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Radon Transforms and Microlocal Analysis in Compton Scattering Tomography RADON TRANSFORMS AND MICROLOCAL ANALYSIS IN COMPTON SCATTERING TOMOGRAPHY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2017 By James W Webber School of Mathematics Contents Abstract 4 Declaration 5 Copyright 6 Acknowledgements 7 Author contributions 8 1 Introduction 9 1.1 The hyperplane Radon transform . 10 1.2 The Funk{Radon transform . 13 1.3 Radon transforms over a general class of hypersurfaces in Rn ... 15 1.4 The spherical Radon transform . 17 1.5 The circular arc Radon transform . 20 1.6 Sobolev space estimates . 23 1.7 Microlocal analysis . 26 1.7.1 Fourier integral operators . 28 1.8 An introduction to the papers . 35 1.8.1 X-ray Compton scattering tomography . 36 1.8.2 Three dimensional Compton scattering tomography . 37 1.8.3 Microlocal analysis of a spindle transform . 38 Bibliography 41 2 X{ray Compton scattering tomography 48 3 Three dimensional Compton scattering tomography 77 2 4 Microlocal analysis of a spindle transform 106 5 Conclusions and further work 133 A Definitions 137 3 Abstract Radon transforms and microlocal analysis in Compton scattering tomography James W Webber A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy, 2017 In this thesis we present new ideas and mathematical insights in the field of Compton Scattering Tomography (CST), an X-ray and gamma ray imaging technique which uses Compton scattered data to reconstruct an electron density of the target. This is an area not considered extensively in the literature, with only two dimensional gamma ray (monochromatic source) CST problems being analysed thus far [50, 48, 46]. The analytic treatment of the polychromatic source case is left untouched and while there are three dimensional acquisition geome- tries in CST which consider the reconstruction of gamma ray source intensities [41, 47, 64, 37], an explicit three dimensional electron density reconstruction from Compton scatter data is yet to be obtained. Noting this gap in the literature, we aim to make new and significant advancements in CST, in particular in answering the questions of the three dimensional density reconstruction and polychromatic source problem. Specifically we provide novel and conclusive results on the sta- bility and uniqueness properties of two and three dimensional inverse problems in CST through an analysis of a disc transform and a generalized spindle torus transform. In the final chapter of the thesis we give a novel analysis of the sta- bility of a spindle torus transform from a microlocal perspective. The practical application of our inversion methods to fields in X-ray and gamma ray imaging are also assessed through simulation work. 4 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 5 Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, De- signs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproduc- tions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/ DocuInfo.aspx?DocID=24420), in any relevant Thesis restriction declara- tions deposited in the University Library, The University Library's regula- tions (see http://www.library.manchester.ac.uk/about/regulations/) and in The University's policy on presentation of Theses 6 Acknowledgements I would like to thank my PhD supervisor Bill Lionheart for his guidance, mo- tivation, inspiration and helpful discussions regarding this thesis and my future career. A special thanks goes to the people at Rapiscan for their funding and support throughout my PhD. In particular I would like to thank Ed Morton and Dan Oberg for their helpful discussions on CT and help with providing data. I want to thank Eric Miller and Brian Tracey from Tufts and Ed Morton again for helping to arrange and fund my US placement, and making me feel very welcome and at home during my time there. I would like to thank my family and friends for their unwavering support throughout the course of my life and PhD. I would also like to thank the EPSRC for their funding support. 7 Author contributions The contributions by the authors to the papers presented in this thesis are stated below: X{ray Compton scattering tomography Authors: James Webber All content by James Webber with helpful suggestions and insight provided by William Lionheart. Three dimensional Compton scattering tomography Authors: James Webber and William Lionheart Idea and analysis of the problem by JW and WL. Written by JW. All proofs and simulations by JW. Micolocal analysis of a spindle transform Authors: James Webber and Sean Holman Idea by JW, SH and WL. Written by JW and SH. Proofs and analyses by JW and SH. Simulations by JW. 8 Chapter 1 Introduction In geometrical inverse problems in tomography we aim to reconstruct an image of a physical quantity (e.g. density or absorption coefficient) represented by a real valued function f : Rn R from its integrals over a set of curves or hypersurfaces ! (n 1 manifolds embedded in Rn), described by a Radon transform − Rf(y) = fdS; (1.1) ZHy n where y parameterises a set of hypersurfaces Hy in R and dS is the surface measure on Hy. For the moment f is a function for which the above integral makes sense. We will visit the types of functions for which various Radon transforms are defined throughout this thesis. The fundamental questions we ask when we perform an analysis of R are listed below: (i) On what classes of functions is R injective? If R is injective, is there an explicit left inverse of R? (ii) What is the range of R? Can we give conditions to determine if a given dataset is in the range of R? Can we fully characterize the range? (iii) How stable is R and if R has a left inverse, how unstable is its left inverse? In what Sobolev spaces is our solution stable, if any? (iv) How are the support of the function f and the support of Rf related? (v) Can we identify the singularities of a function f (image edges) in all di- rections from Rf and if not, in which directions are the singularities the hardest to identify? 9 CHAPTER 1. INTRODUCTION 10 With recent developments in energy sensitive detectors in imaging applications [11, 21], Compton Scattering Tomography (CST) is gaining more relevance, and there are various Radon transforms and imaging ideas already introduced in the literature [48, 46, 41]. In this thesis we introduce a new set of Radon transforms which describe two and three dimensional problems in CST and we provide an analysis of their injectivity, stability and microlocal properties. 1.1 The hyperplane Radon transform The most classical, well studied and well understood example of a Radon trans- form is the hyperplane Radon transform, which gives the integrals of a function n 1 over hyperplanes [45, page 9]. For s R and θ S − we define the hyperplane 2 2 n n n Hs,θ = x R : x θ = s . Then for f S(R ) in the Schwartz space on R f 2 · g 2 (see Appendix A), we define the hyperplane Radon transform Rf(s; θ) = fdS = δ(s x θ)f(x)dx; (1.2) n − · ZHs,θ ZR where δ denotes the Dirac delta function and dS is the surface measure on Hs,θ. n n 1 n+1 Here Rf is defined as a function on the unit cylinder Z = R S − in R . × The hyperplane Radon transform is widely applied in medicine in X-ray CT, emission CT and ultrasound CT, and also has applications in other fields such as astronomy, electron microscopy and nuclear magnetic resonance [19, 34, 16, 10, 62, 17]. The application of the hyperplane Radon transform in two dimensional X-ray CT arises from the Beer-Lambert law [39, 8], which describes the loss in intensity of an incident radiation through a medium fdl I = I0e− L ; (1.3) R where I0 is the initial intensity of a beam travelling along a straight line L, I is the resulting intensity and f is the attenuation coefficient of the medium. Here dl denotes the arc length measure on L. The integral in (1.3) is a value of the X{ ray transform [44] of f, which is defined for attenuation coefficients f S(Rn).
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