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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).
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;
A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a chord diagram.That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such as Wiener measure on the space of real-valued continuous functions on the interval [0, 1]. A measure on ℝ is a Radon measure if and only if it is a locally finite Borel measure. [5] The following are not examples of Radon ...
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.
For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic ...
An example is the linear map that takes any point with coordinates (,) to the point (+,). In this case, the displacement is horizontal by a factor of 2 where the fixed line is the x-axis, and the signed distance is the y-coordinate. Note that points on opposite sides of the reference line are displaced in opposite directions.