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

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.

41 pages

References in corpus (6)

• The geodesic ray transform on Riemannian surfaces with conjugate points
• Broken ray tomography in the disk
• On Radon transforms on tori
• Broken ray transform on a Riemann surface with a convex obstacle
• On the broken ray transform
• Spectral rigidity for spherically symmetric manifolds with boundary

Cited by in corpus (5)

• Unique continuation of the normal operator of the X-ray transform and applications in geophysics
• The Broken Ray Transform: Additional Properties and New Inversion Formula
• Inversion of Integral Models: a Neural Network Approach
• Functions of constant geodesic X-ray transform
• Integral geometry and unique continuation principles