Weighted Radon transforms and their applications

Abstract : This thesis is devoted to studies of inverse problems for weighted Radon tranforms in euclidean spaces. On one hand, our studies are motivated by applications of weighted Radon transforms in different tomographies, for example, in emission tomographies (PET, SPECT), flourescence tomography and optical tomography. In particular, we develop a new reconstruction approach for tomographies in 3D, where data are modelized by weighted ray transforms along rays parallel to some fixed plane. In this connection our results include: formulas for reduction of the aforementioned weighted ray transforms to weghted Radon transforms along planes in 3D; an analog of Chang approximate inversion formula and an analog of Kunyansky-type iterative inversion algorithm for weighted Radon transforms in multidimensions; numercal reconstructions from simulated and real data. On the other hand, our studies are motivated by mathematical problems related to the aforementioned transforms. More precisely, we continue studies of injectivity and non-injectivity of weighted ray and Radon transforms in multidimensions and we construct a series of counterexamples to injectivity for the latter. These counterexamples are interesting and in some sense unexpected because they are close to the setting when the corresponding weighted ray and Radon transforms become injective. In particular, by one ofour constructions we give counterexamples to well-known injectivity theorems for weighted ray transforms (Quinto (1983), Markoe, Quinto (1985), Finch (1986), Ilmavirta (2016)) when the regularity assumptions on weights are slightly relaxed. By this result we show that, in particular, the regularity assumptions on weights are crucial for the injectivity and there is a breakdown of the latter if the assumptions are slightly relaxed.
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Fedor Goncharov. Weighted Radon transforms and their applications. General Mathematics [math.GM]. Université Paris-Saclay, 2019. English. ⟨NNT : 2019SACLX029⟩. ⟨tel-02273044⟩

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