Robust Computed Tomography with Incomplete Data

AbstractThis thesis provides the theoretical background for analysing the ill-posedness of X-ray tomography problems and uses this to analyse the special case of laminar tomography problems.

The first part of the thesis is devoted to describing the ill-posed characteristics of X-ray tomography problems. For this purpose we show that the deconvolution problem is ill-posed. In particular we show that small high frequent perturbations in the values of the convolution can give arbitrarily large changes in solutions to the deconvolution problem. We then relate the standard mathematical model of X-ray tomography, the Radon transform, to the deconvolution problem and show that the ill-posed properties are likely to carry over. Moreover, we show that the discretised mathematical model of X-ray tomography also exhibit the same ill-posed properties.

In the second part of the thesis, we develop a method of analysing the solvability of an X-ray tomography problem by considering how ill-posed it is. In particular we use the method to describe the ill-posed characteristics of several "toy" problems. Finally the work culminates by considering two cases of laminar tomography. Here it is shown, that the characteristics from some of the toy problems carry over to the laminar tomography problems.
TypeBachelor thesis [Academic thesis]
PublisherTechnical University of Denmark, Department of Applied Mathematics and Computer Science
AddressRichard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark,
SeriesDTU Compute B.Sc.-2015
NoteDTU supervisor: Per Christian Hansen,, DTU Compute, co-supervised by Professor Bill Lionheart, The University of Manchester
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IMM Group(s)Scientific Computing