Integral transforms | Theorems in Fourier analysis
Projection-slice theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * 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 * F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above, * P1 is the projection operator (which projects a 2-D function onto a 1-D line), * S1 is a slice operator (which extracts a 1-D central slice from a function), then This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medicalCT scans where a "projection" is an x-rayimage of an internal organ. The Fourier transforms of these images areseen to be slices through the Fourier transform of the 3-dimensionaldensity of the internal organ, and these slices can be interpolated to buildup a complete Fourier transform of that density. The inverse Fourier transformis then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem. (Wikipedia).
