X-ray transform

In mathematics, the X-ray transform (also called ray transform¹ or John transform) is an integral transform introduced by Fritz John in 1938² 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 Gaussian or ordinary hypergeometric function can be written as an X-ray transform.(Gelfand, Gindikin & Graev 2003, 2.1.2).