Magnetic resonance imaging (MRI) is a versatile noninvasive tool for achieving full-field quantitative visualization of biomedical fluid flows. In this study, two MRI velocimetry techniques (spin tagging and phase contrast) are used to obtain velocity measurements in a Poiseuille flow for Reynolds numbers below 1,000. Spin-tagging MRI velocimetry supplies the displacement of tagged grids of nuclear spins from which the velocity field can be inferred, while phase contrast MRI velocimetry directly provides velocity data for every pixel in the field of view. Although the phase contrast method is more accurate for this flow, this technique is more sensitive to errors from magnetic susceptibility gradients, higher order motions, and has limited dynamic range. Spin-tagging MRI velocimetry is a viable alternative if automatic methods for extracting velocity fields from the tags can be found. Optical flow, a technique originally developed for machine vision applications, is proposed here as a postprocessing step to obtain two-dimensional velocity fields from spin-tagging MRI images. Results with artificially generated grids demonstrate the robustness of the optical flow algorithm to noise and indicate that a 7%-10% average error can be expected from the optical flow calculations alone, independent of MRI image artifacts. Experiments on spin-tagging MRI images for a Re=230 Poiseuille flow gave an average error of 6.41%, which was consistent with the measurement error of the generated (synthetic) images with the same level of random noise superimposed.