American Institute of Mathematical Sciences

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. 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. 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. 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. 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. 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.), 47 (2003), 47. [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. 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. 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. doi: 10.1007/s00041-014-9344-3. [10] J. Ilmavirta, Broken ray tomography in the disc,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/3/035008. [11] J. Ilmavirta, A Reflection Approach to the Broken Ray Transform,, preprint, (). [12] C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications,, Anal. PDE, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003. [13] F. Natterer, Inversion of the attenuated Radon transform,, Inverse Problems, 17 (2001), 113. doi: 10.1088/0266-5611/17/1/309. [14] F. Natterer, The Mathematics of Computerized Tomography,, SIAM, (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. 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. 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. doi: 10.1137/0524069. [18] E. T. Quinto, An introduction to X-ray tomography and Radon transforms,, in The Radon transform, 63 (2006), 1. 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. 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. 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. [22] P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging,, Anal. PDE, 1 (2008), 115. 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. doi: 10.2140/apde.2012.5.219. [24] S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005). [25] G. Uhlmann and A. Vasy, The Inverse Problem for the Local Geodesic Ray Transform,, preprint, ().

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References:
 [1] G. Bal, On the attenuated Radon transform with full and partial measurements,, Inverse Problems, 20 (2004), 399. 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. 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. 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. 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. 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.), 47 (2003), 47. [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. 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. 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. doi: 10.1007/s00041-014-9344-3. [10] J. Ilmavirta, Broken ray tomography in the disc,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/3/035008. [11] J. Ilmavirta, A Reflection Approach to the Broken Ray Transform,, preprint, (). [12] C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications,, Anal. PDE, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003. [13] F. Natterer, Inversion of the attenuated Radon transform,, Inverse Problems, 17 (2001), 113. doi: 10.1088/0266-5611/17/1/309. [14] F. Natterer, The Mathematics of Computerized Tomography,, SIAM, (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. 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. 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. doi: 10.1137/0524069. [18] E. T. Quinto, An introduction to X-ray tomography and Radon transforms,, in The Radon transform, 63 (2006), 1. 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. 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. 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. [22] P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging,, Anal. PDE, 1 (2008), 115. 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. doi: 10.2140/apde.2012.5.219. [24] S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005). [25] G. Uhlmann and A. Vasy, The Inverse Problem for the Local Geodesic Ray Transform,, preprint, ().
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