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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.
Sigma algebra. Separable sigma algebra; Filtration (abstract algebra) Borel algebra; Borel measure; Indicator function; Lebesgue measure; Lebesgue integration; Lebesgue's density theorem; Counting measure; Complete measure; Haar measure; Outer measure; Borel regular measure; Radon measure; Measurable function; Null set, negligible set; Almost ...
6.1.1 Positive Radon measures. ... 6.4 Tempered distributions and Fourier transform. ... Download as PDF; Printable version; In other projects
In theory, the inverse Radon transformation would yield the original image. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, f ( x , y ) {\displaystyle f(x,y)} .
In theoretical physics, the Penrose transform, introduced by Roger Penrose (1967, 1968, 1969), is a complex analogue of the Radon transform that relates massless fields on spacetime, or more precisely the space of solutions to massless field equations, to sheaf cohomology groups on complex projective space.
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
In the mathematics of topological vector spaces, Minlos's theorem states that a cylindrical measure on the dual of a nuclear space is a Radon measure if its Fourier transform is continuous. It is named after Robert Adol'fovich Minlos and can be proved using Sazonov's theorem .
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space.
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