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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)} .
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 terms of mathematics, the raw data acquired by the scanner consists of multiple "projections" of the object being scanned. These projections are effectively the Radon transformation of the structure of the object. Reconstruction essentially involves solving the inverse Radon transformation.
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
Using an inverse Radon transformation (the filtered back projection) on (,) leads to the Wigner function, (,), [7] which can be converted by an inverse Fourier transform into the density matrix for the state in any basis. [5] A similar technique is often used in medical tomography.
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.
SAMV (iterative sparse asymptotic minimum variance [1] [2]) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing.
Tomographic reconstruction methods used in USCT systems for transmission information based imaging are classical inverse radon transform and fourier slice theorem and derived algorithms (cone beam etc.). As advanced alternatives, ART-based approaches are also utilized.