An efficient analytical calculation of probability matrix in 2D SPECT

Expectation Maximization iterative reconstruction algorithms are being widely used in PET and SPECT imaging. The system probability matrix is usually calculated by using Monte Carlo simulations, since the analytical calculation is a rather complicated problem, especially in 3D reconstruction; however, realistic Monte Carlo simulations in 3D are time consuming and simplifications are necessary. In this paper, the probability matrix in the case of 2D SPECT is analytically calculated, using only two basic parameters: (i) the total number of image pixels and (ii) the number of projection angles. In the more general phase three more parameters will be taken into account: (i) the distance between the detectors and the object to be imaged; (ii) the collimator parameters; (iii) scintillator cells parameters (in the case of pixilated scintillators) and relative position between scintillator cell and collimator hole. It is shown that the accuracy of the probability matrix affects the quality of reconstructed images, especially in the case of pixilated scintillators, which are used in many dedicated SPECT systems. The methods presented here can be extended to 3D SPECT and also 2D and 3D PET. In addition, this analytically calculated matrix can be a reference matrix in order to be compared with Monte Carlo generated matrices.