- Previous Article
- IPI Home
- This Issue
-
Next Article
Capped $\ell_p$ approximations for the composite $\ell_0$ regularization problem
Stability estimates in tensor tomography
1. | Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden |
2. | Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia |
3. | Novosibirsk State University, 2 Pirogov street, Novosibirsk, 630090, Russia |
We study the X-ray transform $I$ of symmetric tensor fields on a smooth convex bounded domain $Ω\subset{\mathbb R}^n$. The main result is the stability estimate $\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$, where $^{s}f$ is the solenoidal part of the tensor field $f$. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.
References:
[1] |
G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. |
[2] |
G. Fichera, Existence Theorems in Elasticity, Springer, 1972. |
[3] |
K. O. Friedrichs,
On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.
doi: 10.2307/1969180. |
[4] |
J. Gobert,
Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.
|
[5] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983.
doi: 10.1007/978-3-642-96750-4. |
[6] |
V. A. Kondrat'ev and O. A. Oleinik,
Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.
doi: 10.1070/RM1988v043n05ABEH001945. |
[7] |
A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269.
doi: 10.5802/afst.251. |
[8] |
W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
[9] |
S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964. |
[10] |
F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986. |
[11] |
L. E. Payne and H. F. Weinberger,
On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.
doi: 10.1007/BF00277432. |
[12] |
L. N. Pestov and V. A. Sharafutdinov,
Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.
doi: 10.1007/BF00969652. |
[13] |
Yu. G. Reshetnyak,
Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.
|
[14] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
[15] |
V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp).
doi: 10.1088/1361-6420/33/2/025002. |
[16] |
V. Sharafutdinov, M. Skokan and G. Uhlmann,
Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.
doi: 10.1007/BF02930983. |
[17] |
V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
[18] |
P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007.
doi: 10.1088/1742-6596/124/1/012007. |
[19] |
M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4684-9320-7. |
show all references
References:
[1] |
G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. |
[2] |
G. Fichera, Existence Theorems in Elasticity, Springer, 1972. |
[3] |
K. O. Friedrichs,
On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.
doi: 10.2307/1969180. |
[4] |
J. Gobert,
Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.
|
[5] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983.
doi: 10.1007/978-3-642-96750-4. |
[6] |
V. A. Kondrat'ev and O. A. Oleinik,
Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.
doi: 10.1070/RM1988v043n05ABEH001945. |
[7] |
A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269.
doi: 10.5802/afst.251. |
[8] |
W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
[9] |
S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964. |
[10] |
F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986. |
[11] |
L. E. Payne and H. F. Weinberger,
On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.
doi: 10.1007/BF00277432. |
[12] |
L. N. Pestov and V. A. Sharafutdinov,
Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.
doi: 10.1007/BF00969652. |
[13] |
Yu. G. Reshetnyak,
Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.
|
[14] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
[15] |
V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp).
doi: 10.1088/1361-6420/33/2/025002. |
[16] |
V. Sharafutdinov, M. Skokan and G. Uhlmann,
Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.
doi: 10.1007/BF02930983. |
[17] |
V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
[18] |
P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007.
doi: 10.1088/1742-6596/124/1/012007. |
[19] |
M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4684-9320-7. |
[1] |
Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems and Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020 |
[2] |
Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471 |
[3] |
François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems and Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713 |
[4] |
Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems and Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015 |
[5] |
Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089 |
[6] |
Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems and Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 |
[7] |
Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems and Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453 |
[8] |
Nuutti Hyvönen, Martti Kalke, Matti Lassas, Henri Setälä, Samuli Siltanen. Three-dimensional dental X-ray imaging by combination of panoramic and projection data. Inverse Problems and Imaging, 2010, 4 (2) : 257-271. doi: 10.3934/ipi.2010.4.257 |
[9] |
Arun K. Kulshreshth, Andreas Alpers, Gabor T. Herman, Erik Knudsen, Lajos Rodek, Henning F. Poulsen. A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data. Inverse Problems and Imaging, 2009, 3 (1) : 69-85. doi: 10.3934/ipi.2009.3.69 |
[10] |
Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems and Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431 |
[11] |
Weihao Shen, Wenbo Xu, Hongyang Zhang, Zexin Sun, Jianxiong Ma, Xinlong Ma, Shoujun Zhou, Shijie Guo, Yuanquan Wang. Automatic segmentation of the femur and tibia bones from X-ray images based on pure dilated residual U-Net. Inverse Problems and Imaging, 2021, 15 (6) : 1333-1346. doi: 10.3934/ipi.2020057 |
[12] |
Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc. Inverse Problems and Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 |
[13] |
Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov. Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021076 |
[14] |
James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137 |
[15] |
Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27 |
[16] |
Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 |
[17] |
Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 |
[18] |
Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems and Imaging, 2020, 14 (1) : 1-26. doi: 10.3934/ipi.2019061 |
[19] |
Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems and Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009 |
[20] |
François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems and Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]