X-ray transform
Encyclopedia
In mathematics
, the X-ray transform (also called John transform) is an integral transform introduced by 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 line
s rather than over hyperplane
s as in the Radon transform. The X-ray transform derives its name from X-ray tomography
because the X-ray transform of a function ƒ represents the scattering
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 scattering 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 lines in Rn by
where x0 is an initial point on the line and θ is a unit vector 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 Gauss hypergeometric function can be written as an X-ray transform .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the X-ray transform (also called John transform) is an integral transform introduced by that is one of the cornerstones of modern integral geometry
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 transformations from the space of functions on one geometrical space to the...
. It is very closely related to the Radon transform
Radon transform
thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s rather than over hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s as in the Radon transform. The X-ray transform derives its name from X-ray tomography
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
because the X-ray transform of a function ƒ represents the scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
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 scattering data.
In detail, if ƒ is a compactly supported continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
on the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn, then the X-ray transform of ƒ is the function Xƒ defined on the set of all lines lines in Rn by
where x0 is an initial point on the line and θ is a unit vector 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
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
on the Euclidean line L.
The X-ray transform satisfies an ultrahyperbolic wave equation
Ultrahyperbolic wave equation
In the mathematical field of partial differential equations, the ultrahyperbolic equation is a partial differential equation for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form...
called John's equation
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....
.
The Gauss hypergeometric function can be written as an X-ray transform .