A Framework for Fluid Motion Estimation using a Constraint-Based Refinement Approach

The goal of this paper is to formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of the constraint, our results closely approximate the continuity equation based fluid flow. This closeness is theoretically justified through a modified augmented Lagrangian method and validated numerically. Further, along with the continuity constraint, our model can include other geometric constraints as demonstrated. The mathematical well-posedness 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 approach preserves the spatial characteristics of the divergence and the vorticities. We perform several numerical experiments and show the results on different datasets.