On the range of the $ X $-ray transform of symmetric tensors compactly supported in the plane

Kamran Sadiq; Alexandru Tamasan

doi:10.3934/ipi.2022070

Article Contents

2023, Volume 17, Issue 3: 660-685. Doi: 10.3934/ipi.2022070

This issue Previous Article A distance function based cascaded neural network for accurate polyps segmentation and classification Next Article Coupling local and nonlocal fourth-order evolution equations for image denoising

On the range of the $ X $-ray transform of symmetric tensors compactly supported in the plane

Kamran Sadiq^1,2, , and
Alexandru Tamasan^3,

1.
Johann Radon Institute of Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040 Linz, Austria
2.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
3.
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

^*Corresponding author: Kamran Sadiq
^*Corresponding author: Kamran Sadiq

Received: April 2022

Revised: October 2022

Early access: December 2022

Published: June 2023

Abstract / Introduction Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

We find the necessary and sufficient conditions for the Fourier coefficients of a function $ g $ on the torus to be in the range of the $ X $-ray transform of a symmetric tensor of compact support in the plane.

Keywords:

$ X $-ray transform,
Radon transform,
fan-beam coordinates,
Gelfand-Graev-Helgason-Ludwig moment conditions,
$ A $-analytic maps,
Hilbert transform.

Mathematics Subject Classification: Primary: 44A12, 35J56; Secondary: 45E05.

Citation:

Full Text(HTML)

Figure 1. Fan-beam coordinates: $ e^{i \beta}\in \varGamma $, $ e^{i\theta}\in {{\mathbb S}^ 1} $, and $ \boldsymbol \theta = (\cos \theta, \sin \theta) $

Download: Full-size image PowerPoint slide

Figure 2. An even order $ m $-tensor field $ {\mathbf f} $ is determined by the odd negative angular modes on or above the diagonal $ k = -n $ (green region), and the odd negative angular modes (marked red) on the $ \frac{m}{2} $ red lines $ n+2k = -(m+1) $ for $ k\geq0 $. All the odd non-positive angular modes on and below the line $ n+2k = -(m+1) $, and left of the line $ n = -(m+1) $ vanish

Download: Full-size image PowerPoint slide

Figure 3. An odd order $ m $-tensor field $ {\mathbf f} $ is determined by the even negative angular modes on or above the diagonal $ k = -n $ (green region), and the even negative angular modes (marked red) on the $ \frac{m+1}{2} $ red lines $ n+2k = -(m+1) $ for $ k\geq0 $. All the even non-positive angular modes on and below the line $ n+2k = -(m+3) $, and left of the line $ n = -(m+3) $ vanish

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	È. V. Arzubov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math., 8 (1998), 1-20.
[2]	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, 111 (2018), 161-190. doi: 10.1016/j.matpur.2017.09.006.
[3]	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.
[4]	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. doi: 10.1007/BF02922067.
[5]	A. L. Bukhgeim, Inversion Formulas in Inverse Problems, in Linear Operators and Ill-Posed Problems, (eds. M. M. Lavrentev and L. Ya. Savalev), Plenum, New York, (1995).
[6]	G.-H. Chen and S. Leng, A new data consistency condition for fan-beam projection data, Med. Phys., 32 (2005), 961-967. doi: 10.1118/1.1861395.
[7]	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), 8437. doi: 10.1088/0031-9155/58/23/8437.
[8]	R. Clackdoyle, Necessary and sufficient consistency conditions for fan-beam projections along a line, IEEE Transations on Nuclear Science, 60 (2013), 1560-1569. doi: 10.1109/TNS.2013.2251901.
[9]	E. Derevtsov and I. Svetov, Tomography of tensor fields in the plane, Eurasian J. Math. Comput. Appl., 3, (2015), 24-68. doi: 10.32523/2306-6172-2015-3-2-25-69.
[10]	D. V. Finch, The attenuated x-ray transform: Recent developments, in Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47 (2003), Cambridge Univ. Press, Cambridge, 47-66.
[11]	I. M. Gel'fand and M. I. Graev, Integrals over hyperplanes of basic and generalized functions, Soviet Math. Dokl., 1 (1960), 1369-1372.
[12]	S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogenous spaces and Grassmann manifolds, Acta Math., 113 (1965), 153-180. doi: 10.1007/BF02391776.
[13]	S. Helgason, The Radon Transform, 2$^{nd}$ edition, Birkhäuser, Boston, (1999). doi: 10.1007/978-1-4757-1463-0.
[14]	J. Ilmavirta, O. Koskela and J. Railo, Torus computed tomography, SIAM Journal on Applied Mathematics, 80 (2020), 1947-1976. doi: 10.1137/19M1268070.
[15]	S. G. Kazantsev and A. A. Bukhgeim, Singular value decomposition for the 2D fan-beam Radon transform of tensor fields, J. Inverse Ill-Posed Probl., 12 (2004), 245-278. doi: 10.1163/1569394042215865.
[16]	S. G. Kazantsev and A. A. Bukhgeim, The Chebyshev ridge polynomials in 2D tensor tomography, J. Inverse Ill-Posed Probl., 14 (2006), 157-188. doi: 10.1163/156939406777571094.
[17]	S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated X-ray transforms in a unit disc, J. Inverse Ill-Posed Probl., 15 (2007), 735-765. doi: 10.1515/jiip.2007.040.
[18]	V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, II: Range characterization in the Schwartz space, Inverse Problems, 36 (2020), 045009, 33pp. doi: 10.1088/1361-6420/ab6a65.
[19]	P. A. Kuchment and S. Ya. L'vin, Range of the Radon exponential transform, Soviet Math. Dokl., 42 (1991), 183-184.
[20]	W. Li, K. Ren and D. Rim, A range characterization of the single-quadrant ADRT, Math. Comp., 92 (2023), 283-306. doi: 10.1090/mcom/3750.
[21]	D. Ludwig, The Radon transform on euclidean space, Comm. Pure Appl. Math., 19 (1966), 49-81. doi: 10.1002/cpa.3160190207.
[22]	F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459. doi: 10.3934/ipi.2016007.
[23]	F. Monard, Efficient tensor tomography in fan-beam coordinates. II: Attenuated transforms, Inverse Probl. Imaging, 12 (2018), 433-460. doi: 10.3934/ipi.2018019.
[24]	N. I. Muskhelishvili, Singular Integral Equations, 2$^{nd}$ edition, Dover, New York, 2008.
[25]	F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986.
[26]	F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.
[27]	R. G. Novikov, Une formule d'inversion pour la transformation d'un rayonnement X atténué, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 1059-1063. doi: 10.1016/S0764-4442(01)01965-6.
[28]	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.
[29]	D. Omogbhe and K. Sadiq, On the $X$-ray transform of planar symmetric higher order tensors, preprint, 2022, arXiv: 2210.01473.
[30]	E. Yu. Pantyukhina, Description of the image of a ray transformation in the two-dimensional case (in Russian), Methods for Solving Inverse Problems (Russian), (1990) 80-89.
[31]	S. K. Patch, Moment conditions indirectly improve image quality, in Radon Transforms and Tomography, (South Hadley, MA, 2000), 193-205, Contemp. Math., 278, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/conm/278/04605.
[32]	G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chin. Ann. Math. Ser. B., 35 (2014), 399-428. doi: 10.1007/s11401-014-0834-z.
[33]	L. Pestov and G. Uhlmann, On characterization of the range and inversion formulas for the geodesic X-ray transform, Int. Math. Res. Not., 80 (2004), 4331-4347. doi: 10.1155/S1073792804142116.
[34]	J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math-Nat., K 1 69 (1917), 262-267. doi: 10.1090/psapm/027/692055.
[35]	K. Sadiq, O. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors, J. Math. Anal. Appl., 442 (2016), 31-49. doi: 10.1016/j.jmaa.2016.04.018.
[36]	K. Sadiq and A. Tamasan, On the range of the attenuated Radon transform in strictly convex sets, Trans. Amer. Math. Soc., 367 (2015), 5375-5398. doi: 10.1090/S0002-9947-2014-06307-1.
[37]	K. Sadiq and A. Tamasan, On the range characterization of the two dimensional attenuated Doppler transform, SIAM J. Math.Anal., 47 (2015), 2001-2021. doi: 10.1137/140984282.
[38]	K. Sadiq and A. Tamasan, On the range of the planar $X$-ray transform on the Fourier lattice of the torus, preprint, 2022, arXiv: 2201.10926v2.
[39]	V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.
[40]	V. A. Sharafutdinov, Ray Transform on Riemannian manifolds, Lecture Notes at the University of Washington, Spring, 1999.