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 well-posedness 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.

翻译：基于物理的光流模型在捕捉数字图像中流体运动产生的形变方面取得了成功。然而，目前缺乏一个通用的理论框架来分析多种基于物理的模型。为此，我们提出了一种采用基于约束的细化方法进行流体运动估计的通用框架。我们证明，对于特定的约束选择，我们的结果能够紧密逼近经典的基于连续性方程的流体流动方法。这种近似性通过增广拉格朗日方法以一种新颖的方式得到了理论上的证明。Uzawa迭代的收敛性通过改进的有界约束算法得以展示。在希尔伯特空间框架下研究了问题的数学适定性。此外，我们观察到与Cauchy-Riemann算子之间存在令人惊讶的联系，该算子使系统对角化，导致了涉及流场散度和旋度的扩散现象。我们进行了多项数值实验，并在不同数据集上展示了结果。同时，我们证明，一种涉及流场旋度的流动驱动细化方法，在无需对图像数据添加任何额外假设的前提下，其性能优于经典的基于物理的光流方法。