Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds (bibtex)

by Maarten V. de Hoop and 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.

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Reference:

Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds (Maarten V. de Hoop and Joonas Ilmavirta), Inverse Problems, volume 33, number 12, pp. 124003, 2017. (Special issue "100 Years of the Radon Transform".) [show abstract] [hide 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}

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.}
}