  The goal of this paper is to propose two nonlinear variational models for
obtaining a refined motion estimation from an image sequence. Both the proposed
models can be considered as a part of a generalized framework for an accurate
estimation of physics-based flow fields such as rotational and fluid flow. The
first model is novel in the sense that it is divided into two phases: the first
phase obtains a crude estimate of the optical flow and then the second phase
refines this estimate using additional constraints. The correctness of this
model is proved using an evolutionary PDE approach. The second model achieves
the same refinement as the first model, but in a standard manner, using a
single functional. A special feature of our models is that they permit us to
provide efficient numerical implementations through the first-order primal-dual
Chambolle-Pock scheme. Both the models are compared in the context of accurate
estimation of angle by performing an anisotropic regularization of the
divergence and curl of the flow respectively. We observe that, although both
the models obtain the same level of accuracy, the two-phase model is more
efficient. In fact, we empirically demonstrate that the single-phase and the
two-phase models have convergence rates of order $O(1/N^2)$ and $O(1/N)$
respectively.
