# American Institute of Mathematical Sciences

May  2016, 10(2): 433-459. doi: 10.3934/ipi.2016007

## Efficient tensor tomography in fan-beam coordinates

 1 Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, United States

Received  October 2015 Revised  February 2016 Published  May 2016

We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization first appearing in [32] to enforce in a fast way classical moment conditions at all orders. When considering direction-dependent integrands (e.g., tensors), a problem where injectivity no longer holds, we propose a suitable representative (other than the traditionally sought-after solenoidal candidate) to be reconstructed, as well as an efficient procedure to do so. Numerical examples illustrating the method are provided at the end.
Citation: François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems and Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007
##### References:
 [1] Y. E. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., 5 (1997), 487-490. doi: 10.1515/jiip.1997.5.6.487. [2] G.-H. Chen and S. Leng, A new data consistency condition for fan-beam projection data, Med. Phys., 32 (2005), 961-965. doi: 10.1118/1.1861395. [3] R. Clackdoyle and L. Desbat, Full data consistency conditions for cone-beam projections with sources on a plane, Physics in medicine and biology, 58 (2013), p8437. doi: 10.1088/0031-9155/58/23/8437. [4] R. Clarkdoyle, Necessary and sufficient consistency conditions for fan-beam pprojections along a line, IEEE transations on Nuclear Science, 60 (2013), 1560-1569. [5] N. S. Dairbekov and V. Sharafutdinov, On conformal killing symmetric tensor fields on riemannian manifolds, (Russian) Mat. Tr., 13 (2010), 85-145. [6] E. Derevtsov and V. Pickalov, Reconstruction of vector fields and their singularities by ray transforms, Numerical Analysis and Applications, 4 (2011), 21-35. doi: 10.1134/S1995423911010034. [7] E. Derevtsov and I. E. Svetov, Tomography of tensor fields in the plain, Eurasian Journal of Mathematical and computer applications, 3 (2015), 25-69. [8] E. Y. Derevtsov, An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems, J. Inv. Ill-Posed Problems, 13 (2005), 213-246. doi: 10.1515/156939405775199587. [9] C. Epstein, Introduction to the Mathematics of Medical Imaging, $2^{nd}$ edition, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717792. [10] I. Gelfand and M. Graev, Integrals over hyperplanes of basic and generalized functions, Dokl. Akad. Nauk. SSSR, 135 (1960), 1307-1310. English transl., Soviet Math. Dokl., 1 (1960), 1369-1372. [11] C. Guillarmou, Invariant distributions and X-ray transform for Anosov flows, J. Diff. Geom., (to appear) (2016). arXiv:1408.4732 [12] ________, Lens rigidity for manifolds with hyperbolic trapped set, (2014) arXiv:1412.1760 [13] C. Guillarmou, G. Paternain, M. Salo and G. Uhlmann, The X-ray transform for connections in negative curvature, Comm. Math. Phys., 343 (2016), 83-127. doi: 10.1007/s00220-015-2510-x. [14] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4. [15] S. Helgason, The Radon Transform, $2^{nd}$ edition, Birkäuser, 1999. doi: 10.1007/978-1-4757-1463-0. [16] G. T. Herman, A. V. Lakshminarayanan and A. Naparstek, Convolution reconstruction techniques for divergent beams, Comput. Biol. Med., 6 (1976), 259-271. [17] S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic x-ray transform with conjugate points, preprint, 2015. arXiv:1502.06545 [18] J. Ilmavirta, On Radon transforms on compact Lie groups, Proceedings of the American Mathematical Society, 144 (2016), 681-691. [19] ________, On Radon transforms on finite groups. arXiv:1411.3829. [20] ________, On Radon transforms on tori, Journal of Fourier Analysis and Applications, 21 (2015), 370-382. [21] S. Kazantsev and A. Bukhgeim, The chebyshev ridge polynomials in 2d tensor tomography, Journal of Inverse and Ill-posed Problems, 14 (2006), 157-188. doi: 10.1515/156939406777571094. [22] 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. doi: 10.1515/1569394042215865. [23] D. Ludwig, The radon transform on euclidean space, Communications on Pure and Applied Mathematics, 19 (1966), 49-81. [24] F. Monard, Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion, SIAM J. Imaging Sciences, 7 (2014), 1335-1357. doi: 10.1137/130938657. [25] ________, On reconstruction formulas for the X-ray transform acting on symmetric differentials on surfaces, Inverse Problems, 30 (2014), 065001. [26] ________, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal. (to appear), (2015). [27] F. Monard, P. Stefanov and G. Uhlmann, The geodesic X-ray transform on Riemannian surfaces with conjugate points, Comm. Math. Phys., 337 (2015), 1491-1513. doi: 10.1007/s00220-015-2328-6. [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284. [29] S. Patch, Moment conditions indirectly improve image quality, Contemporary Mathematics, 278 (2001), 193-205. doi: 10.1090/conm/278/04605. [30] G. Paternain, M. Salo and G. Uhlmann, On the range of the attenuated ray transform for unitary connections, International Math. Research Notices, 2015 (2015), 873-897. doi: 10.1093/imrn/rnt228. [31] ________, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247. [32] 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. doi: 10.1155/S1073792804142116. [33] ________, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110. [34] S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions, Michigan Math, 50 (2002), 71-87. doi: 10.1307/mmj/1022636751. [35] J.Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte lngs gewisser Mannigfaltigkeiten, Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 69 (1917), 262-277. [36] K. Sadiq, O. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors, preprint, 2015.arXiv:1503.04322 [37] 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. [38] ________, On the range of the attenuated radon transform in strictly convex sets, Trans. Amer. Math. Soc., 367 (2015), 5375-5398. [39] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095. [40] ________, Integral geometry of tensor fields on a surface of revolution, Siberian Math. J., 38 (1997). [41] ________, Variations of Dirichlet-to-Neumann map and deformation boundary rigidity on simple 2-manifolds, J. Geom. Anal., 17 (2007), 147-187. [42] ________, Killing tensor fields on the 2-torus, preprint, 2014. arXiv:1411.4741. [43] P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Analysis and PDE, 5 (2012), 219-260. doi: 10.2140/apde.2012.5.219. [44] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, arXiv:1410.5145 (2014). [45] ________, Boundary rigidity with partial data, Journal of the American Mathematical Society (to appear), (2015). [46] I. E. Svetov, E. Y. Derevtsov, Y. S. Volkov and T. Schuster, A numerical solver based on B-splines for 2D vector field tomography in a refracting medium, Mathematics and Computers in Simulation, 97 (2014), 207-223. doi: 10.1016/j.matcom.2013.10.002. [47] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Math. (online), (2015), 1-38. doi: 10.1007/s00222-015-0631-7. [48] G. Van Gompel, M. Defrise and D. Van Dyck, Elliptical extrapolation of truncated 2d ct projections using helgason-ludwig consistency conditions, Medical Imaging, International Society for Optics and Photonics, 6142 (2006), 61424B-61424B. doi: 10.1117/12.653293. [49] A. Welch, C. Campbell, R. Clackdoyle, F. Natterer, M. Hudson, A. Bromiley, P. Mikecz, F. Chillcot, M. Dodd, P. Hopwood, et al., Attenuation correction in pet using consistency information, IEEE Transactions on Nuclear Science, 45 (1998), 3134-3141. doi: 10.1109/23.737676. [50] J. Xu, K. Taguchi and B. M. Tsui, Statistical projection completion in x-ray ct using consistency conditions, IEEE Transactions on Medical Imaging, 29 (2010), 1528-1540. [51] H. Yu and G. Wang, Data consistency based rigid motion artifact reduction in fan-beam ct, IEEE Transactions on Medical Imaging, 26 (2007), 249-260. doi: 10.1109/TMI.2006.889717.

show all references

##### References:
 [1] Y. E. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., 5 (1997), 487-490. doi: 10.1515/jiip.1997.5.6.487. [2] G.-H. Chen and S. Leng, A new data consistency condition for fan-beam projection data, Med. Phys., 32 (2005), 961-965. doi: 10.1118/1.1861395. [3] R. Clackdoyle and L. Desbat, Full data consistency conditions for cone-beam projections with sources on a plane, Physics in medicine and biology, 58 (2013), p8437. doi: 10.1088/0031-9155/58/23/8437. [4] R. Clarkdoyle, Necessary and sufficient consistency conditions for fan-beam pprojections along a line, IEEE transations on Nuclear Science, 60 (2013), 1560-1569. [5] N. S. Dairbekov and V. Sharafutdinov, On conformal killing symmetric tensor fields on riemannian manifolds, (Russian) Mat. Tr., 13 (2010), 85-145. [6] E. Derevtsov and V. Pickalov, Reconstruction of vector fields and their singularities by ray transforms, Numerical Analysis and Applications, 4 (2011), 21-35. doi: 10.1134/S1995423911010034. [7] E. Derevtsov and I. E. Svetov, Tomography of tensor fields in the plain, Eurasian Journal of Mathematical and computer applications, 3 (2015), 25-69. [8] E. Y. Derevtsov, An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems, J. Inv. Ill-Posed Problems, 13 (2005), 213-246. doi: 10.1515/156939405775199587. [9] C. Epstein, Introduction to the Mathematics of Medical Imaging, $2^{nd}$ edition, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717792. [10] I. Gelfand and M. Graev, Integrals over hyperplanes of basic and generalized functions, Dokl. Akad. Nauk. SSSR, 135 (1960), 1307-1310. English transl., Soviet Math. Dokl., 1 (1960), 1369-1372. [11] C. Guillarmou, Invariant distributions and X-ray transform for Anosov flows, J. Diff. Geom., (to appear) (2016). arXiv:1408.4732 [12] ________, Lens rigidity for manifolds with hyperbolic trapped set, (2014) arXiv:1412.1760 [13] C. Guillarmou, G. Paternain, M. Salo and G. Uhlmann, The X-ray transform for connections in negative curvature, Comm. Math. Phys., 343 (2016), 83-127. doi: 10.1007/s00220-015-2510-x. [14] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4. [15] S. Helgason, The Radon Transform, $2^{nd}$ edition, Birkäuser, 1999. doi: 10.1007/978-1-4757-1463-0. [16] G. T. Herman, A. V. Lakshminarayanan and A. Naparstek, Convolution reconstruction techniques for divergent beams, Comput. Biol. Med., 6 (1976), 259-271. [17] S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic x-ray transform with conjugate points, preprint, 2015. arXiv:1502.06545 [18] J. Ilmavirta, On Radon transforms on compact Lie groups, Proceedings of the American Mathematical Society, 144 (2016), 681-691. [19] ________, On Radon transforms on finite groups. arXiv:1411.3829. [20] ________, On Radon transforms on tori, Journal of Fourier Analysis and Applications, 21 (2015), 370-382. [21] S. Kazantsev and A. Bukhgeim, The chebyshev ridge polynomials in 2d tensor tomography, Journal of Inverse and Ill-posed Problems, 14 (2006), 157-188. doi: 10.1515/156939406777571094. [22] 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. doi: 10.1515/1569394042215865. [23] D. Ludwig, The radon transform on euclidean space, Communications on Pure and Applied Mathematics, 19 (1966), 49-81. [24] F. Monard, Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion, SIAM J. Imaging Sciences, 7 (2014), 1335-1357. doi: 10.1137/130938657. [25] ________, On reconstruction formulas for the X-ray transform acting on symmetric differentials on surfaces, Inverse Problems, 30 (2014), 065001. [26] ________, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal. (to appear), (2015). [27] F. Monard, P. Stefanov and G. Uhlmann, The geodesic X-ray transform on Riemannian surfaces with conjugate points, Comm. Math. Phys., 337 (2015), 1491-1513. doi: 10.1007/s00220-015-2328-6. [28] F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001. doi: 10.1137/1.9780898719284. [29] S. Patch, Moment conditions indirectly improve image quality, Contemporary Mathematics, 278 (2001), 193-205. doi: 10.1090/conm/278/04605. [30] G. Paternain, M. Salo and G. Uhlmann, On the range of the attenuated ray transform for unitary connections, International Math. Research Notices, 2015 (2015), 873-897. doi: 10.1093/imrn/rnt228. [31] ________, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247. [32] 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. doi: 10.1155/S1073792804142116. [33] ________, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110. [34] S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions, Michigan Math, 50 (2002), 71-87. doi: 10.1307/mmj/1022636751. [35] J.Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte lngs gewisser Mannigfaltigkeiten, Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 69 (1917), 262-277. [36] K. Sadiq, O. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors, preprint, 2015.arXiv:1503.04322 [37] 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. [38] ________, On the range of the attenuated radon transform in strictly convex sets, Trans. Amer. Math. Soc., 367 (2015), 5375-5398. [39] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095. [40] ________, Integral geometry of tensor fields on a surface of revolution, Siberian Math. J., 38 (1997). [41] ________, Variations of Dirichlet-to-Neumann map and deformation boundary rigidity on simple 2-manifolds, J. Geom. Anal., 17 (2007), 147-187. [42] ________, Killing tensor fields on the 2-torus, preprint, 2014. arXiv:1411.4741. [43] P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Analysis and PDE, 5 (2012), 219-260. doi: 10.2140/apde.2012.5.219. [44] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, arXiv:1410.5145 (2014). [45] ________, Boundary rigidity with partial data, Journal of the American Mathematical Society (to appear), (2015). [46] I. E. Svetov, E. Y. Derevtsov, Y. S. Volkov and T. Schuster, A numerical solver based on B-splines for 2D vector field tomography in a refracting medium, Mathematics and Computers in Simulation, 97 (2014), 207-223. doi: 10.1016/j.matcom.2013.10.002. [47] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Math. (online), (2015), 1-38. doi: 10.1007/s00222-015-0631-7. [48] G. Van Gompel, M. Defrise and D. Van Dyck, Elliptical extrapolation of truncated 2d ct projections using helgason-ludwig consistency conditions, Medical Imaging, International Society for Optics and Photonics, 6142 (2006), 61424B-61424B. doi: 10.1117/12.653293. [49] A. Welch, C. Campbell, R. Clackdoyle, F. Natterer, M. Hudson, A. Bromiley, P. Mikecz, F. Chillcot, M. Dodd, P. Hopwood, et al., Attenuation correction in pet using consistency information, IEEE Transactions on Nuclear Science, 45 (1998), 3134-3141. doi: 10.1109/23.737676. [50] J. Xu, K. Taguchi and B. M. Tsui, Statistical projection completion in x-ray ct using consistency conditions, IEEE Transactions on Medical Imaging, 29 (2010), 1528-1540. [51] H. Yu and G. Wang, Data consistency based rigid motion artifact reduction in fan-beam ct, IEEE Transactions on Medical Imaging, 26 (2007), 249-260. doi: 10.1109/TMI.2006.889717.
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