Quadratic programming time pickoff method for multi-voltage threshold digitizer in PET

Multi-voltage threshold (MVT) digitization, which has been implemented in our preclinical scanner–Trans-PET BioCaliBurn, is a low-power sampling solution with a reasonable cost for fast scintillation pulses. This MVT digitizing scheme employs a few comparators with programmable reference voltages for determining the time points when the scintillation pulse crosses the user-defined voltage thresholds. After obtaining the digital time samples, use of various sophisticated linear or nonlinear algorithms to improve the accuracy of timing information becomes possible. In the previous implementation of MVT digitizers, the arrival time of a scintillation pulse was determined by the linear fitting (LF) algorithm, which assumes leading edge as a straight line, and not involves the transition time on the falling edge. It is not unreasonable to expect achieving a better timing resolution in MVT-based PET detectors by the combinatorial optimization of MVT samples both on leading and falling edges. In this paper, a novel method, referred to as quadratic programming (QP) method is therefore proposed to mark the time stamps of sampling event pulses. In this method, the arrival time is directly expressed as a parametric combination of the MVT samples. The parameters in the combination are obtained by the quadratic programming which minimizes variation of timing error. To evaluate the performance of QP/MVT method and other time pickoff methods, we setup two gamma ray detectors using $LaBr_{3}:Ce$ crystal and Hamamatsu R9800 PMT. The scintillation pulses are directly read out by Tektronics DPO 71604 digital storage oscilloscope. CTR of 175 ps was obtained by QP/MVT, and 191 ps by LF/MVT, when four thresholds were employed in each of the two channels. The experimental results indicate the potential advantage of QP/MVT in timing determination. Meanwhile, the assumption of linear leading edge, which is the basis of LF method, was demonstrated to be improper in the data analysis. For QP method, probably of even greater significance is the manner to define the parameters rather than the resulting detector-specified parameters.