# American Institute of Mathematical Sciences

April  2018, 12(2): 433-460. doi: 10.3934/ipi.2018019

## Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms

 University of California Santa Cruz, Department of Mathematics, 1156 High St. Santa Cruz, CA 95064, USA

Received  May 2017 Revised  October 2017 Published  February 2018

Fund Project: The author is supported by NSF grant DMS-1712790.

This article extends the author's past work [11] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the attenuated tensor tomography problem on the Euclidean unit disc in fan-beam coordinates. For a tensor of arbitrary order, we propose an equivalent tensor of the same order which can be uniquely and stably reconstructed from its attenuated transform, as well as an explicit and efficient procedure to do so.

Citation: François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems and Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019
##### References:
 [1] G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46. [2] E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. [3] Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform, Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. [4] G. Bal, On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418. [5] G. Bal and A. Tamasan, Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76. [6] J. Boman and J.-O. Strömberg, Novikov's inversion formula for the attenuated radon transform--a new approach, J. Geom. Anal., 14 (2004), 185-198. [7] D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003. [8] S. Holman and P. Stefanov, The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130. [9] S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765. [10] S. G. Kazantsev and A. A. Bukhgeim, Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278. [11] F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459. [12] F. Monard, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177. [13] F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$ -connection, to appear in Journal of Geometric Analysis, arXiv: 1610.09571. [14] F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119. [15] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. [16] R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math., 40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de Mathématiques). [17] G. Paternain, Inside Out II, chapter Inverse problems for connections, 2012. [18] G. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489. [19] G. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247. [20] G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.  doi: 10.4310/jdg/1406137697. [21] G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.  doi: 10.1007/s00208-015-1169-0. [22] G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903. [23] L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347. [24] L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093. [25] K. Sadiq, O. Scherzer and A. Tamasan, On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.  doi: 10.1016/j.jmaa.2016.04.018. [26] K. Sadiq and A. Tamasan, On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.  doi: 10.1137/140984282. [27] K. Sadiq and A. Tamasan, On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398. [28] M. Salo and G. Uhlmann, The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872. [29] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. [30] G. Sparr, K. Strahlen, K. Lindstrom and H. Persson, Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.  doi: 10.1088/0266-5611/11/5/009. [31] P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.  doi: 10.2140/apde.2008.1.115. [32] E. M. Stein and R. Shakarchi, Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005. [33] W. A. Strauss, Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. [34] A. Tamasan, Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.  doi: 10.1088/0266-5611/23/5/022. [35] H. Zhou, Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.  doi: 10.1080/03605302.2017.1295061.

show all references

##### References:
 [1] G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46. [2] E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. [3] Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform, Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. [4] G. Bal, On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418. [5] G. Bal and A. Tamasan, Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76. [6] J. Boman and J.-O. Strömberg, Novikov's inversion formula for the attenuated radon transform--a new approach, J. Geom. Anal., 14 (2004), 185-198. [7] D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003. [8] S. Holman and P. Stefanov, The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130. [9] S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765. [10] S. G. Kazantsev and A. A. Bukhgeim, Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278. [11] F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459. [12] F. Monard, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177. [13] F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$ -connection, to appear in Journal of Geometric Analysis, arXiv: 1610.09571. [14] F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119. [15] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. [16] R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math., 40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de Mathématiques). [17] G. Paternain, Inside Out II, chapter Inverse problems for connections, 2012. [18] G. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489. [19] G. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247. [20] G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.  doi: 10.4310/jdg/1406137697. [21] G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.  doi: 10.1007/s00208-015-1169-0. [22] G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903. [23] L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347. [24] L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093. [25] K. Sadiq, O. Scherzer and A. Tamasan, On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.  doi: 10.1016/j.jmaa.2016.04.018. [26] K. Sadiq and A. Tamasan, On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.  doi: 10.1137/140984282. [27] K. Sadiq and A. Tamasan, On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398. [28] M. Salo and G. Uhlmann, The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872. [29] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. [30] G. Sparr, K. Strahlen, K. Lindstrom and H. Persson, Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.  doi: 10.1088/0266-5611/11/5/009. [31] P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.  doi: 10.2140/apde.2008.1.115. [32] E. M. Stein and R. Shakarchi, Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005. [33] W. A. Strauss, Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. [34] A. Tamasan, Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.  doi: 10.1088/0266-5611/23/5/022. [35] H. Zhou, Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.  doi: 10.1080/03605302.2017.1295061.
Fan-beam coordinates, scattering relation $\mathcal{S}$ and antipodal scattering relation ${{\mathcal{S}}_{A}}$
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