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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.
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
the Radon measure concept of measure as linear functional; the Radon transform, in integral geometry, based on integration over hyperplanes—with application to tomography for scanners (see tomographic reconstruction); Radon's theorem, that d + 2 points in d dimensions may always be partitioned into two subsets with intersecting convex hulls;
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).
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
A central problem of integral geometry is to reconstruct a function from knowledge of its orbital integrals. The Funk transform and Radon transform are two special cases. When G/K is a Riemannian symmetric space, the problem is trivial, since M r ƒ(x) is the average value of ƒ over the generalized sphere of radius r, and
The Radon point of any four points in the plane is their geometric median, the point that minimizes the sum of distances to the other points. [5] [6] Radon's theorem forms a key step of a standard proof of Helly's theorem on intersections of convex sets; [7] this proof was the motivation for Radon's original discovery of Radon's theorem.
Two-sided Laplace transform; Inverse two-sided Laplace transform; Laplace–Carson transform; Laplace–Stieltjes transform; Legendre transform; Linear canonical transform; Mellin transform. Inverse Mellin transform; Poisson–Mellin–Newton cycle; N-transform; Radon transform; Stieltjes transformation; Sumudu transform; Wavelet transform ...