June  2020, 14(3): 423-435. doi: 10.3934/ipi.2020020

On range condition of the tensor x-ray transform in $ \mathbb R^n $

University of Warmia and Mazury, Faculty of Mathematics and Computer Science, ul. Słoneczna 54, 10-710 Olsztyn, Poland

* Corresponding author: Aleksander Denisiuk

Received  April 2019 Revised  November 2019 Published  March 2020

Consider the problem of the range description of the tensor x-ray transform in $ \mathbb R^n $, $ n\ge3 $. In this paper we use the relation between the x-ray transform and the Radon transform to obtain a geometrical interpretation of the range condition and related John differential operator. As a corollary, it is proved that the range of the $ m $-tensor x-ray transform in $ \mathbb R^n $ can be described by $ \binom{n+m-2}{m+1} $ linear differential equations of order $ 2(m+1) $.

Citation: Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020
References:
[1]

A. Denisiuk, Inversion of the x-ray transform for complexes of lines in $\Bbb R^n$, Inverse Problems, 32 (2016), 025007. doi: 10.1088/0266-5611/32/2/025007.  Google Scholar

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A. Denisiuk, Reconstruction in the cone-beam vector tomography with two sources, Inverse Problems, 34 (2018), 124008. doi: 10.1088/1361-6420/aae9ac.  Google Scholar

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A. Denisjuk, Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve, Inverse Problems, 22 (2006), 399-411.  doi: 10.1088/0266-5611/22/2/001.  Google Scholar

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F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted radon transforms along hyperplanes in multidimensions, Inverse Problems, 34 (2018), 054001. doi: 10.1088/1361-6420/aab24d.  Google Scholar

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P. Maass, The x-ray transform: Singular value decomposition and resolution, Inverse Problems, 3 (1987), 729-741.  doi: 10.1088/0266-5611/3/4/016.  Google Scholar

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N. S. Nadirashvili, V. A. Sharafutdinov and S. G. Vlăduţ, The John equation for tensor tomography in three-dimensions, Inverse Problems, 32 (2016), 105013. doi: 10.1088/0266-5611/32/10/105013.  Google Scholar

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E. Y. Pantjukhina, Description of the image of a ray transform in two-dimensional case, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 144 1990, 80–89.  Google Scholar

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V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and ill-posed problems series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar

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M. Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.  Google Scholar

show all references

References:
[1]

A. Denisiuk, Inversion of the x-ray transform for complexes of lines in $\Bbb R^n$, Inverse Problems, 32 (2016), 025007. doi: 10.1088/0266-5611/32/2/025007.  Google Scholar

[2]

A. Denisiuk, Reconstruction in the cone-beam vector tomography with two sources, Inverse Problems, 34 (2018), 124008. doi: 10.1088/1361-6420/aae9ac.  Google Scholar

[3]

A. Denisjuk, Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve, Inverse Problems, 22 (2006), 399-411.  doi: 10.1088/0266-5611/22/2/001.  Google Scholar

[4]

I. M. Gel'fandS. G. Gindikin and M. I. Graev, Integral geometry in affine and projective spaces, Journal of Soviet Mathematics, 18 (1982), 39-167.  doi: 10.1007/BF01098201.  Google Scholar

[5]

I. M. Gel'fand, M. I. Graev and Z. J. Šhapiro, Integral geometry on $k$-dimensional planes, (Russian) Funkcional Anal. i Priložen, 1 (1967), 15–31.  Google Scholar

[6] I. M. Gel'fandM. I. Graev and N. Y. Vilenkin, Integral Geometry and Representation Theory, vol. 5 of Generalized functions, Academic Press, New York-London, 1966.   Google Scholar
[7] I. M. Gel'fand and G. E. Shilov, Generalized Functions. Volume I: Properties and Operations, Academic Press, New York-London, 1964.   Google Scholar
[8]

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted radon transforms along hyperplanes in multidimensions, Inverse Problems, 34 (2018), 054001. doi: 10.1088/1361-6420/aab24d.  Google Scholar

[9]

F. John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J., 4 (1938), 300-322.  doi: 10.1215/S0012-7094-38-00423-5.  Google Scholar

[10]

P. Maass, The x-ray transform: Singular value decomposition and resolution, Inverse Problems, 3 (1987), 729-741.  doi: 10.1088/0266-5611/3/4/016.  Google Scholar

[11]

N. S. Nadirashvili, V. A. Sharafutdinov and S. G. Vlăduţ, The John equation for tensor tomography in three-dimensions, Inverse Problems, 32 (2016), 105013. doi: 10.1088/0266-5611/32/10/105013.  Google Scholar

[12]

E. Y. Pantjukhina, Description of the image of a ray transform in two-dimensional case, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 144 1990, 80–89.  Google Scholar

[13]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and ill-posed problems series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar

[14]

M. Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.  Google Scholar

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