The SNR of positron emission data with Gaussian and non-Gaussian time-of-flight kernels, with application to prompt photon coincidence.

It is well known that measurement of the time-of-flight (TOF) increases the information provided by coincident events in positron emission tomography (PET). This information increase propagates through the reconstruction and improves the signal-to-noise ratio in the reconstructed images. Takehiro Tomitani has analytically computed the gain in variance in the reconstructed image, provided by a particular TOF resolution, for the center of a uniform disk and for a Gaussian TOF kernel. In this paper we extend this result, by computing the signal-to-noise ratio (SNR) contributed by individual coincidence events for two different tasks. One task is the detection of a hot spot in the center of a uniform cylinder. The second one is the same as that considered by Tomitani, i.e. the reconstruction of the central voxel in the image of a uniform cylinder. In addition, we extend the computation to non-Gaussian TOF kernels. It is found that a modification of the TOF-kernel changes the SNR for both tasks in almost exactly the same way. The proposed method can be used to compare TOF-systems with different and possibly event-dependent TOF-kernels, as encountered when prompt photons, such as Cherenkov photons are present, or when the detector is composed of different scintillators. The method is validated with simple 2D simulations and illustrated by applying it to PET detectors producing optical photons with event-dependent timing characteristics.