Projection-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), 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

• F₁ and F₂ are the 1- and 2-dimensional Fourier transform operators mentioned above,
• P₁ is the projection operator (which projects a 2-D function onto a 1-D line),
• S₁ is a slice operator (which extracts a 1-D central slice from a function),

then

${\displaystyle F_{1}P_{1}=S_{1}F_{2}.}$

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 Ronald N. Bracewell in 1956 for a radio-astronomy problem.^[1]