The Broken Ray Transform in $n$ Dimensions with Flat Reflecting Boundary

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 for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result for the square, 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.

22 pages, 5 figures

References in corpus (3)

• The Calderon problem with partial data on manifolds and applications
• The inverse problem for the local geodesic ray transform
• Broken ray tomography in the disk

Cited by in corpus (5)

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• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• On the broken ray transform
• Artifacts in the inversion of the broken ray transform in the plane
• Survey on Broken Ray Transforms