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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)} .
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
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.
Tomosynthesis reconstruction algorithms are similar to CT reconstructions, in that they are based on performing an inverse Radon transform.Due to partial data sampling with very few projections, approximation algorithms have to be used.
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 ...
Alexander G. Ramm (born 1940 in St. Petersburg, Russia) is an American mathematician. His research focuses on differential and integral equations, operator theory, ill-posed and inverse problems, scattering theory, functional analysis, spectral theory, numerical analysis, theoretical electrical engineering, signal estimation, and tomography.
The Mojette transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection operator. The IRCCyN laboratory - UMR CNRS 6597 in Nantes, France has been developing it since 1994. The first characteristic of the Mojette transform is using only additions and subtractions.