Limited Data Problems in X-ray and Polarized Light Tomography

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Abstract

We present new reconstruction results and methods for limited data problems in photoelastic tomography. We begin with a survey of the current state of x-ray tomography. Discussing the Radon transform and its inversion we also consider some stability results for reconstruction in Sobolev spaces. We describe certain limited data problems and ways to tackle these, in particular the Two Step Hilbert reconstruction method. We then move on to photoelastic tomography, where we make use of techniques from scalar tomography to develop new methods for photoelastic tomographic reconstruction. We present the main mathematical model used in photoelasticity, the Truncated Transverse Ray Transform (TTRT). After some initial numerical studies, we extend a recently presented reconstruction algorithm for the TTRT from the Schwartz class to certain Sobolev spaces. We also give some stability results for inversion in these spaces. Moving on from general reconstruction to focus on inversion of some special cases of tensors we consider solenoidal and potential tensor fields. We discuss existing reconstruction methods and present several novel reconstructions and discuss their advantages over using more general machinery. We also extend our new algorithms, as well as existing ones, to certain cases of data truncation. Finally, we present numerical studies of the general reconstruction method. We give the first published results of TTRT reconstruction and go into some detail describing the implementation before presenting our results.

Keyword(s)

photoelasicity; photoelastic tomography; tensor tomography

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