Projection-slice theorem
From Wikipedia, the free encyclopedia
In mathematics, the projection-slice theorem in two dimensions states that the Fourier transform of the projection of a two-dimensional function f(r) onto a line is equal to a slice through the origin of the two-dimensional Fourier transform of that function which is parallel to the projection line. In operator terms:
where F1 and F2 are the 1- and 2-dimensional Fourier transform operators, P1 is the projection operator, which projects a 2-D function onto a 1-D line, and S1 is a slice operator which extracts a 1-D central slice from a function. This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CAT 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 slice 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.
Contents |
[edit] The projection-slice theorem in N dimensions
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. In operator terms:
[edit] Proof in two dimensions
The projection-slice theorem is easily proven for the case of two dimensions. Without loss of generality, we can take the projection line to be the x-axis. If f(x, y) is a two-dimensional function, then the projection of f(x) onto the x axis is p(x) where
The Fourier transform of f(x,y) is
The slice is then s(kx)
which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example.
[edit] The FHA cycle
If the two-dimensional function f(r) is circularly symmetric, it may be represented as f(r) where r = |r|. In this case the projection onto any projection line will be the Abel transform of f(r). The two-dimensional Fourier transform of f(r) will be a circularly symmetric function given by the zeroth order Hankel transform of f(r), which will therefore also represent any slice through the origin. 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.
[edit] See also
[edit] References
- Bracewell, R.N. (1990). "Numerical Transforms". Science 248: 697-704. doi: .
- Gaskill, Jack D. (1978). Linear Systems, Fourier Transforms, and Optics. John Wiley & Sons, New York. ISBN 0-471-29288-5.