Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 27, Issue 4

Issues

On solenoidal-injective and injective ray transforms of tensor fields on surfaces

Venkateswaran P. Krishnan
  • Centre For Applicable Mathematics, Tata Institute of Fundamental Research, Post Bag No. 6503, GKVK Post Office, Sharada Nagar, Chikkabommasandra, Bangalore 560065, India
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Rohit K. Mishra / François MonardORCID iD: https://orcid.org/0000-0003-2913-7644
Published Online: 2019-01-30 | DOI: https://doi.org/10.1515/jiip-2018-0067

Abstract

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms, which are fully injective over tensor fields. The first approach is by construction of appropriate weights, which vary along the geodesic flow, generalizing the moment transforms. The second one is by changing the pairing with the tensor field to generate a collection of transverse ray transforms.

Keywords: Geodesic X-ray transform; tensor tomography; attenuated ray transform; moment transform; transverse ray transform; reconstruction

MSC 2010: 35R30; 35A22; 44A12

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. (2018), 10.1007/s00041-018-09649-7. Web of ScienceGoogle Scholar

  • [2]

    Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform, J. Math. Pures Appl. (9) 111 (2018), 161–190. Web of ScienceCrossrefGoogle Scholar

  • [3]

    R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer, New York, 1997. Google Scholar

  • [4]

    E. Y. Derevtsov and S. V. Maltseva, Reconstruction of a singular support of a tensor field given in refractive medium from its ray transform, Sib. Zh. Ind. Mat. 18 (2015), no. 3, 11–25. Google Scholar

  • [5]

    E. Y. Derevtsov and V. V. Pickalov, Reconstruction of vector fields and their singularities from ray transform, Numer. Anal. Appl. 4 (2011), no. 1, 21–35. CrossrefGoogle Scholar

  • [6]

    E. Y. Derevtsov and I. E. Svetov, Tomography of tensor fields in the plane, Eurasian J. Math. Comp. 3 (2015), no. 2, 24–68. Google Scholar

  • [7]

    E. Y. Derevtsov, I. E. Svetov, Y. S. Volkov and T. Schuster, Numerical B-spline solution of emission and vector 2D-tomography problems for media with absorption and refraction, 2008 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering—SIBIRCON-08, IEEE Press, Piscataway (2008), 212–217. Google Scholar

  • [8]

    V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology 19 (1980), no. 3, 301–312. CrossrefGoogle Scholar

  • [9]

    H. Hammer and B. Lionheart, Application of Sharafutdinov’s ray transform in integrated photoelasticity, J. Elasticity 75 (2004), no. 3, 229–246. CrossrefGoogle Scholar

  • [10]

    J. Ilmavirta, J. Lehtonen and M. Salo, Geodesic x-ray tomography for piecewise constant functions on nontrapping manifolds, Math. Proc. Cambridge Philos. Soc. (2017), 10.1017/S0305004118000543. Google Scholar

  • [11]

    J. Ilmavirta and F. Monard, Integral geometry on manifolds with boundary and applications, preprint 2018, https://arxiv.org/abs/1806.06088; to appear in Inverse Problems.

  • [12]

    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), no. 3, 245–278. CrossrefGoogle Scholar

  • [13]

    F. Monard, On reconstruction formulas for the ray transform acting on symmetric differentials on surfaces, Inverse Problems 30 (2014), no. 6, Article ID 065001. Web of ScienceGoogle Scholar

  • [14]

    F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging 10 (2016), no. 2, 433–459. CrossrefWeb of ScienceGoogle Scholar

  • [15]

    F. Monard, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal. 48 (2016), no. 2, 1155–1177. Web of ScienceCrossrefGoogle Scholar

  • [16]

    F. Monard, Efficient tensor tomography in fan-beam coordinates. II: Attenuated transforms, Inverse Probl. Imaging 12 (2018), no. 2, 433–460. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    V. Palamodov, On reconstruction of strain fields from tomographic data, Inverse Problems 31 (2015), no. 8, Article ID 085002. Web of ScienceGoogle Scholar

  • [18]

    G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math. 193 (2013), no. 1, 229–247. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Math. Ann. 363 (2015), no. 1–2, 305–362. Web of ScienceCrossrefGoogle Scholar

  • [20]

    G. P. Paternain, M. Salo and G. Uhlmann, On the range of the attenuated ray transform for unitary connections, Int. Math. Res. Not. IMRN 2015 (2015), no. 4, 873–897. CrossrefGoogle Scholar

  • [21]

    M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Differential Geom. 88 (2011), no. 1, 161–187. CrossrefGoogle Scholar

  • [22]

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

  • [23]

    V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1994. Google Scholar

  • [24]

    V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, preprint (2018), https://arxiv.org/abs/1808.00768; to appear in Inverse Probl. Imaging.

  • [25]

    V. A. Sharafutdinov, Integral geometry of a tensor field on a surface of revolution, Sibirsk. Mat. Zh. 38 (1997), no. 3, 697–714. Google Scholar

  • [26]

    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. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2018-07-28

Revised: 2018-11-27

Accepted: 2018-11-30

Published Online: 2019-01-30

Published in Print: 2019-08-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1616564

Award identifier / Grant number: DMS-1814104

V. P. Krishnan is partially supported by NSF grant DMS-1616564 and a SERB Matrics grant. F. Monard is partially funded by NSF grant DMS-1814104 and a Hellmann Fellowship.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 4, Pages 527–538, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2018-0067.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Rohit Kumar Mishra
Journal of Inverse and Ill-posed Problems, 2019, Volume 0, Number 0

Comments (0)

Please log in or register to comment.
Log in