New Consistency Equation for Time-of-Flight PET

The redundancy in 3D time-of-flight (TOF) PET data can be exploited to reduce data storage or to estimate unmeasured data samples caused by defective or missing detectors. Mathematically, redundancy is expressed by consistency conditions which can be expressed either in terms of the 3D Fourier transform of the data or as a pair of partial differential equations (PDE). The benefit of the latter is that the PDEs are local and therefore can be applied even if some data samples are missing. This paper describes a new consistency PDE for 3D TOF PET, which only involves data within a single "segment" (data subset with fixed polar angle). The PDE is applied to rebin 3D TOF data onto 3D non-TOF data. The proposed rebinning algorithm reduces to the methods based on the most likely annihilation point in the limit where the TOF resolution tends to zero. Numerical results with simulated and phantom data illustrate the performance of the algorithm.