In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
In operator terms, if
then
F1P1=S1F2.
This idea can be extended to higher dimensions.
This theorem is used, for example, in the analysis of medicalCT 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 slices 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. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem.[1]
In N dimensions, the projection-slice theorem states that theFourier transform of the projection of an N-dimensional functionf(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:
FmPm=SmFN.
In addition to generalizing to N dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis.[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. Then the generalized Fourier-slice theorem can be stated as
FmPmB=Sm
| B-T | |
| |B-T| |
FN
where
B-T=(B-1)T
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.There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds.
If f(x, y) is a two-dimensional function, then the projection of f(x, y) onto the x axis is p(x) where
| infty | |
| p(x)=\int | |
| -infty |
f(x,y)dy.
The Fourier transform of
f(x,y)
F(kx,ky)=\int
| infty | |
| -infty |
| infty f(x,y)e | |
| \int | |
| -infty |
| -2\pii(xkx+yky) | |
dxdy.
The slice is then
s(kx)
s(kx)=F(kx,0) =\int
| infty | |
| -infty |
| infty | |
| \int | |
| -infty |
| -2\piixkx | |
| f(x,y)e |
dxdy
| infty | |
| =\int | |
| -infty |
| -2\piixkx | |
| f(x,y)dy\right]e |
dx
| infty | |
| =\int | |
| -infty |
| -2\piixkx | |
| p(x)e |
dx
which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example.
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
F1A1=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-transformoperator, and H represents the zeroth-order Hankel-transform operator.
The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.[3]