Nested cages

Abstract

Many tasks in geometry processing and physical simulation benefit from multiresolution hierarchies. One important characteristic across a variety of applications is that coarser layers strictly encage finer layers, nesting one another. Existing techniques such as surface mesh decimation, voxelization, or contouring distance level sets do not provide sufficient control over the quality of the output surfaces while maintaining strict nesting. We propose a solution that enables use of application-specific decimation and quality metrics. The method constructs each next-coarsest level of the hierarchy, using a sequence of decimation, flow, and contact-aware optimization steps. From coarse to fine, each layer then fully encages the next while retaining a snug fit. The method is applicable to a wide variety of shapes of complex geometry and topology. We demonstrate the effectiveness of our nested cages not only for multigrid solvers, but also for conservative collision detection, domain discretization for elastic simulation, and cage-based geometric modeling.

Generalized Matryoshka: Computational Design of Nesting Objects

“…Recently, Sacht et al presented a flow‐based method to progressively fit a triangle mesh within a simplification thereof [SVJ15]. These “nested cages” are tightly fitting, but solve a complementary problem: they continuously deform the outer object to fit the inner object as tightly as possible.…”

“…Unlike traditional bounding volumes [SVJ15], we define valid self‐nesting to occur when a solid shape 𝒜 transformed by a non‐reflecting similarity transform T : ℝ 3 → ℝ 3 lies strictly inside itself ( T (𝒜) → 𝒜) and the parts of 𝒜 above and below a cut plane P can be removed by translating linearly along some vectors a + ∊ ℝ 3 and a − ∊ ℝ 3 (resp.) without colliding into the transformed shape.…”

Proxy‐guided Image‐based Rendering for Mobile Devices

“…The recent Nested Cages algorithm [SVJ15] is able to produce strictly encaging simplifications. However, source code is not available.…”

A Multilevel SPH Solver with Unified Solid Boundary Handling

“…For mesh coarsening, Müller [Mül08] proposed to successively coarsen finer-levels preserving only representative particles, and Sacht et al [SVJ15] presented a method for constructing the hierarchy of surface meshes such that coarser levels are contained by their finer levels. For mesh coarsening, Müller [Mül08] proposed to successively coarsen finer-levels preserving only representative particles, and Sacht et al [SVJ15] presented a method for constructing the hierarchy of surface meshes such that coarser levels are contained by their finer levels.…”

“…Constructing the hierarchical structures for particle-based methods is difficult mainly because of irregular particle positions and changing particle neighbors (i.e., connectivities). For mesh coarsening, Müller [Mül08] proposed to successively coarsen finer-levels preserving only representative particles, and Sacht et al [SVJ15] presented a method for constructing the hierarchy of surface meshes such that coarser levels are contained by their finer levels. However, these approaches are essentially designed for unstructured meshes with no frequent connectivity changes, and thus they are computationally expensive for particle-based methods.…”