KAHIBARO
19.2. Coincidence Sorting
Table of Contents
Pairing singles
Coincidence sorting is the step that takes individual detector events, called singles, and decides which of them belong together as coincidence events. In PET, a coincidence represents two detected photons that are assumed to come from the same positron annihilation. GATE does not know this automatically. You must configure a coincidence sorter that inspects the singles list and groups events based on their detection times.
Conceptually, coincidence sorting works on a time-ordered list of singles. Each single has at least a detection time, a detector identifier, and an energy. The sorter scans this list and forms pairs that satisfy some conditions. The most important condition is temporal proximity, which is controlled by the coincidence time window, but you can also apply energy and detector constraints before or after sorting. Some implementations use a sliding time window, where you look at all singles that fall within a given time difference of a reference single, and select valid partners among them.
In GATE, coincidence sorting is usually part of the digitizer chain. First, a hits collection is converted into singles, including energy summation in a crystal, energy and time blurring, and application of energy thresholds. Then, a coincidence module reads these singles and outputs coincidence events. Each coincidence event typically contains information about both members of the pair, such as their detector positions, energies, times, and sometimes additional tags like event ID or scatter state if available.
It is important to remember that coincidence sorting does not know which pairs are physically true, scattered, or random. It just applies a temporal rule. The classification of coincidences into true, scattered, and random is usually done later in the analysis, sometimes with the help of extra information recorded in the simulation, such as a true event ID or a flag indicating Compton scattering. From the sorter point of view, any two singles that fit its rules form a coincidence, regardless of their physical origin.
In practical simulations, you must also think about how multiple singles in a short time span are handled. For example, three singles very close in time could, in principle, form up to three different pairs. A coincidence sorter needs a policy for these situations. Some sorters only allow each single to belong to at most one coincidence, usually by pairing the first valid partner it finds. Others allow multiple pairs, which can be useful for studying randoms, but increases the output size and complexity. The specific behavior depends on the configuration of the coincidence digitizer in your simulation.
The quality of coincidence pairing has a direct impact on PET performance metrics such as sensitivity, scatter fraction, and random fraction. If your pairing rule is too strict, you may lose true coincidences. If it is too loose, you will increase random coincidences. When you later analyze PET performance, always keep in mind exactly how the coincidence sorter was configured, because all your quantitative results depend on this step.
Coincidence windows
The core parameter of coincidence sorting is the coincidence time window. This is the maximum allowed time difference between two singles for them to be considered a coincidence. If $t_1$ and $t_2$ are the detection times of two singles, the basic condition is
$$
\lvert t_1 - t_2 \rvert \le \Delta t_\text{coinc},
$$
where $\Delta t_\text{coinc}$ is the coincidence time window.
Coincidence condition:
$$
\lvert t_1 - t_2 \rvert \le \Delta t_\text{coinc}
$$
Any pair of singles that fails this condition must not be counted as a coincidence.
In a PET simulation, the coincidence time window is usually chosen to reflect the timing capabilities of the scanner that you want to model. Conventional non TOF PET systems often use windows on the order of several nanoseconds. TOF PET systems, which have better timing resolution, can use narrower windows. In a GATE simulation, you set this window directly in the coincidence digitizer configuration, typically in units such as $ns$.
The choice of window width is a trade off. A very small window reduces the number of random coincidences, because unrelated singles are unlikely to happen so close in time, but it also risks losing true coincidences if your detector timing resolution is not perfect or if time blurring is significant. A very large window captures almost all true coincidences, but also accepts many random pairs, which degrades image quality and makes quantitative analysis more difficult.
In most simulations, the coincidence window works together with an energy window that you have already applied when creating singles. The energy window restricts singles to a range around the photopeak, for example around 511 keV. This removes many scattered or low energy events before coincidence sorting. Then, the time window selects which of these remaining singles are close enough in time to be considered coincident. Even though the coincidence sorter mainly uses time, your effective coincidence selection is a combination of both time and energy constraints.
The definition of the time difference can also be important. Usually, the sorter works in a global simulation time, where each single has an absolute timestamp from the start of the acquisition. The coincidence module typically uses this global time directly. When you later perform TOF analysis, you often compute the time difference again from the two recorded timestamps, but the coincidence window used during sorting has already determined which pairs exist in your output.
For complex acquisition scenarios, such as high activity sources or systems with many detectors, the coincidence window must also be considered from a computational perspective. A very large window forces the sorter to compare more singles with each other, which can increase processing time and memory use. A well chosen window that is physically realistic for your scanner and activity level will keep the coincidence output manageable and more relevant for your analysis.
When you design or interpret a PET simulation in GATE, always document the coincidence window you used. Its value directly affects measured count rates, random fractions, and many derived performance metrics. If you compare different scanner configurations or experimental data, make sure that the coincidence window is consistent, or explicitly account for differences, otherwise the comparison can be misleading.
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