In mathematics, the X-ray transform (also called ray transform or John transform) is an

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...

introduced by

Fritz John Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a ...

in 1938 that is one of the cornerstones of modern

integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformati ...

. It is very closely related to the

Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...

, 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 Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, astrophysics, ...

(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 In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...

continuous function on the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

R^''n'', then the X-ray transform of ''ƒ'' is the function ''Xƒ'' defined on the set of all lines in R^''n'' by :

Xf(L) = \int_L f = \int_ f(x_0+t\theta)dt

where ''x''₀ is an initial point on the line and ''θ'' is a unit vector in R^''n'' 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 In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function of variables of the form \frac + \cdots + \frac - \frac - \cdots - \frac = 0. More ...

called

John's equation John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John. Given a function f\colon\mathbb^n \rightarrow \mathbb with compact support the ''X-ray transform'' is ...

. The

Gauss hypergeometric function Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

can be written as an X-ray transform .

References

*. * * *{{Citation , last1=Helgason , first1=Sigurdur , url=http://www-math.mit.edu/~helgason/Radonbook.pdf , title=The Radon Transform , publisher= Birkhauser , location=Boston, M.A. , edition=2nd , series=Progress in Mathematics , year=1999 Integral geometry Integral transforms X-ray computed tomography