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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.
The right-hand side in the Crofton formula is sometimes called the Favard length. [ 1 ] In general, the space of oriented lines in R n {\displaystyle \mathbb {R} ^{n}} is the tangent bundle of S n − 1 {\displaystyle S^{n-1}} , and we can similarly define a kinematic measure d φ ∧ d p {\displaystyle d\varphi \wedge dp} on it, which is also ...
Some authors use the preceding approach to define positive Radon measures to be the positive linear forms on K (X). [4] In this set-up it is common to use a terminology in which Radon measures in the above sense are called positive measures and real-valued Radon measures as above are called (real) measures.
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.
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.
But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for < <, the classical Hilbert transform is defined on (,) where denotes the unit circle and the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on (,).
A group is called unimodular if the modular function is identically , or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are abelian groups , compact groups , discrete groups (e.g., finite groups ), semisimple Lie groups and connected nilpotent Lie groups .
The fourth is a modified wave-field transform base on frequency decomposition and slant stacking, performed by Xia et al. (2007). [9] The fifth is a high-resolution Linear Radon transformation performed by Luo et al. (2008). [10] In performing a wave-field transformation, a slant stack is done, followed by a Fourier transform. The way in which ...