Mathematics > Analysis of PDEs
[Submitted on 25 Feb 2013 (v1), last revised 12 Jun 2013 (this version, v2)]
Title:The Broken Ray Transform On The Square
View PDFAbstract:We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for $C_{0}^{2}$ perturbations of the constant unit weight. Given an open subset $E$ of the boundary, we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule. Using the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order -1 plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel, which leads to the desired result.
Submission history
From: Mark Hubenthal [view email][v1] Mon, 25 Feb 2013 18:56:03 UTC (31 KB)
[v2] Wed, 12 Jun 2013 16:28:06 UTC (35 KB)
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