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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 practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm. [2] With a sampled discrete system, the inverse Radon transform is
F 1 and F 2 are the 1- and 2-dimensional Fourier transform operators mentioned above, P 1 is the projection operator (which projects a 2-D function onto a 1-D line), S 1 is a slice operator (which extracts a 1-D central slice from a function), then =. This idea can be extended to higher dimensions.
The history of X-ray computed tomography (CT) dates back to at least 1917 with the mathematical theory of the Radon transform. [1] [2] In the early 1900s an Italian radiologist named Alessandro Vallebona invented tomography (named "stratigrafia") which used radiographic film to see a single slice of the body.
In conventional CT machines, an X-ray tube and detector are physically rotated behind a circular shroud (see the image above right). An alternative, short lived design, known as electron beam tomography (EBT), used electromagnetic deflection of an electron beam within a very large conical X-ray tube and a stationary array of detectors to achieve very high temporal resolution, for imaging of ...
However, essential mathematics and reconstruction algorithms used for CT and OPT are similar; for example, radon transform or iterative reconstruction based on projection data are used in both medical CT scan and OPT for 3D reconstruction. Both medical CT and OPT compute 3D volumes based on transmission of the photon through the material of ...
In mathematics, the X-ray transform (also called ray transform [1] or John transform) is an integral transform introduced by Fritz John in 1938 [2] that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions.
In modern (helical) CT, the source/detector makes at least a complete 180-degree rotation about the subject obtaining a complete set of data from which images may be reconstructed. Digital tomosynthesis, on the other hand, only uses a limited rotation angle (e.g., 15-60 degrees) with a lower number of discrete exposures (e.g., 7-51) than CT.