Projection-slice theorem: Difference between revisions

Revision as of 08:21, 31 August 2020

Revision as of 08:22, 31 August 2020

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== 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

: <math>F_1A_1F_1 A_1 = H\,</math>

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 -transform

operator, and ''H'' represents the zeroth -order Hankel -transform operator.

== Extension to fan beam or cone-beam CT ==