Geodesic Tomography Problems on Riemannian Manifolds
Julkaistu sarjassa
JYU dissertationsTekijät
Päivämäärä
2019Tekijänoikeudet
© The Author & University of Jyväskylä
This dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such information determines a function uniquely and stably. A new numerical model for computed tomography imaging is created as a part of the dissertation. The introduction of the dissertation contains an introduction to inverse problems and mathematical models associcated to computed tomography. The main focus is in definitions of integral geometry problems, survey of the related literature, and introducing the main results of the dissertation. A list of important open problems in integral geometry is given. In the first article of the dissertation, it is shown that a symmetric solenoidal tensor field can be determined uniquely from its geodesic ray transform on Cartan-Hadamard manifolds, when certain geometric decay conditions are satisfied. The studied integral transforms appear in inverse scattering theory in quantum physics and general relativity. In the second article of the dissertation, it is shown that a piecewise constant vector-valued function can be determined uniquely from its geodesic ray transform with a continuous and non-singular matrix weight on Riemannian manifolds that admit a strictly convex function and have a strictly convex boundary. These integral transforms can be used to model attenuated ray transforms and inverse problems for connections and Higgs fields. The third and fourth articles of the dissertation study the geodesic ray transform over closed geodesics on flat tori when the functions have low regularity assumptions. The fourth article studies a generalization of the geodesic ray transform when the integrals of a function are known over lower dimensional isometrically embedded flat tori. New inversion formulas, regularization strategies and stability estimates are proved in the articles. The new results have applications in different computational tomography methods.
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Julkaisija
Jyväskylän yliopistoISBN
978-951-39-7958-4ISSN Hae Julkaisufoorumista
2489-9003Julkaisuun sisältyy osajulkaisuja
- Artikkeli I: Lehtonen, J., Railo, J., & Salo, M. (2018). Tensor tomography on Cartan-Hadamard manifolds. Inverse Problems, 34 (4), 044004. DOI: 10.1088/1361-6420/aaaf85
- Artikkeli II: Ilmavirta, J., Railo, J. (2020). Geodesic ray transform with matrix weights for piecewise constant functions. Annales Academiae Scientiarum Fennicae-Mathematica, 45 (2), 1095-1102. DOI: 10.5186/aasfm.2020.4558
- Artikkeli III: Ilmavirta, J., Koskela, O., Railo, J. (2020). Torus Computed Tomography. SIAM Journal on Applied Mathematics, 80 (4), 1947-1976. DOI: 10.1137/19M1268070
- Artikkeli IV: Railo, J. (2020). Fourier Analysis of Periodic Radon Transforms. Journal of Fourier Analysis and Applications, 26 (4), 64. DOI: 10.1007/s00041-020-09775-1
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- JYU Dissertations [748]
- Väitöskirjat [3426]
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Pestov identities and X-ray tomography on manifolds of low regularity
Ilmavirta, Joonas; Kykkänen, Antti (American Institute of Mathematical Sciences (AIMS), 2023)We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds (M, g) with g ∈ C1,1. In addition to a proof, we produce a redefinition of simplicity ... -
Pestov identities and X-ray tomography on manifolds of low regularity
Ilmavirta, Joonas; Kykkänen, Antti (American Institute of Mathematical Sciences (AIMS), 2023)We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds (M, g) with g ∈ C1,1. In addition to a proof, we produce a redefinition of simplicity ... -
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Optimal recovery of a radiating source with multiple frequencies along one line
Brander, Tommi; Ilmavirta, Joonas; Piiroinen, Petteri; Tyni, Teemu (American Institute of Mathematical Sciences (AIMS), 2020)We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam ...
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