Projection-slice theorem

In mathematics, the projection-slice theorem in two dimensionsstates that the Fourier transform of the projectionof a two-dimensional function "f"(r) onto a lineis 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 "F"₁ and "F"₂ are the 1- and 2-dimensional Fourier transform operators, "P"₁ is the projection operator, which projects a2-D function onto a 1-D line, and "S"₁ is a slice operator which extracts a1-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-rayimage of an internal organ. The Fourier transforms of these images areseen to be slices through the Fourier transform of the 3-dimensionaldensity of the internal organ, and these slice can be interpolated to buildup a complete Fourier transform of that density. The inverse Fourier transformis 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 submanifoldis equal to an m-dimensional slice of the "N"-dimensional Fourier transform of thatfunction 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

:$F(k\_x,k\_y)=int\_\{-infty\}^infty\; int\_\{-infty\}^inftyf(x,y),e^\{-2pi\; i(xk\_x+yk\_y)\},dxdy$

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

:$s(k\_x)=F(k\_x,0)=int\_\{-infty\}^infty\; int\_\{-infty\}^infty\; f(x,y),e^\{-2pi\; ixk\_x\},dxdy$:::$=int\_\{-infty\}^inftyleft\; [int\_\{-infty\}^infty\; f(x,y),dy\; ight]\; ,e^\{-2pi\; ixk\_x\}\; dx$:::$=int\_\{-infty\}^infty\; p(x),e^\{-2pi\; ixk\_x\}\; dx$

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 linewill be the Abel transform of "f"("r"). The two-dimensional Fourier transformof "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 "A"₁ represents the Abel transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, "F"₁ represents the 1-D Fourier transformoperator, and "H" represents the zeroth order Hankel transform operator.

See also

* Radon Transform

References

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