X-ray transform

In mathematics, the X-ray transform (also called John transform) is an integral transform introduced by Fritz John in 1938[1] 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 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

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 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

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) [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, 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 (PDF), Progress in Mathematics (2nd ed.), Boston, M.A.: Birkhauser

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[1] 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.