Editing Projection-slice theorem (section)

{{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>