Abel transforms with low regularity with applications to x-ray tomography on spherically symmetric manifolds. (8th November 2017)

• Record Type: Journal Article
• Authors: de Hoop, Maarten V; Ilmavirta, Joonas
• Is Part Of: Inverse problems. Volume 33:Number 12(2017:Dec.)
• Publication Date: 2017-11-08
• Subjects: geodesic x-ray tomography -- geophysical imaging -- Abel transforms -- Broken ray tomography -- spherical symmetry; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa9423
• Languages: English
• ISSNs: 0266-5611

Abstract

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.