Projection-slice theorem

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:

$F\_1\; P\_1=S\_1\; F\_2,$

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.

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:

$F\_mP\_m=S\_mF\_N,$

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

$p(x)=int\_\{-infty\}^infty\; f(x,y),dy$

The Fourier transform of $f(x,y)$ is

$$

F(k_x,k_y)=int_{-infty}^infty int_{-infty}^infty f(x,y),e^{-2pi i(xk_x+yk_y)},dxdy

The slice is then $s(k\_x)$

$s(k\_x)=F(k\_x,0)$

=int_{-infty}^infty int_{-infty}^infty f(x,y),e^{-2pi ixk_x},dxdy

$=int\_\{-infty\}^infty$

left[int_{-infty}^infty f(x,y),dyright],e^{-2pi ixk_x} dx

$=int\_\{-infty\}^infty\; p(x),e^\{-2pi\; ixk\_x\}\; dx$

which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example.

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

$F\_1A\_1=H,$

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.

See also

• Radon Transform

References

• Bracewell, R.N. (1990). "Numerical Transforms". Science 248 697–704.
• Gaskill, Jack D. (1978). Linear Systems, Fourier Transforms, and Optics. John Wiley & Sons, New York. ISBN 0-471-29288-5.