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Radon transform. Maps f on the (x, y)-domain to Rf on the (α, s)-domain.. 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 line integral of the function over that line.
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, based on the work of Minkowski (1904).
Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if
If X is a manifold, M f will be a manifold of dimension one higher, and it is said to "fiber over the circle". As a simple example, let X {\displaystyle X} be the circle, and f {\displaystyle f} be the inversion e i x ↦ e − i x {\displaystyle e^{ix}\mapsto e^{-ix}} , then the mapping torus is the Klein bottle.
In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm. [ 2 ] With a sampled discrete system, the inverse Radon transform is
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 ...
A probability measure assigns length one to the circle. The 1/2π is necessary to get a total length of one, so the two definitions are equivalent. I can attest that this is precisely how Helgason defines the dual Radon transform, but there may be other normalization conventions in the literature (I don't know).
For example, the product of the unit circle (with its usual topology) and the real line with the discrete topology is a locally compact group with the product topology and a Haar measure on this group is not inner regular for the closed subset {} [,].