# American Institute of Mathematical Sciences

• Previous Article
A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification
• IPI Home
• This Issue
• Next Article
Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry
August  2022, 16(4): 787-826. doi: 10.3934/ipi.2021076

## Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

 1 TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Yelahanka New Town, Bangalore, India 2 Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia

Dedicated to the memory of our teacher Yuri Grigor'evich Reshetnyak

Received  June 2021 Revised  October 2021 Published  August 2022 Early access  December 2021

Fund Project: The first author was supported by India SERB Matrics Grant MTR/2017/000837, and the second author was supported by RFBR, Grant 20-51-15004 (joint French – Russian grant)

For an integer $r\ge0$, we prove the $r^{\mathrm{th}}$ order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on ${{\mathbb R}}^n$. Roughly speaking, for a tensor field $f$, the order $r$ refers to $L^2$-integrability of higher order derivatives of the Fourier transform $\widehat f$ over spheres centered at the origin. Certain differential operators $A^{(m,r,l)}\ (0\le l\le r)$ on the sphere ${{\mathbb S}}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.

Citation: Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov. Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas. Inverse Problems and Imaging, 2022, 16 (4) : 787-826. doi: 10.3934/ipi.2021076
##### References:
 [1] I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York-London, (1966). [2] S. Helgason, The Radon Transform, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4757-1463-0. [3] F. John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J., 2 (1938), 300-322.  doi: 10.1215/S0012-7094-38-00423-5. [4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, London, Sydney, 1963. [5] V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679-701.  doi: 10.3934/ipi.2019031. [6] R. Seeley, Complex powers of an elliptic operator, In Proceeding of Symposia in Pure Mathematics, Vol. X, Singular Integrals, American Mathematical Society, Providence, R.I., 1967,288–307. [7] V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht (1994). doi: 10.1515/9783110900095. [8] V. A. 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. [9] V. A. Sharafutdinov, X-ray transform on Sobolev spaces, Inverse Problems, 37 (2021), 015007, 25 pp. doi: 10.1088/1361-6420/abb5e0. [10] V. A. Sharafutdinov, Radon transform on Sobolev spaces, Siberian Math. J., 62 (2021), 560-580.

show all references

##### References:
 [1] I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York-London, (1966). [2] S. Helgason, The Radon Transform, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4757-1463-0. [3] F. John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J., 2 (1938), 300-322.  doi: 10.1215/S0012-7094-38-00423-5. [4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, London, Sydney, 1963. [5] V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679-701.  doi: 10.3934/ipi.2019031. [6] R. Seeley, Complex powers of an elliptic operator, In Proceeding of Symposia in Pure Mathematics, Vol. X, Singular Integrals, American Mathematical Society, Providence, R.I., 1967,288–307. [7] V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht (1994). doi: 10.1515/9783110900095. [8] V. A. 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. [9] V. A. Sharafutdinov, X-ray transform on Sobolev spaces, Inverse Problems, 37 (2021), 015007, 25 pp. doi: 10.1088/1361-6420/abb5e0. [10] V. A. Sharafutdinov, Radon transform on Sobolev spaces, Siberian Math. J., 62 (2021), 560-580.
 [1] 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 [2] 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 [3] 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 [4] 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 [5] Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79. [6] 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 [7] 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 [8] 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 [9] Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems and Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 [10] Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems and Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013 [11] 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 [12] Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619 [13] Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems and Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143 [14] James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems and Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013 [15] Ruixue Zhao, Jinyan Fan. Quadratic tensor eigenvalue complementarity problems. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022073 [16] Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems and Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 [17] 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 [18] Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049 [19] Wei-Kang Xun, Shou-Fu Tian, Tian-Tian Zhang. Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021259 [20] Yuan Li, Shou-Fu Tian. Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation. Communications on Pure and Applied Analysis, 2022, 21 (1) : 293-313. doi: 10.3934/cpaa.2021178

2021 Impact Factor: 1.483

Article outline