This paper proposes a new method for reconstructing SPECT data, which builds on the EM approach to maximum likelihood estimation but aims to maximize the posterior probability, incorporating prior beliefs about the smoothness of isotope concentration. The method is illustrated using brain scan data. The introduction explains the background of SPECT, the challenges of tomographic data due to the random nature of radioactive decay, and the limitations of standard algorithms. The paper follows a Bayesian paradigm, constructing two probability models: one for the detected photon counts and another for the prior distribution of isotope concentration patterns. The first model accounts for Poisson variation in counts, while the second models prior knowledge about the patterns. The reconstruction is achieved by considering the posterior distribution derived from these two models. The paper also discusses the modeling of photon counts and weights, the reconstruction without prior information, and the modeling of prior information. The EM algorithm is adapted for Bayesian reconstruction, and the one-step-late (OSL) approximation is used to improve convergence. The method is applied to real data, showing improved results compared to standard methods. The paper concludes with discussions on diagnostics, model improvement, and implementation details.This paper proposes a new method for reconstructing SPECT data, which builds on the EM approach to maximum likelihood estimation but aims to maximize the posterior probability, incorporating prior beliefs about the smoothness of isotope concentration. The method is illustrated using brain scan data. The introduction explains the background of SPECT, the challenges of tomographic data due to the random nature of radioactive decay, and the limitations of standard algorithms. The paper follows a Bayesian paradigm, constructing two probability models: one for the detected photon counts and another for the prior distribution of isotope concentration patterns. The first model accounts for Poisson variation in counts, while the second models prior knowledge about the patterns. The reconstruction is achieved by considering the posterior distribution derived from these two models. The paper also discusses the modeling of photon counts and weights, the reconstruction without prior information, and the modeling of prior information. The EM algorithm is adapted for Bayesian reconstruction, and the one-step-late (OSL) approximation is used to improve convergence. The method is applied to real data, showing improved results compared to standard methods. The paper concludes with discussions on diagnostics, model improvement, and implementation details.