X-ray transform explained

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

Xf(L) = \int _{L} f = \int _{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..

References

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Notes and References

1. Book: Natterer, Frank . Mathematical Methods in Image Reconstruction . SIAM . 2001 . Philadelphia . Wübbeling . Frank . 10.1137/1.9780898718324.fm.
2. Fritz . John . The ultrahyperbolic differential equation with four independent variables . Duke Mathematical Journal . 1938 . 4 . 300–322 . 10.1215/S0012-7094-38-00423-5 . 23 January 2013.