Particle tracking velocimetry (PTV) gives quantitative estimates of fluid flow velocities from images. But particle tracking is a complicated problem, and it often produces results that need substantial post-processing. We propose a novel Gaussian process regression-based post-processing step for PTV: The method smooths ("denoises") and densely interpolates velocity estimates while rejecting track irregularities. The method works under a large range of particle densities and fluid velocities. In addition, the method calculates standard deviances (error bars) for the velocity estimates, opening the possibility of propagating the standard deviances through subsequent processing and analysis. The accuracy of the method is experimentally evaluated using Optical Coherence Tomography images of particles in laminar flow in a pipe phantom. Following this, the method is used to quantify cilia-driven fluid flow and vorticity patterns in a developing embryo.