Startseite Mathematik Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn
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Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn

  • Rohit Kumar Mishra ORCID logo EMAIL logo
Veröffentlicht/Copyright: 2. Oktober 2019

Abstract

We show that a vector field in n can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve γn. The question of reconstruction of a symmetric m-tensor field from the knowledge of the first m+1 integral moments was posed by Sharafutdinov [Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994, p. 78]. In this work, we provide an answer to Sharafutdinov’s question for the case of vector fields from restricted data comprising of the first two integral moment transforms.

MSC 2010: 46F12; 44A12

Funding statement: The author benefited from the support of the Airbus Group Corporate Foundation Chair “Mathematics of Complex Systems” established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India.

Acknowledgements

The author would like to thank his advisor Venkateswaran P. Krishnan for his guidance and many inspiring discussions.

References

[1] A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, J. Fourier Anal. Appl. 25 (2019), no. 4, 1487–1512. 10.1007/s00041-018-09649-7Suche in Google Scholar

[2] J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J. 55 (1987), no. 4, 943–948. 10.1215/S0012-7094-87-05547-5Suche in Google Scholar

[3] J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic line complexes in three-space, Trans. Amer. Math. Soc. 335 (1993), no. 2, 877–890. 10.1090/S0002-9947-1993-1080733-8Suche in Google Scholar

[4] H. Braun and A. Hauck, Tomographic reconstruction of vector fields, IEEE Trans. Signal Process. 39 (2002), 464–471. 10.1109/78.80830Suche in Google Scholar

[5] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, J. Appl. Phys. 34 (1963), 2722–2727. 10.1063/1.1729798Suche in Google Scholar

[6] A. Denisiuk, Reconstruction in the cone-beam vector tomography with two sources, Inverse Problems 34 (2018), no. 12, Article ID 124008. 10.1088/1361-6420/aae9acSuche in Google Scholar

[7] A. Denisjuk, Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve, Inverse Problems 22 (2006), no. 2, 399–411. 10.1088/0266-5611/22/2/001Suche in Google Scholar

[8] A. S. Denisyuk, Inversion of the generalized Radon transform, Applied Problems of Radon Transform, Amer. Math. Soc. Transl. Ser. 2 162, American Mathematical Society, Providence (1994), 19–32. 10.1090/trans2/162/02Suche in Google Scholar

[9] F. Eriksson, On the measure of solid angles, Math. Mag. 63 (1990), no. 3, 184–187. 10.1080/0025570X.1990.11977515Suche in Google Scholar

[10] G. W. Faris and R. L. Byer, Three-dimensional beam-deflection optical tomography of a supersonic jet, Appl. Optics 27 (1988), 5202–5212. 10.1364/AO.27.005202Suche in Google Scholar PubMed

[11] D. Finch, I.-R. Lan and G. Uhlmann, Microlocal analysis of the x-ray transform with sources on a curve, Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ. 47, Cambridge University, Cambridge (2003), 193–218. Suche in Google Scholar

[12] A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. 10.1215/S0012-7094-89-05811-0Suche in Google Scholar

[13] R. Griesmaier, R. K. Mishra and C. Schmiedecke, Inverse source problems for Maxwell’s equations and the windowed Fourier transform, SIAM J. Sci. Comput. 40 (2018), no. 2, A1204–A1223. 10.1137/17M1150943Suche in Google Scholar

[14] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. An Introduction, London Math. Soc. Lecture Note Ser. 196, Cambridge University, Cambridge, 1994. 10.1017/CBO9780511721441Suche in Google Scholar

[15] V. Guillemin and S. Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), no. 4, 915–955. 10.2307/2373923Suche in Google Scholar

[16] S. Helgason, The Radon Transform, 2nd ed., Progr. Math. 5, Birkhäuser, Boston, 1999. 10.1007/978-1-4757-1463-0Suche in Google Scholar

[17] S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Probl. Imaging 4 (2010), no. 1, 111–130. 10.3934/ipi.2010.4.111Suche in Google Scholar

[18] B. M. Howe, P. F. Worcester and R. C. Spindel, Ocean acoustic tomography: Mesoscale velocity, J. Geophys. Res. Oceans 92 (1987), 3785–3805. 10.1029/JC092iC04p03785Suche in Google Scholar

[19] A. Katsevich, Microlocal analysis of an FBP algorithm for truncated spiral cone beam data, J. Fourier Anal. Appl. 8 (2002), no. 5, 407–425. 10.1007/s00041-002-0020-7Suche in Google Scholar

[20] A. Katsevich, Improved cone beam local tomography, Inverse Problems 22 (2006), no. 2, 627–643. 10.1088/0266-5611/22/2/015Suche in Google Scholar

[21] A. Katsevich and T. Schuster, An exact inversion formula for cone beam vector tomography, Inverse Problems 29 (2013), no. 6, Article ID 065013. 10.1088/0266-5611/29/6/065013Suche in Google Scholar

[22] V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Probl. Imaging 13 (2019), no. 3, 679–701. 10.3934/ipi.2019031Suche in Google Scholar

[23] V. P. Krishnan and R. K. Mishra, Microlocal analysis of a restricted ray transform on symmetric m-tensor fields in n, SIAM J. Math. Anal. 50 (2018), no. 6, 6230–6254. 10.1137/18M1178530Suche in Google Scholar

[24] V. P. Krishnan, R. K. Mishra and F. Monard, On solenoidal-injective and injective ray transforms of tensor fields on surfaces, J. Inverse Ill-Posed Probl. 27 (2019), no. 4, 527–538. 10.1515/jiip-2018-0067Suche in Google Scholar

[25] D. Ludwig, The Radon transform on euclidean space, Comm. Pure Appl. Math. 19 (1966), 49–81. 10.1002/cpa.3160190105Suche in Google Scholar

[26] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), no. 1, 71–83. 10.1007/BF01389295Suche in Google Scholar

[27] W. Munk and C. Wunsch, Ocean acoustic tomography: A scheme for large scale monitoring, Deep Sea Res. Part A. Oceanographic Res. Papers 26 (1979), 123–161. 10.1016/0198-0149(79)90073-6Suche in Google Scholar

[28] V. Palamodov, Reconstruction of a differential form from doppler transform, SIAM J. Math. Anal. 41 (2009), no. 4, 1713–1720. 10.1137/090753917Suche in Google Scholar

[29] L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sibirsk. Mat. Zh. 29 (1988), no. 3, 114–130, 221. 10.1007/BF00969652Suche in Google Scholar

[30] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, 75 years of Radon Transform (Vienna 1992), Conf. Proc. Lecture Notes Math. Phys. 4, International Press, Cambridge (1994), 324–339. 10.1090/psapm/027/692055Suche in Google Scholar

[31] K. Ramaseshan, Microlocal analysis of the Doppler transform on 3, J. Fourier Anal. Appl. 10 (2004), no. 1, 73–82. 10.1007/s00041-004-8004-4Suche in Google Scholar

[32] T. Sato, H. Aoki and O. Ikeda, Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data, J. Acoust. Soc. Amer. 77 (1985), 2104–2106. 10.1121/1.391734Suche in Google Scholar

[33] T. Schuster, The 3D Doppler transform: elementary properties and computation of reconstruction kernels, Inverse Problems 16 (2000), no. 3, 701–722. 10.1088/0266-5611/16/3/311Suche in Google Scholar

[34] T. Schuster, An efficient mollifier method for three-dimensional vector tomography: Convergence analysis and implementation, Inverse Problems 17 (2001), no. 4, 739–766. 10.1088/0266-5611/17/4/312Suche in Google Scholar

[35] V. A. Sharafutdinov, A problem of integral geometry for generalized tensor fields on n, Dokl. Akad. Nauk SSSR 286 (1986), no. 2, 305–307. Suche in Google Scholar

[36] V. A. Sharafutdinov, Integral geometry of a tensor field on a manifold with upper-bounded curvature, Sibirsk. Mat. Zh. 33 (1992), no. 3, 192–204, 221. 10.1007/BF00970902Suche in Google Scholar

[37] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994. 10.1515/9783110900095Suche in Google Scholar

[38] V. A. Sharafutdinov, Slice-by-slice reconstruction algorithm for vector tomography with incomplete data, Inverse Problems 23 (2007), no. 6, 2603–2627. 10.1088/0266-5611/23/6/021Suche in Google Scholar

[39] G. Sparr and K. Strahlen, Vector field tomography: An overview, Technical Report, Centre for Mathematical Sciences, Lund Institute of Technology, Lund, 1998. Suche in Google Scholar

[40] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (2004), no. 3, 445–467. 10.1215/S0012-7094-04-12332-2Suche in Google Scholar

[41] P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (2005), no. 4, 975–1003. 10.1090/S0894-0347-05-00494-7Suche in Google Scholar

[42] P. Stefanov and G. Uhlmann, Integral geometry on tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math. 130 (2008), no. 1, 239–268. 10.1353/ajm.2008.0003Suche in Google Scholar

[43] P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, J. Anal. Math. 136 (2018), no. 1, 151–208. 10.1007/s11854-018-0058-3Suche in Google Scholar

[44] I. E. Svetov, Reconstruction of the solenoidal part of a three-dimensional vector field by its ray transforms along straight lines parallel to coordinate planes, Numer. Anal. Appl. 5 (2012), no. 3, 271–283. 10.1134/S1995423912030093Suche in Google Scholar

[45] 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, Math. Comput. Simulation 97 (2014), 207–223. 10.1016/j.matcom.2013.10.002Suche in Google Scholar

[46] M. Tanaka, S. A. Johnson, J. F. Greenleaf and G. Flandro, Acoustical Holography. Vol. 7, Springer, New York, 1977. Suche in Google Scholar

[47] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math. 43 (1983), no. 3, 546–552. 10.1137/0143035Suche in Google Scholar

[48] G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Invent. Math. 205 (2016), no. 1, 83–120. 10.1007/s00222-015-0631-7Suche in Google Scholar

[49] L. B. Vertgeim, Integral geometry problems for symmetric tensor fields with incomplete data, J. Inverse Ill-Posed Probl. 8 (2000), no. 3, 355–364. 10.1515/jiip.2000.8.3.355Suche in Google Scholar

Received: 2018-03-31
Revised: 2019-08-26
Accepted: 2019-09-09
Published Online: 2019-10-02
Published in Print: 2020-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.3.2026 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2018-0028/html
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