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The transform was introduced in 1917 by Johann Radon, [1] who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform).
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)} .
Abel transform can be viewed as the Radon transform of an isotropic 2D function f(r). As f(r) is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.
(1)Results of measurement, i.e. a series of images obtained by transmitted light are expressed (modeled) as a function p (s,θ) obtained by performing radon transform to μ(x, y), and (2)μ(x, y) is restored by performing inverse radon transform to measurement results.
Radon transform: Rƒ ( + ) ... In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function.
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
Inverse two-sided Laplace transform; Laplace–Carson transform; Laplace–Stieltjes transform; Legendre transform; Linear canonical transform; Mellin transform. Inverse Mellin transform; Poisson–Mellin–Newton cycle; N-transform; Radon transform; Stieltjes transformation; Sumudu transform; Wavelet transform (integral) Weierstrass transform ...
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.