Complex-Analytic Methods in Reconstructive Integral Geometry
Nicholas McMurray Hoell
- Complex-Analytic Methods in Reconstructive Integral Geometry
- Hoell, Nicholas McMurray
- Thesis Advisor(s):
- Bal, Guillaume
- Applied Physics and Applied Mathematics
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- Ph.D., Columbia University.
- Each chapter is self-contained yet thematically dependent. A review of some of the main objectives and techniques in reconstructive integral geometry is presented. The inverse problem central to the author's work, namely reconstructing a function of position from its averages over a class of curves in the unit disc, is then introduced. We give several new results on that front and present explicit ltered backprojection inversion formulae for the attenuated and non-attenuated X-ray transform over a wide class of curves in a simply-connected region of 2-dimensional Euclidean space. The method used to derive these formulae is based on the complexication of the vector felds defing the particle transport, thereby making the problem amenable to complex-analytic techniques. The remainder of the thesis can largely be considered to be variations on this theme. The proof of the reconstruction procedure we give demands the vector fields governing the transport satisfy a somewhat stringent condition we call condition H. A thorough investigation of this condition is then presented culminating in results applicable to a certain subset of the space of vector fields with polynomial coecients. This is done by explicitly looking at the complexification and appealing to the logarithmic Poisson-Jensen formula as well as some results on quasi-conformal mapping theory. These results are used in establishing a stability estimate on a natural generalization of this polynomial space in real-analytic functions in an attempt to address approximation/truncation concerns. Finally, we take up a variant of this problem on 2-dimensional, simple and compact Riemannian manifolds with boundary. In this case, we deal with data that is of a fanbeam type and thereby have to concern ourselves with the boundary of the domain we are interested in probing. A related but somewhat less direct means of complexification is used in this case on local trivializations of the unitized bundle of Hamiltonian coordinates. A novel derivation of an existing formula is then presented using a similar approach as was considered in earlier chapters. This serves as a prelude to a new result for an explicit formula for inverting the attenuated ray transform in such settings modulo a Fredholm error. We close with an appendix on containing all useful geometrical jargon used throughout.
- Applied mathematics
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