# American Institute of Mathematical Sciences

February  2015, 9(1): 143-161. doi: 10.3934/ipi.2015.9.143

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

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

Received  November 2013 Revised  September 2014 Published  January 2015

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.
Citation: Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems & Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Ilmavirta, Broken ray tomography in the disc, Inverse Problems, 29 (2013), 035008, 17pp. doi: 10.1088/0266-5611/29/3/035008.  Google Scholar [11] J. Ilmavirta, A Reflection Approach to the Broken Ray Transform,, preprint, ().   Google Scholar [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.  Google Scholar [13] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar [14] F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [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.  Google Scholar [16] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Ark. Mat., 40 (2002), 145-167. doi: 10.1007/BF02384507.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J., 34 (2008), 67-112.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005.  Google Scholar [25] G. Uhlmann and A. Vasy, The Inverse Problem for the Local Geodesic Ray Transform,, preprint, ().   Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Ilmavirta, Broken ray tomography in the disc, Inverse Problems, 29 (2013), 035008, 17pp. doi: 10.1088/0266-5611/29/3/035008.  Google Scholar [11] J. Ilmavirta, A Reflection Approach to the Broken Ray Transform,, preprint, ().   Google Scholar [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.  Google Scholar [13] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar [14] F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [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.  Google Scholar [16] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Ark. Mat., 40 (2002), 145-167. doi: 10.1007/BF02384507.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J., 34 (2008), 67-112.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005.  Google Scholar [25] G. Uhlmann and A. Vasy, The Inverse Problem for the Local Geodesic Ray Transform,, preprint, ().   Google Scholar
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