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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 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)} .
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 a Fourier transform changes x-t data into x-ω (ω is angular frequency) data shows why phase velocity dominates surface wave ...
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 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 ...
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.
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 ...
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