Constraint Based Refinement of Optical Flow

The goal of this paper is to formulate a general framework for a constraint-based refinement of the optical flow using variational methods. We demonstrate that for a particular choice of the constraint, formulated as a minimization problem with the quadratic regularization, our results are close to the continuity equation based fluid flow. This closeness to the continuity model is theoretically justified through a modified augmented Lagrangian method and validated numerically. Further, along with the continuity constraint, our model can include geometric constraints as well. The correctness of our process is studied in the Hilbert space setting. Moreover, a special feature of our system is the possibility of a diagonalization by the Cauchy-Riemann operator and transforming it to a diffusion process on the curl and the divergence of the flow. Using the theory of semigroups on the decoupled system, we show that our process preserves the spatial characteristics of the divergence and the vorticities. We perform several numerical experiments and show the results on different datasets.