Projection-slice theorem

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.

[1] 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 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.

[2] 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.

There is no loss of generality because if we use a shifted and rotated line, the law still applies.

The projection-slice theorem then states that the Fourier transform of the projection equals the slice or where A1 represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, F1 represents the 1-D Fourier-transform operator, and H represents the zeroth-order Hankel-transform operator.

The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections.

The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.

Fourier slice theorem
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 x axis, which is parallel to the x axis and labelled s ( k x ). The projection-slice theorem states that p ( x ) and s ( k x ) are 1-dimensional Fourier transform pairs.