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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).
Tomographic reconstruction: Projection, Back projection and Filtered back projection. Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections.
Because, again, the formula is additive over concatenation of line segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the unit circle. The proof for the generalized version proceeds exactly as above.
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 ...
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Abstractly, the Penrose transform operates on a double fibration of a space Y, over two spaces X and Z. In the classical Penrose transform, Y is the spin bundle, X is a compactified and complexified form of Minkowski space (which as a complex manifold is (,)) and Z is the twistor space (which is ).
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).