Article Contents
Article Contents

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

• * Corresponding author: Aleksander Denisiuk
• 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 $2(m+1)$.

Mathematics Subject Classification: Primary: 44A12; Secondary: 53A45.

 Citation:

•  [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. [2] A. Denisiuk, Reconstruction in the cone-beam vector tomography with two sources, Inverse Problems, 34 (2018), 124008. doi: 10.1088/1361-6420/aae9ac. [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. [4] I. M. Gel'fand, S. 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. [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. [6] I. M. Gel'fand,  M. I. Graev and  N. Y. Vilenkin,  Integral Geometry and Representation Theory, vol. 5 of Generalized functions, Academic Press, New York-London, 1966. [7] I. M. Gel'fand and  G. E. Shilov,  Generalized Functions. Volume I: Properties and Operations, Academic Press, New York-London, 1964. [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. [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. [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. [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. [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. [13] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and ill-posed problems series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095. [14] M. Spivak, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.