Subdivision‐Specialized Linear Algebra Kernels for Static and Dynamic Mesh Connectivity on the GPU

Abstract

Subdivision surfaces have become an invaluable asset in production environments. While progress over the last years has allowed the use of graphics hardware to meet performance demands during animation and rendering, high‐performance is limited to immutable mesh connectivity scenarios. Motivated by recent progress in mesh data structures, we show how the complete Catmull‐Clark subdivision scheme can be abstracted in the language of linear algebra. While this high‐level formulation allows for a fully parallel implementation with significant performance gains, the underlying algebraic operations require further specialization for modern parallel hardware. Integrating domain knowledge about the mesh matrix data structure, we replace costly general linear algebra operations like matrix‐matrix multiplication by specialized kernels. By further considering innate properties of Catmull‐Clark subdivision, like the quad‐only structure after refinement, we achieve an additional order of magnitude in performance and significantly reduce memory footprints. Our approach can be adapted seamlessly for different use cases, such as regular subdivision of dynamic meshes, fast evaluation for immutable topology and feature‐adaptive subdivision for efficient rendering of animated models. In this way, patchwork solutions are avoided in favor of a streamlined solution with consistent performance gains throughout the production pipeline. The versatility of the sparse matrix linear algebra abstraction underlying our work is further demonstrated by extension to other schemes such as and Loop subdivision.

A Halfedge Refinement Rule for Parallel Catmull‐Clark Subdivision

“…Therefore, we separated creased‐mesh performances from the other ones for fair comparisons. Note that Nießner et al [NLMD12] and Mlakar et al [MWS∗20] claim support for semi‐sharp creases in their respective papers, but we were unable to reproduce such results using their source code. Since our implementation is publicly available and provides straightforward support for semi‐sharp creases, we position ourselves as the only practical alternative to OpenSubdiv.…”

“…We also emphasize that each aforementioned publication requires a custom data‐structure, while ours relies on the popular halfedge data‐structure. Interestingly, halfedges have been discarded for being too complex and/or costly for parallel processors [SJP05,MWS∗20]. This is probably because halfedges are often understood as linked‐lists; in Section 3.3, we provide a pointerless approach that hopefully clarifies this misunderstanding.…”

“…Memory Allocation Our implementation only requires memory for topology ℋ 1 , ⃛, ℋ D and vertex points 𝒱 1 , ⃛, 𝒱 D . Note that unlike the approaches of [PEO09] and [MWS∗20], we do not require any extra temporary memory as our buffers provide all the necessary information to compute the subdivision. For the topology, we allocate a halfedge buffer that stores the operators T win , V ert , and E dge as signed, 32‐bit integers; we neglect the remaining operators N ext , P rev , and F ace as we compute them analytically as shown in Figure 2 (b).…”

“…First, the geometric complexity of modern scenes makes it hard for a sequential implementation to cope with the exponential complexity induced by the algorithm. Second, there is a strong desire within the industry for interactive GPU‐based implementations supporting both rendering and modeling scenarios [BFK∗16,MWS∗20]. In this work, we contribute to this research area via a novel parallel method for uniform subdivision on halfedge meshes suitable for both CPUs and GPUs.…”

RXMesh

“…Nonetheless we believe that our data structure, with its focus on mesh partitioning, is well suited for an evolution toward dynamic capability because parallelizing across partitions helps ease the problems with concurrency. We are further encouraged by recent advances in mesh subdivision on the GPU [Mlakar et al 2020] and dynamic GPU graph data structures [Awad et al 2020;Winter et al 2018].…”