The Central Section Theorem

The central section theorem (Eq. 10) can be generalized to 3D, and states that

where

is the 2D Fourier transform of a parallel projection and F is the 3D Fourier transform of the image. Note that as the integral in Eq. (59) is over the whole projection plane n\ the central section theorem is only valid for non-truncated parallel projections.

Geometrically, this theorem means that a projection of direction n allows the recovery of the Fourier transform of the image on the central plane orthogonal to n in 3D frequency space. A corollary is that the image can be reconstructed in a stable way from a set of non-truncated projections n Q C S2 if and only if the set Q has an intersection with any equatorial circle on the unit sphere S2. This condition is due to Orlov [70]. The equatorial band Q(0max) in Eq. (52) satisfies Orlov's condition for any 0max > 0.

The direct 3D Fourier reconstruction algorithm is a direct implementation of Eq. (58) [71]. This technique involves a complex interpolation in frequency space, and has not so far been used in practice. However, Matej [15] recently demonstrated a significant gain of reconstruction time compared to the standard FBP.

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