, the typical challenges to overcome is the sensitivity of the solution with respect to noise in the measurements, insufficient measurements, and inaccuracy in the mathematical models. These challenges all contribute to error and uncertainty in the solution of the inverse problem, which in the worst case can lead to a wrong solution that for example could lead to a misdiagnosis of cancer.
To solve the challenges of inverse problems one can, apply a Bayesian approach to the inverse problem, where the observations and the unknown quantity are modelled as random variables. This approach makes it possible to impose prior knowledge to the problem through the prior distribution of the unknown. The measurement noise and various sources of errors in the problem are described using the likelihood function, and then we can quantify the uncertainty of the solution by computing the posterior distribution.
The prior distribution of the unknown describes our belief about the solution without taking measurements into account. For example, we could assume the solution to be smooth, piecewise constant, or having a complicated non-smooth pattern. The prior distribution is a valuable tool for solving ill-posed inverse problems where the choice of prior is crucial for reducing error and uncertainty in the solution. Most of the research for Bayesian inverse problems focuses on Gaussian priors. Gaussian priors are applicable for many problems and are well understood and has fast computational properties. The problem with Gaussian priors is that it is a poor model of non-smooth behavior that occurs for example in the detection of edges in image reconstruction problems.
One way to overcome the non-smoothing problem of Gaussian priors is to use Besov priors, which is the focus of this PhD project. The Besov prior is constructed as a generalization of the Gaussian prior, and the core is that Besov functions can be expanded as a weighted wavelet expansion which through different choices of Besov parameters can be used to characterize smooth and non-smooth functions.
In this PhD project we want to investigate the flexibility of the Besov prior in the choice of parameters and basis, to understand the limitations and practical use for Besov priors. In addition, we want to extend the definition of the Besov measure to possibly other domains and expansions to make it applicable to more problems, while keeping the important properties of the Besov prior.