The authors develop the image reconstruction problem in emission computed tomography (ECT) from the photon transport equation. Based on this model, two possible inversion strategies are discussed. The first uses additional transmission measurements to alleviate the difficulty of the inverse problem. The second uses the emission data and is mathematically demanding. The authors conduct numerical experiments, which are based on the second strategy. They present the theoretical setting of ECT and discuss some new numerical strategies based on Tikhonov type regularization. They also include experiments to compare some of the numerical approaches.