  In this paper, we provide a precise mathematical model of crystal-to-crystal
response which is used to generate the white image - a necessary compensation
model needed to overcome the physical limitations of the PET scanner. We
present a closed-form solution, as well as several accurate approximations, due
to the complexity of the exact mathematical expressions. We prove,
experimentally and analytically, that the difference between the best
approximations and real crystal-to-crystal response is insignificant. The
obtained responses are used to generate the white image compensation model. It
can be written as a single closed-form expression making it easy to implement
in known reconstruction methods. The maximum likelihood expectation
maximization (MLEM) algorithm is modified and our white image model is
integrated into it. The modified MLEM algorithm is not based on the system
matrix, rather it is based on ray-driven projections and back-projections. The
compensation model provides all necessary information about the system.
Finally, we check our approach on synthetic and real data. For the real-world
acquisition, we use the Raytest ClearPET camera for small animals and the NEMA
NU 4-2008 phantom. The proposed approach overperforms competitive,
non-compensated reconstruction methods.
