Projection-slice theorem

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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.

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[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:

F_mP_m=S_mF_N\,

[edit] Proof in two dimensions

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

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^{-2\pi i(xk_x+yk_y)}\,dxdy

The slice is then s(kx)

s(k_x)=F(k_x,0) =\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,e^{-2\pi ixk_x}\,dxdy
=\int_{-\infty}^\infty \left[\int_{-\infty}^\infty f(x,y)\,dy\right]\,e^{-2\pi ixk_x} dx
=\int_{-\infty}^\infty p(x)\,e^{-2\pi ixk_x} dx

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

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.

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