KAHIBARO
19.5. Random Coincidences
Table of Contents
Unrelated detection events
In PET simulations, a coincidence event is created when two detector singles occur within the coincidence time window. Not every coincidence corresponds to two photons from the same annihilation. When the paired singles are unrelated in origin, the result is called a random coincidence.
Random coincidences arise when two gamma photons, each produced by a different positron annihilation, happen to be detected within the coincidence timing window by two detectors that form a valid pair. The coincidence sorter does not know the true origin of each photon. It only checks time and detector geometry. Any pair of singles that passes these checks becomes a coincidence event. If those singles originate from different annihilation events, you have a random.
It is useful to think of randoms as pure chance alignments in time. At high activity or with a wide coincidence timing window, the probability that two independent detection events fall within the window becomes significant. Randoms contribute counts that do not correspond to any real line of response through a single annihilation position and therefore degrade image quality and quantitative accuracy.
In a simulation, you can conceptually label each primary annihilation with a unique event or history identifier. True coincidences contain two photons with the same annihilation ID. Random coincidences contain photons with different annihilation IDs. In real measurements you cannot see this label, but in Monte Carlo you can use it to study and quantify randoms.
The detection rate of random coincidences depends on the singles count rates in the two detectors and the coincidence time window width $\Delta t$. If $R_1$ and $R_2$ are the singles rates in two detectors that can form coincidences, a common analytical approximation for the random coincidence rate between that detector pair is
$$
R_{\text{random}} \approx 2 R_1 R_2 \Delta t.
$$
The factor 2 appears because either detector can register the first photon in time. For a system with many detector pairs you sum over all valid detector pairs or integrate over the whole angular field of view. Simulation with GATE allows you to compare this analytical expectation to the random rates you observe in the coincidence output.
Random coincidences are coincidence events built from two singles that come from different annihilation events. They are unrelated in origin but fall within the coincidence timing window by chance. An approximate formula for the random rate for one detector pair is
$$R_{\text{random}} \approx 2 R_1 R_2 \Delta t.$$
Randoms have several important consequences for PET imaging. They increase the apparent coincidence count rate, which can saturate detector electronics and distort standard performance curves. They add a nearly uniform background component in sinogram space, which reduces contrast and signal to noise ratio. At high activity, the fraction of random coincidences can become comparable to or even exceed the true coincidence rate, especially if the timing window is not sufficiently narrow.
From a simulation point of view, controlling randoms involves both acquisition parameters and data processing. Narrowing the coincidence timing window in the digitizer configuration reduces the probability that two independent singles will be classified as coincident, but it also risks rejecting some true coincidences that have larger time differences due to detector timing resolution. Improving the modeled timing resolution also reduces randoms because fewer singles appear to fall within the effective window.
In GATE, you examine random coincidences by inspecting coincidence output and using the simulation truth information. If the coincidence output includes tags such as an annihilation or history identifier for each photon, you can classify coincidences in post processing as true, scatter, or random by comparing these identifiers. Random coincidences are those where the two photons have different annihilation identifiers. This classification is essential when you want to compute PET performance metrics such as noise equivalent count rate or to validate random correction methods.
Understanding random coincidences at this level allows you to design simulations that test how activity, timing window, and detector timing resolution influence the random fraction and overall image quality.
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