The broken ray transform in $n$ dimensions with flat reflecting boundary

Mark Hubenthal

doi:10.3934/ipi.2015.9.143

Article Contents

2015, Volume 9, Issue 1: 143-161. Doi: 10.3934/ipi.2015.9.143

This issue Previous Article Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type Next Article Overlapping domain decomposition methods for linear inverse problems

The broken ray transform in $n$ dimensions with flat reflecting boundary

Mark Hubenthal^1,

1.
University of Houston Department of Mathematics, Department of Mathematics, 641 PGH, Houston, TX 77204-3008

Received: November 2013

Revised: September 2014

Published: February 2015

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Abstract

We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann ([7]) for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result in [9], although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order $-1$ plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.

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Mathematics Subject Classification: Primary: 45Q05, 47G30; Secondary: 53C65.

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