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:

* Take a two-dimensional function ''f''('''r'''), [[Projection (mathematics)|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

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== 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 [[Euclidean space|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:

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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>

where <math>B^{-T}=(B^{-1})^T</math> is the transpose of the inverse of the change of basis transform.

== Proof in two dimensions ==

[[Image:ProjectionSlice.png|frame|center|512px|A graphical illustration of the projection slice theorem in two dimensions. ''f''('''r''') and ''F''('''k''') are 2-dimensional Fourier transform pairs. The projection of ''f''('''r''') onto the ''x''-axis is the integral of ''f''('''r''') along lines of sight parallel to the ''y''-axis and is labelled ''p''(''x''). The slice through ''F''('''k''') is on the ''k''_''x'' axis, which is parallel to the ''x'' axis and labelled ''s''(''k''_''x''). The projection-slice theorem states that ''p''(''x'') and ''s''(''k''_''x'') are 1-dimensional Fourier transform pairs.]]

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.

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== Extension to fan beam or cone-beam CT ==

The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.<ref name="ZhaoFSliceThoerem">{{cite book |author = Zhao S.R. and H.Halling |title = A1995 NewIEEE FourierNuclear TransformScience MethodSymposium forand fanMedical BeamImaging TomographyConference Record |journalchapter = PublishedA innew 1995Fourier Nuclearmethod Sciencefor Symposiumfan andbeam Medical Imaging Conference Recordreconstruction |volume = 2 |year = 1995 |pages = 1287–91 |doi = 10.1109/NSSMIC.1995.510494 |isbn = 978-0-7803-3180-8 |s2cid = 60933220 }}</ref>

== See also ==

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== Further reading ==

* {{cite journal |last = Bracewell |first = Ronald N. |authorlinkauthor-link = Ronald N. Bracewell |title = Numerical Transforms |journal = Science |year = 1990 |volume = 248 |pages = 697–704 |doi = 10.1126/science.248.4956.697 |pmid = 17812072 |issue = 4956 |bibcode = 1990Sci...248..697B |s2cid = 5643835 }}

* {{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 |doi-access = free }}

* {{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 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 }}