by Maarten V. de Hoop, Joonas Ilmavirta
Abstract:
We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.
Reference:
Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds (Maarten V. de Hoop, Joonas Ilmavirta), Inverse Problems, volume 33, number 12, pp. 124003, 2017. (Special issue "100 Years of the Radon Transform".)
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We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.
[arXiv]
Bibtex Entry:
@article{rough-radial-xrt,
author = {Maarten V. de Hoop and Joonas Ilmavirta},
title = {{Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds}},
journal = {Inverse Problems},
note = {Special issue "100 Years of the Radon Transform".},
doi = {10.1088/1361-6420/aa9423},
month = nov,
year = {2017},
volume = {33},
number = {12},
pages = {124003},
arxiv = {1702.07625},
gsid = {3482031548206785194},
url={http://users.jyu.fi/~jojapeil/pub/rough-radial-xrt.pdf},
abstract = {We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.}
}