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Concept
Projection-slice theorem
Résumé
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:
Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection.
Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line.
In operator terms, if
F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above,
P1 is the projection operator (which projects a 2-D function onto a 1-D line),
S1 is a slice operator (which extracts a 1-D central slice from a function),
then
This idea can be extended to higher dimensions.
This theorem is used, for example, in the analysis of medical
CT 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 slices 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. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem.
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:
In addition to generalizing to N dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis. 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.
Source officielle
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