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- Projection-slice_theorem abstract "In mathematics, the projection-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 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) and 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 Bracewell (1956) for a radio astronomy problem.".
- Projection-slice_theorem thumbnail ProjectionSlice.png?width=300.
- Projection-slice_theorem wikiPageID "1499590".
- Projection-slice_theorem wikiPageRevisionID "546503908".
- Projection-slice_theorem hasPhotoCollection Projection-slice_theorem.
- Projection-slice_theorem subject Category:Fourier_analysis.
- Projection-slice_theorem subject Category:Image_processing.
- Projection-slice_theorem subject Category:Integral_transforms.
- Projection-slice_theorem subject Category:Theorems_in_analysis.
- Projection-slice_theorem type Abstraction100002137.
- Projection-slice_theorem type Communication100033020.
- Projection-slice_theorem type Message106598915.
- Projection-slice_theorem type Proposition106750804.
- Projection-slice_theorem type Statement106722453.
- Projection-slice_theorem type Theorem106752293.
- Projection-slice_theorem type TheoremsInAnalysis.
- Projection-slice_theorem comment "In mathematics, the projection-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.".
- Projection-slice_theorem label "Projection-slice theorem".
- Projection-slice_theorem label "Teorema da projeção de fatia".
- Projection-slice_theorem sameAs Teorema_da_projeção_de_fatia.
- Projection-slice_theorem sameAs m.0565kn.
- Projection-slice_theorem sameAs Q7249438.
- Projection-slice_theorem sameAs Q7249438.
- Projection-slice_theorem sameAs Projection-slice_theorem.
- Projection-slice_theorem wasDerivedFrom Projection-slice_theorem?oldid=546503908.
- Projection-slice_theorem depiction ProjectionSlice.png.
- Projection-slice_theorem isPrimaryTopicOf Projection-slice_theorem.
