Marie Foged Schmidt: Reconstruction problems, where one wants to determine phenomena that can only be measured by observing indirect effects, can be formulated as inverse problems.
This PhD project is part of the project High-Definition Tomography funded by Advanced Grant No. 291405 from the European Research Council.
Such problems appear in e.g. medical imaging (for example computed tomography), geophysics, and material science. However, the algorithms for the reconstructions suffer from high sensibility to noise and errors in the measurements and the corresponding inverse problems are called ill-posed. In order to transform the problems from ill-posed to well-posed, regularization is introduced. The design of the regularization can be based on prior knowledge about the object to reconstruct. This could be, e.g., that the solution is smooth or piecewise constant, depending on the phenomena of interest.
One way to find an approximate solution to an inverse problem is through a variational framework. In this case one minimizes a functional consisting of a data-fidelity term and a regularization term. The purpose of the data-fidelity term is to fit the solution to the possibly noisy measurements, whereas the regularization term makes sure to incorporate prior knowledge into the solution. A regularization parameter balances the weight between the two terms. However, variational methods quite often lead to solutions suffering from a loss of contrast. Instead one can use Bregman iteration, which is actually iteration over modified variational methods. In this project we study a time-continuous version of Bregman iteration which is called the inverse scale space flow. The inverse scale space flow is an evolution process for approximating a solution to an inverse problem. In numerical tests, solutions of inverse scale space flows have shown superior properties to solutions obtained through variational methods.
The aim of the project is to characterize solutions of different kinds of inverse scale space flows and to develop algorithms for reconstruction based on the flows. The flows can also lead to a different view on regularization and possibly then lead to the formulation of new prior-driven regularization functionals. In particular we are interested in applications in medical imaging including computed tomography.
Effective start/end date 01/12/2014 → 30/11/2017
Supervisors: Kim Knudsen, Yiqiu Dong
Section for Scientific Computing