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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
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
The history of X-ray computed tomography (CT) dates back to at least 1917 with the mathematical theory of the Radon transform. [ 1 ] [ 2 ] In the early 1900s an Italian radiologist named Alessandro Vallebona invented tomography (named "stratigrafia") which used radiographic film to see a single slice of the body.
Geometric tomography is a mathematical field that focuses on problems of reconstructing homogeneous (often convex) objects from tomographic data (this might be X-rays, projections, sections, brightness functions, or covariograms). More precisely, according to R.J. Gardner (who introduced the term), "Geometric tomography deals with the retrieval ...
The following objects are central in geometric measure theory: Hausdorff measure and Hausdorff dimension; Rectifiable sets (or Radon measures), which are sets with the least possible regularity required to admit approximate tangent spaces. Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after German-American mathematician Fritz John . Given a function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } with compact support the X-ray transform is the integral over all lines ...