We present a method combining generalized Tikhonov regularization with a finite element approximation for reconstructing smooth velocity and velocity gradient fields from spatially scattered and noisy velocity data in a two-dimensional complex flow domain. Synthetic velocity data for a cross-slot geometry are generated using the Oldroyd-B solution, subsequently perturbed by random noise. Performances of diverse finite element continuity-regularization criterion combinations are tested against noise-free data, while the optimum regularization parameter is determined using generalized cross-validation. The best performance is achieved for the velocity field and its gradients simultaneously by C-2 continuous Hermite finite elements and minimization of a norm of the velocity's third derivative. The standard regularization criterion based on the second derivative is shown to lead to systematic distortions in boundary regions, allowing therefore a lower reduction in the statistical error. Furthermore, optical fields are calculated by applying a differential constitutive equation directly to the reconstructed flow kinematics; high quality velocity gradient fields are shown to be an essential prerequisite for their reliable prediction. Overall, the method is expedient to implement and does not require boundary conditions. (C) 2011 The Society of Rheology. [DOI: 10.1122/1.3539986]