## Axial Sampling

The sinograms formed in PET are composed of projections p (s, 8, z). In the 2D case all data are sampled (or assumed to be sampled) with polar angle 8 = 0°. In the 3D case this is extended to measuring projections at polar angles 8 > 0°. According to Orlov's criteria, the data acquired in 2D are sufficient for reconstructing the entire volume [4, 5]. However, in the 3D case all projections formed from angles with 8 ^ 0° are redundant, as the object can be completely described by the 2D projections. The 8 ^ 0° data are useful, however, as they contribute an increase in sensitivity and hence improve the signal-to-noise ratio of the reconstructed data. The redundancy of the oblique lines of response was exploited in the 3D reprojection algorithm [6, 7]. This is discussed further in the next chapter.

A convenient graphical representation was introduced by the Belgian scientist Christian Michel to illustrate the plane definitions used in a large multi-ring PET system, showing how the planes can be combined to optimize storage space and data-handling requirements. They have become known as "Michelogram" representations. Different modes of acquisition are shown in the Michelograms in Fig. 3.7 for a simple eight-ring tomograph.

The situation gets far more complicated for a larger number of rings, and when operating in 3D mode.

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Figure 3.7. The graphical Michelogam is shown for three different acquisition modes on a simple eight-ring tomograph. Each point in the graph represents a plane of response defined between two sets of opposed detectors (a sinogram). In the graph on the left (a simple 2D acquisition with no "inter-planes"), the first plane defined is ring 0 in coincidence with the opposing detectors in the same ring, 0; ring 1 in coincidence with ring 1; etc, for all rings, resulting in a total of eight sinograms. In the middle graph, the same planes are acquired with the addition of a set of "inter-planes" formed between the rings with a ring difference of ±1 ring (ring 0 with ring 1, ring 1 with ring 0, etc). These planes are added together to form a single plane, indicated by the line joining them. This would lead to approximately twice the count rate in this plane compared with the adjacent plane which contains data from one ring only. Physically, this plane is positioned half way between detector rings 0 and 1. While the data come from adjacent rings they are assumed to be acquired with a polar angle of 0° for the purposes of reconstruction. This pattern is repeated for the rest of the rings. This results in 15 (i.e., 2N - 1) sinograms. This is a conventional 2D acquisition mode, resulting in almost twice the number of planes as the previous mode, improving axial sampling, and contributing over 2.5 times as many acquired events. In the graph on the right, a fully 3D acquisition is shown with each plane of data being stored separately (64 in total). The 3D mode would require a fully 3D reconstruction or some treatment of the data, such as a rebinning algorithm, to form 2D projections prior to reconstruction (see Ch. 4).