Editing Projection-slice theorem

{{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
* ''F''<sub>1</sub> and  ''F''<sub>2</sub> are the 1- and 2-dimensional Fourier transform operators mentioned above,
* ''P''<sub>1</sub> is the projection operator (which projects a 2-D function onto a 1-D line),
* ''S''<sub>1</sub> is a slice operator (which extracts a 1-D central slice from a function),
then
: <math>F_1 P_1 = S_1 F_2.</math>

This idea can be extended to higher dimensions.

This theorem is used, for example, in the analysis of medical
[[computed axial tomography|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.<ref>{{cite journal |last = Bracewell |first = Ronald N. |title = Strip integration in radio astronomy |journal = Australian Journal of Physics |year = 1956 |url = https://www.publish.csiro.au/ph/pdf/ph560198 |volume = 9 |issue = 2 |pages = 198–217 |doi = 10.1071/PH560198 |bibcode = 1956AuJPh...9..198B |doi-access = free }}</ref>

== 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:

:<math>F_mP_m=S_mF_N.\,</math>

==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|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''<sub>''x''</sub> axis, which is parallel to the ''x'' axis and labelled ''s''(''k''<sub>''x''</sub>). The projection-slice theorem states that ''p''(''x'') and ''s''(''k''<sub>''x''</sub>) 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.

If ''f''(''x'',&nbsp;''y'') is a two-dimensional function, then the projection of ''f''(''x'',&nbsp;''y'') onto the ''x'' axis is ''p''(''x'') where

:<math>p(x)=\int_{-\infty}^\infty f(x,y)\,dy.</math>

The Fourier transform of <math>f(x,y)</math> is

:<math>
F(k_x,k_y)=\int_{-\infty}^\infty \int_{-\infty}^\infty
f(x,y)\,e^{-2\pi i(xk_x+yk_y)}\,dxdy.
</math>

The slice is then <math>s(k_x)</math>

:<math>s(k_x)=F(k_x,0)
=\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,e^{-2\pi ixk_x}\,dxdy
</math>
:::<math>=\int_{-\infty}^\infty
\left[\int_{-\infty}^\infty f(x,y)\,dy\right]\,e^{-2\pi ixk_x} dx
</math>
:::<math>=\int_{-\infty}^\infty p(x)\,e^{-2\pi ixk_x} dx
</math>

which is just the Fourier transform of ''p''(''x''). The proof for higher dimensions is easily generalized from the above example.

== The FHA cycle ==

If the two-dimensional function ''f''('''r''') is circularly symmetric, it may be represented as ''f''(''r''), where ''r''&nbsp;=&nbsp;|'''r'''|. In this case the projection onto any projection line
will be the [[Abel transform]] of ''f''(''r''). The two-dimensional [[Fourier transform]]
of ''f''('''r''') will be a circularly symmetric function given by the zeroth-order [[Hankel transform]] of ''f''(''r''), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or

: <math>F_1 A_1 = H,</math>

where ''A''<sub>1</sub> represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, ''F''<sub>1</sub> represents the 1-D Fourier-transform
operator, and ''H'' represents the zeroth-order Hankel-transform operator.

== 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 = 1995 IEEE Nuclear Science Symposium and Medical Imaging Conference Record |chapter = A new Fourier method for fan beam reconstruction |volume = 2 |year = 1995 |pages = 1287–91 |doi = 10.1109/NSSMIC.1995.510494 |isbn = 978-0-7803-3180-8 |s2cid = 60933220 }}</ref>

== See also ==
* [[Radon transform#Relationship with the Fourier transform|Radon transform § Relationship with the Fourier transform]]

== References ==
{{Reflist}}

== Further reading ==
* {{cite journal |last = Bracewell |first = Ronald N. |author-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 |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 }}
* {{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 }}
* {{cite journal |last1 = Garces |first1 = Daissy H. |last2 = Rhodes |first2 = William T. |last3 = Peña |first3 = Néstor |title = The Projection-Slice Theorem: A Compact Notation |journal =  Journal of the Optical Society of America A|year = 2011 |volume = 28 |issue = 5 |pages = 766–769 |doi  = 10.1364/JOSAA.28.000766 |pmid = 21532686 |bibcode = 2011JOSAA..28..766G }}

== External links ==
* {{cite AV media |date = September 10, 2015 |title = Fourier Slice Theorem  |medium = video |institution = [[University of Antwerp]] |series = Part of the "Computed Tomography and the ASTRA Toolbox" course |url = https://www.youtube.com/watch?v=YIvTpW3IevI }}
 
[[Category:Theorems in Fourier analysis]]
[[Category:Integral transforms]]
[[Category:Image processing]]