# 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 & 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. 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. 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). 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. [5] N. S. Dairbekov and V. Sharafutdinov, On conformal killing symmetric tensor fields on riemannian manifolds,, (Russian) Mat. Tr., 13 (2010), 85. [6] E. Derevtsov and V. Pickalov, Reconstruction of vector fields and their singularities by ray transforms,, Numerical Analysis and Applications, 4 (2011), 21. 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. [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. doi: 10.1515/156939405775199587. [9] C. Epstein, Introduction to the Mathematics of Medical Imaging,, $2^{nd}$ edition, (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. [11] C. Guillarmou, Invariant distributions and X-ray transform for Anosov flows,, J. Diff. Geom., (2016). [12] ________, Lens rigidity for manifolds with hyperbolic trapped set,, (2014) , (2014). [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. 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. doi: 10.1016/0040-9383(80)90015-4. [15] S. Helgason, The Radon Transform,, $2^{nd}$ edition, (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. [17] S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic x-ray transform with conjugate points,, preprint, (2015). [18] J. Ilmavirta, On Radon transforms on compact Lie groups,, Proceedings of the American Mathematical Society, 144 (2016), 681. [19] ________, On Radon transforms on finite groups., , (). [20] ________, On Radon transforms on tori,, Journal of Fourier Analysis and Applications, 21 (2015), 370. [21] S. Kazantsev and A. Bukhgeim, The chebyshev ridge polynomials in 2d tensor tomography,, Journal of Inverse and Ill-posed Problems, 14 (2006), 157. 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. doi: 10.1515/1569394042215865. [23] D. Ludwig, The radon transform on euclidean space,, Communications on Pure and Applied Mathematics, 19 (1966), 49. [24] F. Monard, Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion,, SIAM J. Imaging Sciences, 7 (2014), 1335. doi: 10.1137/130938657. [25] ________, On reconstruction formulas for the X-ray transform acting on symmetric differentials on surfaces,, Inverse Problems, 30 (2014). [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. 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. 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. doi: 10.1093/imrn/rnt228. [31] ________, Tensor tomography on surfaces,, Inventiones Math., 193 (2013), 229. [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. doi: 10.1155/S1073792804142116. [33] ________, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid,, Annals of Mathematics, 161 (2005), 1093. [34] S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions,, Michigan Math, 50 (2002), 71. 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, 69 (1917), 262. [36] K. Sadiq, O. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors,, preprint, (2015). [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. doi: 10.1137/140984282. [38] ________, On the range of the attenuated radon transform in strictly convex sets,, Trans. Amer. Math. Soc., 367 (2015), 5375. [39] V. Sharafutdinov, Integral Geometry of Tensor Fields,, VSP, (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. [42] ________, Killing tensor fields on the 2-torus,, preprint, (2014). [43] P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics,, Analysis and PDE, 5 (2012), 219. doi: 10.2140/apde.2012.5.219. [44] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors,, , (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. 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. 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, 6142 (2006). 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. 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. [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. 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. 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. 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). 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. [5] N. S. Dairbekov and V. Sharafutdinov, On conformal killing symmetric tensor fields on riemannian manifolds,, (Russian) Mat. Tr., 13 (2010), 85. [6] E. Derevtsov and V. Pickalov, Reconstruction of vector fields and their singularities by ray transforms,, Numerical Analysis and Applications, 4 (2011), 21. 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. [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. doi: 10.1515/156939405775199587. [9] C. Epstein, Introduction to the Mathematics of Medical Imaging,, $2^{nd}$ edition, (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. [11] C. Guillarmou, Invariant distributions and X-ray transform for Anosov flows,, J. Diff. Geom., (2016). [12] ________, Lens rigidity for manifolds with hyperbolic trapped set,, (2014) , (2014). [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. 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. doi: 10.1016/0040-9383(80)90015-4. [15] S. Helgason, The Radon Transform,, $2^{nd}$ edition, (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. [17] S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic x-ray transform with conjugate points,, preprint, (2015). [18] J. Ilmavirta, On Radon transforms on compact Lie groups,, Proceedings of the American Mathematical Society, 144 (2016), 681. [19] ________, On Radon transforms on finite groups., , (). [20] ________, On Radon transforms on tori,, Journal of Fourier Analysis and Applications, 21 (2015), 370. [21] S. Kazantsev and A. Bukhgeim, The chebyshev ridge polynomials in 2d tensor tomography,, Journal of Inverse and Ill-posed Problems, 14 (2006), 157. 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. doi: 10.1515/1569394042215865. [23] D. Ludwig, The radon transform on euclidean space,, Communications on Pure and Applied Mathematics, 19 (1966), 49. [24] F. Monard, Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion,, SIAM J. Imaging Sciences, 7 (2014), 1335. doi: 10.1137/130938657. [25] ________, On reconstruction formulas for the X-ray transform acting on symmetric differentials on surfaces,, Inverse Problems, 30 (2014). [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. 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. 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. doi: 10.1093/imrn/rnt228. [31] ________, Tensor tomography on surfaces,, Inventiones Math., 193 (2013), 229. [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. doi: 10.1155/S1073792804142116. [33] ________, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid,, Annals of Mathematics, 161 (2005), 1093. [34] S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions,, Michigan Math, 50 (2002), 71. 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, 69 (1917), 262. [36] K. Sadiq, O. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors,, preprint, (2015). [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. doi: 10.1137/140984282. [38] ________, On the range of the attenuated radon transform in strictly convex sets,, Trans. Amer. Math. Soc., 367 (2015), 5375. [39] V. Sharafutdinov, Integral Geometry of Tensor Fields,, VSP, (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. [42] ________, Killing tensor fields on the 2-torus,, preprint, (2014). [43] P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics,, Analysis and PDE, 5 (2012), 219. doi: 10.2140/apde.2012.5.219. [44] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors,, , (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. 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. 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, 6142 (2006). 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. 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. [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. doi: 10.1109/TMI.2006.889717.
 [1] François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019 [2] Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471 [3] François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713 [4] Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453 [5] Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems & Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015 [6] Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649 [7] Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27 [8] Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 [9] Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879 [10] Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029 [11] Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009 [12] Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 [13] Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089 [14] Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems & Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693 [15] Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373 [16] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [17] Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721 [18] Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 [19] Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013 [20] Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems & Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619

2018 Impact Factor: 1.469