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

TL;DR

This paper introduces a general framework for fluid motion estimation using constraint-based refinement, connecting physics-based models with classical methods and demonstrating improved flow estimation through numerical experiments.

Contribution

It formulates a unified theoretical framework for fluid motion estimation, linking physics-based models with classical methods via a novel augmented Lagrangian approach.

Findings

Flow-driven refinement outperforms classical methods

The framework closely approximates the continuity equation-based method

Numerical experiments validate the theoretical insights

Abstract

Physics-based optical flow models have been successful in capturing the deformities in fluid motion arising from digital imagery. However, a common theoretical framework analyzing several physics-based models is missing. In this regard, we formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. This closeness is theoretically justified by augmented Lagrangian method in a novel way. The convergence of Uzawa iterates is shown using a modified bounded constraint algorithm. The mathematical wellposedness is studied in a Hilbert space setting. Further, we observe a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. Several numerical experiments are performed and the results are shown on different datasets. Additionally, we demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data.