mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

integral transform In mathematics, an integral transform is a type of transform that 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 charac ...

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 ...

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, 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

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

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, cosmochemistry, ast ...

(used in

CT scan A computed tomography scan (CT scan), formerly called computed axial tomography scan (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 or ...

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 Inversion or inversions may refer to: Arts * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ''Inversions'' (novel) by Iain M. Bank ...

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 of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set ...

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

on the

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

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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

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 g ...

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 German-American mathematician Fritz John. Given a function f\colon\mathbb^n \rightarrow \mathbb with compact s ...

. The Gaussian or ordinary

hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...

can be written as an X-ray transform..

References

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