In this paper, we solve the problem of estimating dense and accurate depth maps from a single moving camera. A probabilistic depth measurement is carried out in real time on a per-pixel basis and the computed uncertainty is used to reject erroneous estimations and provide live feedback on the reconstruction progress. Our contribution is a novel approach to depth map computation that combines Bayesian estimation and recent development on convex optimization for image processing. We demonstrate that our method outperforms state-of-the-art techniques in terms of accuracy, while exhibiting high efficiency in memory usage and computing power. We call our approach REMODE (REgularized MOnocular Depth Estimation). Our CUDA-based implementation runs at 30Hz on a laptop computer and is released as open-source software.