The natural parameterization of PET data uses the indices (da, db) of the two detectors in coincidence, as in Eq. (1). However, there are several reasons to modify this parameterization:

• The natural parameterization is often poorly adapted to analytic algorithms. This is why raw data are usually interpolated into an alternative sinogram parameterization described below.

• The number of recorded coincidences Nevents in a given scan may be too small to take full advantage of the nominal spatial resolution of the scanner. In such a case, undersampling by grouping neighboring LORs reduces the data storage requirements and the reconstruction time without significantly affecting the reconstructed spatial resolution, which is primarily limited by the low count density.

Another approach to reduce data storage and processing time when NLOR » Nevents consists of recording the coordinates (da,db) of each coincident event in a sequential data stream called a list-mode data set. Additional information such as the time or the energy of each detected photon can also be stored. In contrast to undersampling, list-mode acquisition does not compromise the accuracy of the spatial localization of each event. But the fact remains that the number of measured coincidences may be too low to exploit the full resolution of the scanner.

Let us define the standard parameterization of 2D PET data into sinograms. Consider a transaxial section z = z0 measured using a ring of detectors. Figure 4.1 defines the variables s and 0 used to parameterize a straight line (an LOR) with respect to a Cartesian coordinate system (x, y) in the plane. The radial variable s is

1 When the depth of interaction is accounted for, LORs are defined by connecting photon interaction points projected on the long axis of the crystals [1].

Figure 4.1. Schematic representation of a ring scanner. A tube of response between two detectors da and db is represented in grey with the corresponding LOR, which connects the center of the front face of the two detectors. The sinogram variables s and 0 define the location and orientation of the LOR.

p(s, (,z0) = \2dtf(= s co( t s(n y = s sin(+ t co( =z z0 )

where t, the integration variable, is the coordinate along the line. In the presentation of the 2D reconstruction problem below, we will omit the z arguments in the functions p and f

The next section describes how a function f(x,y) can be reconstructed from its line integrals measured for |s| < Rf and 0 < 0 < n. The mathematical operator mapping a function f(x,y) onto its line integrals p(s, 0) is called the x-ray transform(2\ and this operator will be denoted X, so that p(s, 0) = (Xf)(s, 0). The function p(s, 0) is referred to as a sinogram, and the variables (s, 0) are called sinogram variables. This name was coined in 1975 by the Swedish scientist Paul Edholm because the set of LORs containing a fixed point (x0, y0) are located along a sinusoid s = x0 cos 0 + y0 sin 0 in the (s, 0) plane, as can be seen from Eq. (3). For a fixed angle 0 = 00, the set of parallel line integrals p(s, 00) is a 1D parallel projection of f.

At the line integral approximation, and after data pre-correction, the PET data provide estimates of the x-ray transform for all LORs connecting two detectors, i.e., pda,db ~ p(s, 0), where the parameters (s, 0) correspond to the radial position and angle of £-da,db- Thus, the geometrical arrangement of discrete detectors in a scanner determines a set of samples (s, 0) in sinogram space. The most common arrangement is a ring scanner: an even number Nd of detectors uniformly spaced along a circle of radius Rd > RF(3). Each detector, in coincidence with an arc of detectors on the opposite side of the ring, defines a fan of LORs (figure 3.6), and the corresponding sampling of the sinogram is:

Figure 4.1. Schematic representation of a ring scanner. A tube of response between two detectors da and db is represented in grey with the corresponding LOR, which connects the center of the front face of the two detectors. The sinogram variables s and 0 define the location and orientation of the LOR.

the signed distance between the LOR and the center of the coordinate system (usually the center of the detector ring). The angular variable 0 specifies the orientation of the LOR. Line integrals of the tracer distribution are then defined as

where the pair of indices j, k corresponds to the coincidences between the two detectors with indices da = j - k and db = k. Due to the curvature of the ring, each parallel projection j is sampled non-uniformly in the radial variable, with a sampling distance As — 2nRd/Nd near the center of the FOV (i.e. for s — 0). The radial samples of two adjacent parallel projections j and j +1 are shifted by approximately As/2, as can be seen by shifting only one end of a LOR (Fig. 4.2).

For practical and historical reasons, it is customary in PET to reorganize the data on a rectangular sampling grid sk = kAs

Figure 4.2. Representation of the sinogram sampling for a ring scanner with 20 detectors. The interleaved pattern provided by the LORs connecting detector pairs is shown by +'s. Note the decrease of the radial sampling distance at large values of s, which is exaggerated here because the plot extends to 90% of the ring radius. PET acquisition systems reorganize these data into the rectangular sampling pattern (see equation (5)) shown by x's.

2 In 2D, the x-ray transform coincides with the Radon transform, see [2].

3 If the depth of interaction is not measured, an effective value of Rd is used that accounts for the mean penetration of the 511 keV gamma rays into the crystal.

with A j = 2n/Nd, N< = Nd/2, and a uniform radial sampling interval As = Rdn/Nd equal to half the spacing between adjacent detectors in the ring. The parallel-beam sampling defined by Eq. (5) will be used in the rest of the chapter. In this scheme the line defined by a sample (j, k) no longer coincides with a measured LOR connecting two detectors. The reorganization into parallel-beam data therefore requires an interpolation (usually linear interpolation) to redistribute the counts on the rectangular sampling grid (Eq. (5)). This interpolation entails a loss of resolution, which is usually negligible owing to the relatively low SNR in PET(4). In addition, the geometry of some scanners is not circular, but hexagonal or octagonal. Resampling is then needed anyway if standard analytic algorithms are to be used.

When the average number of detected coincidences per sinogram sample is small, undersampling is often applied to reduce the storage and computing requirements. Angular undersampling (increasing Aj) is called transaxial mashing in the PET jargon. The mashing factor defined by m = AjNd/(2n) is usually an integer so that undersampling simply amounts to summing groups of m consecutive rows (j's) in the sinogram. Angular undersampling results in a loss of resolution, which is smallest at the center of the FOV and maximum at its edge. Therefore, the maximum allowed mashing factor depends not only on the SNR but also on the radius RF of the reconstructed FOV: for a fixed SNR, we can allow more mashing for a brain scan than for a whole-body study. Radial undersampling (increasing As) tends to generate more severe artifacts, and is rarely used. A rule of thumb to match the radial and angular sampling is the relation A j — As/RF, which is derived using Shannon's sampling theory [2].

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