Positron emission tomography (PET) has been used for in vivo treatment verification, mainly for range verification, in proton therapy. Evaluating the direct dose from PET measurements remains challenging; however, it is highly desirable from a clinical perspective. In this study, a method for estimating the dose distribution from the positron emitter distributions was developed using the maximum likelihood expectation maximization algorithm. The 1D spatial relationship between positron emitter distributions and a dose distribution in an inhomogeneous target was inputted into the system matrix based on a filter framework. In contrast, spatial resolution of the PET system and total variation regularization (as prior knowledge for dose distribution) were considered in the 3D image-space. The dose estimation was demonstrated using Monte Carlo simulated PET activity distributions with substantial noise in a head and neck phantom. This mimicked the single field irradiation of the spread-out Bragg peak beams at clinical dose levels. Besides the simple implementation of the algorithm, this strategy achieved a high-speed calculation (30 s for a 3D dose estimation) and accurate dose and range estimations (less than 10% and 2 mm errors at 1-σ values, respectively). The proposed method could be key for using PET for in vivo dose monitoring.