We present a new computational method for reconstructing a vector velocity field from scattered, pulsed-wave ultrasound Doppler data. The main difficulty is that the Doppler measurements are incomplete, for they do only capture the velocity component along the beam direction. We thus propose to combine measurements from different beam directions. However, this is not yet sufficient to make the problem well posed because 1) the angle between the directions is typically small and 2) the data is noisy and nonuniformly sampled. We propose to solve this reconstruction problem in the continuous domain using regularization. The reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared difference between the Doppler data and the projected velocities 2) a quadratic regularization functional that imposes some smoothness on the velocity field. We express our solution for this minimization problem in a B-spline basis, obtaining a sparse system of equations that can be solved efficiently. Using synthetic phantom data, we demonstrate the significance of tuning the regularization according to the a priori knowledge about the physical property of the motion. Next, we validate our method using real phantom data for which the ground truth is known. We then present reconstruction results obtained from clinical data that originate from 1) blood flow in carotid bifurcation and 2) cardiac wall motion.