Volumetric Flow Estimation for Incompressible Fluids Using the Stationary Stokes Equations

In experimental fluid dynamics, the flow in a volume of fluid is observed by injecting high-contrast tracer particles and tracking them in multi-view video. Fluid dynamics researchers have developed variants of space-carving to reconstruct the 3D particle distribution at a given time-step, and then use relatively simple local matching to recover the motion over time. On the contrary, estimating the optical flow between two consecutive images is a long-standing standard problem in computer vision, but only little work exists about volumetric 3D flow. Here, we propose a variational method for 3D fluid flow estimation from multi-view data. We start from a 3D version of the standard variational flow model, and investigate different regularization schemes that ensure divergence-free flow fields, to account for the physics of incompressible fluids. Moreover, we propose a semi-dense formulation, to cope with the computational demands of large volumetric datasets. Flow is estimated and regularized at a lower spatial resolution, while the data term is evaluated at full resolution to preserve the discriminative power and geometric precision of the local particle distribution. Extensive experiments reveal that a simple sum of squared differences (SSD) is the most suitable data term for our application. For regularization, an energy whose Euler-Lagrange equations correspond to the stationary Stokes equations leads to the best results. This strictly enforces a divergence-free flow and additionally penalizes the squared gradient of the flow.