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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 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 * ''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) and * ''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. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal organ, and these slices can be interpolated to build up a complete Fourier transform of that density. The inverse Fourier transform is 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. == The projection-slice theorem in ''N'' dimensions == In ''N'' dimensions, the projection-slice theorem states that the Fourier transform of the projection of an ''N''-dimensional function ''f''(r) onto an ''m''-dimensional linear submanifold is equal to an ''m''-dimensional slice of the ''N''-dimensional Fourier transform of that function consisting of an ''m''-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projection-slice theorem」の詳細全文を読む スポンサード リンク
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