In mathematics, the **projection-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**), project 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*_{1}and*F*_{2}are the 1- and 2-dimensional Fourier transform operators mentioned above,*P*_{1}is the projection operator (which projects a 2-D function onto a 1-D line) and*S*_{1}is a slice operator (which extracts a 1-D central slice from a function),

then:

This idea can be extended to higher dimensions.

This theorem is used, for example, in the analysis of medical 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 Bracewell (1956) for a radio astronomy problem.

Read more about Projection-slice Theorem: The Projection-slice Theorem in *N* Dimensions, Proof in Two Dimensions, The FHA Cycle, Extension To N-dimension Signal

### Famous quotes containing the word theorem:

“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)