The standard approach to densely reconstruct the motion in a volume of fluid is to inject high-contrast tracer particles and record their motion with multiple high-speed cameras. Almost all existing work processes the acquired multi-view video in two separate steps, utilizing either a pure Eulerian or pure Lagrangian approach. Eulerian methods perform a voxel-based reconstruction of particles per time step, followed by 3D motion estimation, with some form of dense matching between the precomputed voxel grids from different time steps. In this sequential procedure, the first step cannot use temporal consistency considerations to support the reconstruction, while the second step has no access to the original, high-resolution image data. Alternatively, Lagrangian methods reconstruct an explicit, sparse set of particles and track the individual particles over time. Physical constraints can only be incorporated in a post-processing step when interpolating the particle tracks to a dense motion field. We show, for the first time, how to jointly reconstruct both the individual tracer particles and a dense 3D fluid motion field from the image data, using an integrated energy minimization. Our hybrid Lagrangian/Eulerian model reconstructs individual particles, and at the same time recovers a dense 3D motion field in the entire domain. Making particles explicit greatly reduces the memory consumption and allows one to use the high-resolution input images for matching. Whereas the dense motion field makes it possible to include physical a-priori constraints and account for the incompressibility and viscosity of the fluid. The method exhibits greatly ( $${\approx }\,70\%$$ ≈ 70 % ) improved results over our recently published baseline with two separate steps for 3D reconstruction and motion estimation. Our results with only two time steps are comparable to those of state-of-the-art tracking-based methods that require much longer sequences.

中文翻译：

---

具有集成粒子重建的 3D 流体流动估计

在大量流体中密集重建运动的标准方法是注入高对比度示踪粒子并用多个高速摄像机记录它们的运动。几乎所有现有工作都使用纯欧拉或纯拉格朗日方法分两个单独的步骤处理获取的多视图视频。欧拉方法对每个时间步长的粒子进行基于体素的重建，然后是 3D 运动估计，在不同时间步长的预计算体素网格之间进行某种形式的密集匹配。在这个顺序过程中，第一步不能使用时间一致性考虑来支持重建，而第二步无法访问原始的高分辨率图像数据。或者，拉格朗日方法重建一个明确的，稀疏的粒子集并随着时间的推移跟踪单个粒子。物理约束只能在将粒子轨迹内插到密集运动场时纳入后处理步骤。我们首次展示了如何使用集成能量最小化从图像数据中联合重建单个示踪粒子和密集的 3D 流体运动场。我们的混合拉格朗日/欧拉模型重建单个粒子，同时在整个域中恢复密集的 3D 运动场。使粒子显式大大减少了内存消耗，并允许使用高分辨率输入图像进行匹配。而密集运动场可以包括物理先验约束并考虑流体的不可压缩性和粘性。该方法通过两个单独的步骤进行 3D 重建和运动估计，比我们最近发布的基线表现出极大的（$${\approx }\,70\%$$ ≈ 70 %）改进的结果。我们只有两个时间步长的结果与需要更长序列的最先进的基于跟踪的方法的结果相当。