On solenoidal-injective and injective ray transforms of tensor fields on surfaces
-
Venkateswaran P. Krishnan
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.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1616564
Award Identifier / Grant number: DMS-1814104
Funding statement: 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.
Acknowledgements
The authors would like to thank Gunther Uhlmann and the Institute of Advanced Studies of the Hong Kong University of Science and Technology for their hospitality, as this work was initiated during the “Inverse Problems, Imaging and Partial Differential Equations” workshop organized there in December 2016. The authors also thank the anonymous referees for useful comments.
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. 10.1007/s00041-018-09649-7Suche in Google 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. 10.1016/j.matpur.2017.09.006Suche in Google Scholar
[3] R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer, New York, 1997. 10.1007/978-1-4612-0653-8Suche in 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. Suche in 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. 10.1134/S1995423911010034Suche in Google 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. 10.32523/2306-6172-2015-3-2-25-69Suche in 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. 10.1109/SIBIRCON.2008.4602618Suche in Google Scholar
[8] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology 19 (1980), no. 3, 301–312. 10.1016/0040-9383(80)90015-4Suche in Google Scholar
[9] H. Hammer and B. Lionheart, Application of Sharafutdinov’s ray transform in integrated photoelasticity, J. Elasticity 75 (2004), no. 3, 229–246. 10.1007/s10659-004-7191-1Suche in Google 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. 10.1017/S0305004118000543Suche in 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. 10.1515/9783110560855-004Suche in Google Scholar
[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. 10.1515/1569394042215865Suche in Google 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. 10.1088/0266-5611/30/6/065001Suche in Google Scholar
[14] F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging 10 (2016), no. 2, 433–459. 10.3934/ipi.2016007Suche in Google 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. 10.1137/15M1016412Suche in Google Scholar
[16] F. Monard, Efficient tensor tomography in fan-beam coordinates. II: Attenuated transforms, Inverse Probl. Imaging 12 (2018), no. 2, 433–460. 10.3934/ipi.2018019Suche in Google Scholar
[17] V. Palamodov, On reconstruction of strain fields from tomographic data, Inverse Problems 31 (2015), no. 8, Article ID 085002. 10.1088/0266-5611/31/8/085002Suche in Google Scholar
[18] G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math. 193 (2013), no. 1, 229–247. 10.1007/s00222-012-0432-1Suche in Google 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. 10.1007/s00208-015-1169-0Suche in Google 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. 10.1093/imrn/rnt228Suche in Google Scholar
[21] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Differential Geom. 88 (2011), no. 1, 161–187. 10.4310/jdg/1317758872Suche in Google Scholar
[22]
V. A. Sharafutdinov,
A problem of integral geometry for generalized tensor fields on
[23] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 1994. 10.1515/9783110900095Suche in 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. 10.3934/ipi.2019031Suche in Google Scholar
[25] V. A. Sharafutdinov, Integral geometry of a tensor field on a surface of revolution, Sibirsk. Mat. Zh. 38 (1997), no. 3, 697–714. 10.1007/BF02683847Suche in 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. 10.1016/j.matcom.2013.10.002Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Numerical solution of a source identification problem: Almost coercivity
- Invertibility and stability for a generic class of radon transforms with application to dynamic operators
- A class of homotopy with regularization for nonlinear ill-posed problems in Hilbert space
- Inverse nodal problems for integro-differential operators with a constant delay
- Unique continuation for a reaction-diffusion system with cross diffusion
- On solenoidal-injective and injective ray transforms of tensor fields on surfaces
- The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius
- Identification of point sources in an elliptic equation from interior measurements: Application to a seawater intrusion problem
- Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints
- On the travel time tomography problem in 3D
Artikel in diesem Heft
- Frontmatter
- Numerical solution of a source identification problem: Almost coercivity
- Invertibility and stability for a generic class of radon transforms with application to dynamic operators
- A class of homotopy with regularization for nonlinear ill-posed problems in Hilbert space
- Inverse nodal problems for integro-differential operators with a constant delay
- Unique continuation for a reaction-diffusion system with cross diffusion
- On solenoidal-injective and injective ray transforms of tensor fields on surfaces
- The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius
- Identification of point sources in an elliptic equation from interior measurements: Application to a seawater intrusion problem
- Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints
- On the travel time tomography problem in 3D
