The three-dimensional image-reconstruction problem solved here for optical-sectioning microscopy is to estimate the fluorescence intensity lambda(x), where x epsilon R3, given a series of Poisson counting process measurements [Mj(dx)]jJ = 1, each with intensity [formula: see text] with [formula: see text] being the point spread of the optics focused to the jth plane and sj(y) the detection probability for detector point y at focal depth j. A maximum a posteriori reconstruction generated by inducing a prior distribution on the space of images via Good's three-dimensional rotationally invariant roughness penalty [formula: see text] It is proven that the sequence of iterates that is generated by using the expectation maximization algorithm is monotonically increasing in posterior probability, with stable points of the iteration satisfying the necessary maximizer conditions of the maximum a posteriori solution. The algorithms were implemented on the DECmpp-SX, a 64 x 64 parallel processor, running at < 2 s/(64(3), 3-D iteration). Results are demonstrated from simulated as well as amoebae and volvox data. We study performance comparisons of the algorithms for the missing-data problems corresponding to fast data collection for rapid motion studies in which every other focal plane is removed and for imaging with limited detector areas and efficiency.