Robust Timing Calibration for PET Using L1-Norm Minimization

Positron emission tomography (PET) relies on accurate timing information to pair two 511-keV photons into a coincidence event. Calibration of time delays between detectors becomes increasingly important as the timing resolution of detector technology improves, as a calibration error can quickly become a dominant source of error. Previous work has shown that the maximum likelihood estimate of these delays can be calculated by least squares estimation, but an approach is not tractable for complex systems and degrades in the presence of randoms. We demonstrate the original problem to be solvable iteratively using the LSMR algorithm. Using the LSMR, we solve for 60 030 delay parameters, including energy-dependent delays, in 4.5 s, using 1 000 000 coincidence events for a two-panel system dedicated to clinical locoregional imaging. We then extend the original least squares problem to be robust to random coincidences and low statistics by implementing 11-norm minimization using the alternating direction method of the multipliers (ADMM) algorithm. The ADMM algorithm converges after six iterations, or 20.6 s, and improves the timing resolution from 64.7 ± 0.1s full width at half maximum (FWHM) uncalibrated to 15.63 ± 0.02ns FWHM. We also demonstrate this algorithm's applicability to commercial systems using a GE Discovery 690 PET/CT. We scan a rotating transmission source, and after subtracting the 511-keV photon time-of-flight due to the source position, we calculate 13 824 per-crystal delays using 5 000 000 coincidence events in 3.78 s with three iterations, while showing a timing resolution improvement that is significantly better than previous calibration methods in the literature.