Projection-slice theorem: Difference between revisions

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{{Short description|Theorem in mathematics}}

{{technical|date=April 2025}}

[[File:Fourier Slice Theorem.png|thumb|Fourier 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:

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==The generalized Fourier-slice theorem==

In addition to generalizing to ''N'' dimensions, the projection-slice theorem can be further generalized with an arbitrary [[change of basis]].<ref name="NgFourierSlicePhotography">{{cite journal |last = Ng |first = Ren |title = Fourier Slice Photography |journal = ACM Transactions on Graphics |year = 2005 |url = https://graphics.stanford.edu/papers/fourierphoto/fourierphoto-600dpi.pdf |volume = 24 |issue = 3 |pages = 735–744 |doi = 10.1145/1073204.1073256 }}</ref> For convenience of notation, we consider the change of basis to be represented as ''B'', an ''N''-by-''N'' [[invertible matrix]] operating on ''N''-dimensional column vectors. Then the generalized Fourier-slice theorem can be stated as

: <math>F_m P_m B = S_m \frac{B^{-T}}{|B^{-T}|} F_N</math>

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The projection-slice theorem is easily proven for the case of two dimensions.

[[Without loss of generality]], we can take the projection line to be the ''x''-axis.

There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds.