mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the X-ray transform (also called ray transform or John transform) is an integral transform 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 1 ...

in 1938 that is one of the cornerstones of modern integral geometry. 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 l ...

, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

s rather than over

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s 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, quantu ...

(used in

CT scan A computed tomography scan (CT scan; formerly called computed axial tomography scan or CAT scan) is a medical imaging technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called radiographers ...

s) 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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

on the

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

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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

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