The future evolution of image reconstruction in PET will most probably lead to the generalized utilization of iterative algorithms, both for 2D and for 3D data. As shown in the next section, it is straightforward to extend iterative methods, such as OSEM, to fully 3D scanning. These algorithms have the potential to model accurately the data acquisition, the measurement noise, and also the prior information on the tracer distribution. In contrast, analytic algorithms are bound to the line integral representation of the data. Even though some physical effects can be incorporated in pre- or post-processing steps, an accurate modeling of the Poisson statistics of the data is difficult with analytic methods. To date, however, the computational burden of fully 3D iterative algorithms remains a major issue for some applications involving multiple acquisitions, or for research scanners such as the HRRT which sample a very large number of LORs. The current practice of undersampling these data (see above) to accelerate reconstruction is contradictory with the aim of accurate modeling claimed by iterative methods.
This limitation has led to the application of hybrid algorithms for 3D PET data [41, 66, 91]. These algorithms first rebin the 3D data into a multi-slice set of ordinary sinogram data, using e.g. the SSRB method, or, more often, FORE. Each rebinned sinogram is then reconstructed using some 2D iterative algorithm. This hybrid approach provides a significant time gain with respect to fully 3D iterative reconstruction.
The two components of hybrid algorithms, rebinning and iterative methods, have been discussed in previous sections. In this section, we briefly discuss the interplay between these two elements, the main difficulty being to model the rebinned data that are presented to the 2D iterative algorithm. We focus on the application of FORE followed by a 2D OSEM reconstruction but the same problems would arise with other combinations, such as SSRB followed by an iterative minimization of a 2D penalized weighted least-square (PWLS) cost function .
One of the major benefits of iterative reconstruction arises from a correct modeling of the data statistics, which allows to weight each LOR according to its variance. This is the reason why improved image quality is obtained by reconstructing the raw, uncorrected data with a system matrix incorporating the effects of attenuation, normalisation and scatter, rather than by reconstructing pre-corrected data with a system matrix modeling only the detector's geometric response. Ideally, therefore, we would like to develop a hybrid algorithm in which un-corrected rebinned data are reconstructed by means of a 2D iterative algorithm including the effects of attenuation, etc. This approach is impossible because the FORE Eq. (69) must be applied to fully pre-corrected data as discussed at the end of the previous section. The rebinned data must then be reconstructed with a 2D iterative algorithm which does not model the pre-corrected physical effects.
One solution to improve the statistical model is to de-correct the data for the physical effects after the rebinning. This de-correction restores Poisson-like statistics to the rebinned data, and the physical effects can then be reintroduced in the system matrix. If we hypothesize that the most important effect is that of attenuation, we can decorrect for attenuation only and then reconstruct the de-corrected rebinned data with AW-OSEM (see Eq. (43)). This approach is referred to as the FORE+OSEM(AW) algorithm. Note that this algorithm is still approximate: even in the absence of attenuation and scatter, the rebinned sinograms are not independent Poisson variables because of the complex linear combination of the 3D data during FORE rebinning. Strictly speaking, it is inappropriate to reconstruct the rebinned data using the OSEM algorithm derived for independent Poisson data, and it is preferable to use a weighted least-square method  or the NEC scaling technique  (Eq. (44)). In each case, one needs to estimate the variance of the rebinned data  and also, ideally, the covariance .
Finally, modeling the shift-variant detector response (e.g. due to crystal penetration) has not yet been attempted with hybrid methods. One approach would be to apply sinogram restoration prior to rebinning.
A related problem occurs with scanners such as the Siemens/CPS HRRT , which has gaps between adjacent flat panel detector heads. Since Fourier rebinning requires complete sinogram data, these gaps must be filled before rebinning. Gap filling techniques may range in complexity from linear interpolation to forward projection of an image reconstructed from the 2D segment by using a system matrix which accounts for the missing data . In general, however, a 3D iterative reconstruction is preferable to an hybrid one because the gap filling procedure followed by the rebinning is sensitive to noise propagated from regions with high attenuation.
Despite these difficulties, fast hybrid algorithms such as FORE+OSEM(AW) have been applied to whole-body FDG scans, and shown to provide for these studies an image quality comparable to fully 3D iterative reconstruction (see [90, 92] and the example in Fig. 4.13 below).
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