Computing the directed Hausdorff distance between two triangle meshes is a fundamental operation in geometry processing and simulation. While existing certified branch-and-bound (B&B) methods are efficient for well-separated geometry, they can become prohibitively expensive on large models under tight tolerances and near-zero distance configurations where pruning is limited. We present a GPU-accelerated certified B&B algorithm that explicitly maintains enclosing lower and upper bounds on the directed Hausdorff distance and terminates once their normalized gap, measured with respect to the bounding-box diagonal of the source mesh, meets a user-prescribed tolerance. To map the inherently prioritized search to SIMT (single-instruction, multiple-thread) hardware, we replace priority queues and recursion with a sorted, double-buffered wavefront pipeline built from bulk-parallel worklists for bound evaluation, culling, subdivision, and compaction. To mitigate loose bounds on thin primitives while preserving predictable stream behavior, we introduce a fixed-cardinality adaptive subdivision scheme that selectively applies double longest-edge bisection. To remain robust in deep-refinement regimes, we add a resource-aware deferral mechanism that enforces a device-capacity invariant by prioritizing candidates likely to be culled while postponing expensive ones. Finally, we improve numerical robustness under FP32 (single precision) via triangle-local coordinate transforms and other conservative numerical safeguards, and enhance coherence by spatially ordering the active set and traversing the BVH (bounding volume hierarchy) in triangle packets. Under the same stopping tolerance, experiments on an NVIDIA RTX 5090 show that our GPU solver remains numerically consistent with the FP64 CPU baseline, with normalized cross-platform deviation below 0.01% in over 99.9% of cases. Our method achieves millisecond-scale runtimes capable of supporting interactive frame rates, even on models with millions of triangles. Across the comparison set, it delivers throughput speedups of 836× on the Thingi10K/TetWild benchmark ( A → B ) and 709× on the Thingi10K/Decimation benchmark. Code and data for this paper are available at https://github.com/fhp-transient/gpu-hausdorff.

Abstract We propose a new method to accurately approximate the Pompeiu‐Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not contain the maximizer of the distance to B and subdivide the others for further processing. In contrast to previous methods, we use four upper bounds instead of only one, three of which newly proposed by us. Many triangles are discarded using the simpler bounds, while the most difficult cases are dealt with by the other bounds. Exhaustive testing determines the best ordering of the four upper bounds. A collection of experiments shows that our method is faster than all previous accurate methods in the literature.

We propose new quasi-interpolators for the continuous reconstruction of sampled images, combining a narrowly supported piecewise-polynomial kernel and an efficient digital filter. In other words, our quasi-interpolators fit within the generalized sampling framework and are straightforward to use. We go against standard practice and optimize for approximation quality over the entire Nyquist range, rather than focusing exclusively on the asymptotic behavior as the sample spacing goes to zero. In contrast to previous work, we jointly optimize with respect to all degrees of freedom available in both the kernel and the digital filter. We consider linear, quadratic, and cubic schemes, offering different tradeoffs between quality and computational cost. Experiments with compounded rotations and translations over a range of input images confirm that, due to the additional degrees of freedom and the more realistic objective function, our new quasi-interpolators perform better than the state of the art, at a similar computational cost.

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.

Abstract Decades of research have culminated in a robust geometry processing pipeline for surfaces. Most steps in this pipeline, like deformation, smoothing, subdivision and decimation, may create self‐intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet‐meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self‐intersections in the input surface. This enables volumetric processing on self‐intersecting models. We leverage conformalized mean‐curvature flow, which removes self‐intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.

This work presents a method for obtaining perceptually natural panoramic videos, that is, videos composed of wide-angle frames where straight lines and object shapes are preserved. The shape preservation for moving objects has a special treatment that ensures temporal coherence. Energies describing these properties are obtained for the case of a fixed omni-directional camera, and an optimization procedure is presented. Our optimization works per-frame, which makes the method scalable for arbitrarily long scenes.