Projection-slice theorem: Difference between revisions

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[[File:Fourier Slice Theorem.png|thumb|A visual explanation of the Fourier Slice Theorem: The parallel 2-D projection in spatial ___domain and subsequent 1-D Fourier transform is identical to a slice through 2-D Fourier space.<ref name=Maier2018>{{cite book |author = Andreas Maier, Stefan Steidl, Vincent Christlein, Joachim Hornegger |title = Medical Imaging Systems - An Introductory Guide |publisher = Springer, Heidelberg |year = 2018 |isbn = 978-3-319-96520-8 | url = https://link.springer.com/book/10.1007%2F978-3-319-96520-8 }}</ref>]]

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'''), [[Projection (mathematics)|project]] it onto a (one-dimensional) line, and do a [[Fourier transform]] of that projection.{{Dubious |Talk section Misleading Use of term "Projection"|reason=Inappropriately general definition of projection|date=August 2018}}

* 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,

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:<math>F_1 P_1=S_1 F_2\,</math>

This idea can be extended to higher dimensions.

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* {{cite journal |last = Bracewell |first = Ronald N. |title = Strip Integration in Radio Astronomy |journal = Aust. J. Phys. |year = 1956 |volume = 9 |pages = 198 |doi = 10.1071/PH560198 |pmid = |issue = 2 |bibcode = 1956AuJPh...9..198B }}

* {{cite book |author = Gaskill, Jack D. |title = Linear Systems, Fourier Transforms, and Optics |publisher = John Wiley & Sons, New York |year = 2005 |isbn = 978-0-471-29288-3 }}

* {{cite book |author = Andreas Maier, Stefan Steidl, Vincent Christlein, Joachim Hornegger |title = Medical Imaging Systems - An Introductory Guide |publisher = Springer, Heidelberg |year = 2018 |isbn = 978-3-319-96520-8 | url = https://link.springer.com/book/10.1007%2F978-3-319-96520-8 }}

* {{cite journal |last = Ng |first = Ren |title = Fourier Slice Photography |journal = ACM Transactions on Graphics |year = 2005 |volume = 24 |issue = 3 |pages = 735–744 |doi = 10.1145/1073204.1073256 }}

* {{cite journal |last1 = Zhao |first1 = Shuang-Ren |last2 = Halling |first2 = Horst |title = Reconstruction of Cone Beam Projections with Free Source Path by a Generalized Fourier Method |journal = Proceedings of the 1995 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine |year = 1995 |pages = 323–7 }}