The approximate Fourier rebinning algorithm [81] is more accurate than the SSRB algorithm and extends the range of 3D PET studies that can be processed using rebinning algorithms. The main characteristics of FORE is that it proceeds via the 2D Fourier transform of each oblique sinogram, defined as

where k is the azimuthal Fourier index. Rebinning is based on the following relation between the Fourier transforms of oblique and direct sinograms:

Ps (v,k ,z ,0) ^ Ps (v,k ,Z = z + k tan0/(2nv),0) (69)

Like all analytic algorithms, FORE assumes that the data ps (s, <j, i, 0) are line integrals of the tracer distribution and that each oblique sinogram is sampled over the whole range (s, <j) e [-RF,RF ] x [0, n]. Therefore, the raw data must be corrected for all effects including detector efficiency variations, attenuation, and scattered and random coincidences, before applying FORE. Also, when the data are incomplete due to gaps in the detector assembly, the sinograms must be filled as discussed in the section on properties of the inverse 2D radon transform (above). Refer to [81] for a detailed description and for the derivation of FORE.

In practice, FORE is sufficiently accurate when the axial aperture 0max is smaller than about 20°, though the limit depends on the radius of the FOV and on the type of image. Beyond 20°, artifacts similar to those observed with SSRB (at lower apertures) appear [84]: degraded image quality at increasing distance from the axis. Two variations of FORE, the FOREJ and FOREX rebinning algorithms [82, 83], are exact in the limit of continuous sampling, and have been shown to overcome this loss of axial resolution when reconstructing high statistics data acquired with a large aperture scanner [85]. However, the current implementation of the FOREJ algorithm [82] is more sensitive to noise than FORE since the correction term involves a second derivative of the data with respect to the axial coordinate £, and the application to low statistics data remains questionable.

For each 0 such that the oblique sinogram i, 0 is measured (see Eq. (54)), the RHS yields an independent estimate of the direct data 0 = 0. FORE then averages all these estimates to optimize the SNR. The accuracy of the approximation (Eq. (69)) breaks down at low frequencies v. Therefore, for all frequencies below some small threshold, the Fourier transform of the rebinned data is estimated using the SSRB approximation. The main steps of the FORE algorithm are:

(i) Initialize a stack of Fourier transformed sinograms Pf0re(v, k, z),

(ii) For each oblique sinogram i, 0

a. Calculate the 2D Fourier transform Ps(v, k, i, 0), b. For each frequency component (v, k), increment PfoJv, k, i-k tan 0/(2nv)) by Ps(v, k, i, 0),

(iii) Normalize Pfore(v, k, z) for the varying number of contributions it has received,

(iv) Take the 2D inverse Fourier transform to get the rebinned data pfore(s, <j, z).

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