In mathematics, the X-ray transform (also called John transform) is an integral transform introduced by Fritz John in 1938 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 because the X-ray transform of a function ƒ represents the scattering 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 scattering data.
where x0 is an initial point on the line and θ is a unit vector 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.
- Fritz, John (1938). "The ultrahyperbolic differential equation with four independent variables". Duke Mathematical Journal 4: 300–322. doi:10.1215/S0012-7094-38-00423-5. Retrieved 23 January 2013.
- Berenstein, Carlos A. (2001), "X-ray transform", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
- Gelfand, I. M.; Gindikin, S. G.; Graev, M. I. (2003) , Selected topics in integral geometry, Translations of Mathematical Monographs 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 39 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4530-1, MR 2463854
- Helgason, Sigurdur (1999), The Radon Transform, Progress in Mathematics (2nd ed.), Boston, M.A.: Birkhauser