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[[File:Fourier Slice Theorem.png|thumb|Fourier
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.▼
▲* 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.
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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.
== The projection-slice theorem in ''N'' dimensions ==
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