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: October 31, 2013

Revised: August 31, 2014

Published: January 2015

Abstract Related Papers Cited by

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.

Keywords:

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

Citation:

Related Papers

Cited by

References

[1]	G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.doi: 10.1088/0266-5611/20/2/006.
[2]	J. Boman, Novikov's inversion formula for the attenuated Radon transform-a new approach, J. Geom. Anal., 14 (2004), 185-198.doi: 10.1007/BF02922067.
[3]	E. Chappa, On the characterization of the kernel of the geodesic X-ray transform, Trans. Amer. Math. Soc., 358 (2006), 4793-4807.doi: 10.1090/S0002-9947-06-04059-1.
[4]	G. Eskin, Inverse boundary value problems in domains with several obstacles, Inverse Problems, 20 (2004), 1497-1516.doi: 10.1088/0266-5611/20/5/011.
[5]	D. Finch, Uniqueness for the attenuated x-ray transform in the physical range, Inverse Problems, 2 (1986), 197-203.doi: 10.1088/0266-5611/2/2/010.
[6]	D. Finch, The attenuated x-ray transform: recent developments, in Inside out: inverse problems and applications (series Math. Sci. Res. Inst. Publ.), Cambridge Univ. Press, 47 (2003), 47-66.
[7]	B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108.doi: 10.1007/s12220-007-9007-6.
[8]	E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301.doi: 10.1016/S0393-0440(01)00039-0.
[9]	M. Hubenthal, The broken ray transform on the square, J. Fourier Anal. Appl., 20 (2014), 1050-1082.doi: 10.1007/s00041-014-9344-3.
[10]	J. Ilmavirta, Broken ray tomography in the disc, Inverse Problems, 29 (2013), 035008, 17pp.doi: 10.1088/0266-5611/29/3/035008.
[11]	J. Ilmavirta, A Reflection Approach to the Broken Ray Transform, preprint, arXiv:1306.0341.
[12]	C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Anal. PDE, 6 (2013), 2003-2048.doi: 10.2140/apde.2013.6.2003.
[13]	F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119.doi: 10.1088/0266-5611/17/1/309.
[14]	F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001.doi: 10.1137/1.9780898719284.
[15]	R. G. Novikov, On the range characterization for the two-dimensional attenuated x-ray transformation, Inverse Problems, 18 (2002), 677-700.doi: 10.1088/0266-5611/18/3/310.
[16]	R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Ark. Mat., 40 (2002), 145-167.doi: 10.1007/BF02384507.
[17]	E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $R^2$ and $R^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.doi: 10.1137/0524069.
[18]	E. T. Quinto, An introduction to X-ray tomography and Radon transforms, in The Radon transform, inverse problems, and tomography" (series Proc. Sympos. Appl. Math.), Amer. Math. Soc., 63 (2006), 1-23.doi: 10.1090/psapm/063/2208234.
[19]	P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.doi: 10.1215/S0012-7094-04-12332-2.
[20]	P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.doi: 10.1090/S0894-0347-05-00494-7.
[21]	P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J., 34 (2008), 67-112.
[22]	P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging, Anal. PDE, 1 (2008), 115-126.doi: 10.2140/apde.2008.1.115.
[23]	P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260.doi: 10.2140/apde.2012.5.219.
[24]	S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005.
[25]	G. Uhlmann and A. Vasy, The Inverse Problem for the Local Geodesic Ray Transform, preprint, arXiv:1210.2084.