Detector Modeling with 4D Filtering in PET Bal´azsT´oth 1 , Mil´an Magdics 1 ,L´aszl´oSzirmay-Kalos 1 , and Anton Penzov 2 1 BME IIT, Hungary tbalazs@iit.bme.hu 2 Bulgarian Academy of Sciences, Bulgaria apenzov@yahoo.com Abstract. This paper presents a fast algorithm to simulate inter-crystal scattering and detector response in Positron Emission Tomography (PET). Theoretically, inter-crystal scattering simulation would require the solu- tion of a Fredholm-type integral equation, which is quite time consum- ing. However, most of this calculation can be ported to a pre-processing phase, taking advantage of the fact that the structure of the detector is fixed. Pre-computing the scattering probabilities inside the crystals, the final system response is the convolution of the geometric response ob- tained with the assumption that crystals are ideal absorbers and the crys- tal transport probability matrix. This convolution is four-dimensional which poses complexity problems as the complexity of the naive convo- lution evaluation grows exponentially with the dimension of the domain. We use Monte Carlo method to attack the curse of dimension. We demon- strate that with these techniques the detector model can be incorporated into the reconstruction process paying just negligible overhead. 1 Introduction In iterative tomography reconstruction, forward projections and corrective back projections alternate. Forward projection simulates the physical process and computes the expected detector responses from the actually estimated activ- ity distribution. Then, in back projection, the computed detector responses are compared to the measured responses and the current activity estimation is cor- rected accordingly. In positron emission tomography (PET), we should find the spatial emission intensity distribution of positron–electron annihilations [5, 1]. The actual emission density estimate x(v) describes the number of photon pairs (i.e. the annihilation events) born in a unit volume around point v. To represent the unknown function with finite data, it is approximated in a finite element form: x(v)= N voxel ∑ V =1 x V b V (v), where x 1 ,...,x N voxel are the unknown coefficients and b V (v)(V =1,...,N voxel ) are pre-defined basis functions.