The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

3D registration is a fundamental step to obtain the correspondences between surfaces. Traditional mesh alignment methods tackle this problem through non-rigid deformation, mostly accomplished by applying ICP-based (Iterative Closest Point) optimization. The embedded deformation method is proposed for the purpose of acceleration, which enables various real-time applications. However, it regularizes on an underlying simplified structure, which could be problematic for intricate cases when the simplified graph doesn't fully represent the surface attributes. Moreover, without elaborate parameter-tuning, deformation usually performs suboptimally, leading to slow convergence or a local minimum if all regions on the surface are assumed to share the same rigidity during the optimization. In this article, we propose a novel solution that decouples regularization from the underlying deformation model by explicitly managing the rigidity of vertex clusters. We further design an efficient two-step solution that alternates between isometric deformation and embedded deformation with cluster-based regularization. Our method can easily support region-adaptive regularization with cluster refinement and execute efficiently. Extensive experiments demonstrate the effectiveness of our approach for mesh alignment tasks even under large-scale deformation and imperfect data. Our method outperforms state-of-the-art methods both numerically and visually.

Meshes generated by range scanners are often incomplete and contain complex holes due to limited input coverage and occlusion. In this paper, we present an effective method to fill the gap regions on meshes by leveraging the templates inferred from the learning-based method. We first segment both source and template models into corresponding parts. Each part will be aligned with non-rigid deformation. We then modify the gap regions by “virtual” depth maps rendered using the aligned parts from newly selected viewpoints. Comparing with the template-based mesh completion approaches, our algorithm can generate natural appearances without any user interaction. Comparing with the state-of-the-art volumetric fusion methods, our approach supports selective blending, which only modifies the regions of interest and prevents bad template inference from impacting the source.