In mathematics, the X-ray transform (also called ray transform^[1] or John transform) is an integral transform introduced by Fritz John in 1938^[2] that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ƒ from its known attenuation data.

In detail, if ƒ is a compactly supported continuous function on the Euclidean space Rⁿ, then the X-ray transform of ƒ is the function Xƒ defined on the set of all lines in Rⁿ by

${\displaystyle Xf(L)=\int _{L}f=\int _{\mathbf {R} }f(x_{0}+t\theta )dt}$

where x₀ is an initial point on the line and θ is a unit vector in Rⁿ giving the direction of the line L. The latter integral is not regarded in the oriented sense: it is the integral with respect to the 1-dimensional Lebesgue measure on the Euclidean line L.

The X-ray transform satisfies an ultrahyperbolic wave equation called John's equation.

The Gauss hypergeometric function can be written as an X-ray transform (Gelfand, Gindikin & Graev 2003, 2.1.2).

References

• Berenstein, Carlos A. (2001) [1994], "X-ray transform", Encyclopedia of Mathematics, EMS Press.
• Gelfand, I. M.; Gindikin, S. G.; Graev, M. I. (2003) [2000], Selected topics in integral geometry, Translations of Mathematical Monographs, vol. 220, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2932-5, MR 2000133
• Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4530-1, MR 2463854
• Helgason, Sigurdur (1999), The Radon Transform (PDF), Progress in Mathematics (2nd ed.), Boston, M.A.: Birkhauser