arxiv: 2605.03235 · v1 · submitted 2026-05-05 · 💻 cs.GR

Recognition: unknown

ADS: Random Sampling of Occupancy Functions using Adaptive Delaunay Scaffolding

Alla Sheffer, Leo Foord-Kelcey, Nicholas Vining, Oliver Oxford, Suzuran Takikawa

Pith reviewed 2026-05-07 12:39 UTC · model grok-4.3

classification 💻 cs.GR

keywords occupancy functionsrandom samplingDelaunay tetrahedralizationisosurface extractionimplicit surfacesadaptive refinementmesh generation

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The pith

Adaptive Delaunay scaffolding samples occupancy function surfaces with random points and connecting meshes using far fewer evaluations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method to both randomly sample points on surfaces defined by implicit occupancy functions and generate a mesh connecting those points. It does this by building and refining a Delaunay tetrahedralization in 3D space, focusing refinements on edges that cross the surface to place points close to it. Samples are taken at the surface intersections of these refined edges, and marching tetrahedra creates the mesh from them. Normal estimation then adds more points near detailed areas. This combination allows the task to be done with about ten times fewer calls to the occupancy function than previous approaches, which is useful for applications that need dense, unbiased samples along with surface connectivity.

Core claim

Adaptive Delaunay Sampling (ADS) uses a progressively computed Delaunay tetrahedralization as a scaffold that is refined by repeatedly identifying and subdividing crossing edges whose endpoints lie on opposite sides of the occupancy surface. The intersections of these fine crossing edges with the surface provide the pseudo-random samples, while marching tetrahedra applied to the tetrahedralization produces the isosurface mesh. Subsequent normal-based densification improves sampling near high-curvature features, all while requiring significantly fewer function evaluations than ray-shooting or grid-based alternatives.

What carries the argument

The Adaptive Delaunay Scaffolding, a Delaunay tetrahedralization that is iteratively refined at crossing edges to guide sampling toward the implicit surface and enable mesh extraction.

Load-bearing premise

Repeated refinement of crossing edges in the Delaunay tetrahedralization combined with normal-based densification produces sufficiently uniform and unbiased pseudo-random samples without systematic artifacts.

What would settle it

Measuring the area coverage and nearest-neighbor distances of ADS samples on a unit sphere and checking whether they match the statistics of true uniform random surface points within expected variance.

Figures

Figures reproduced from arXiv: 2605.03235 by Alla Sheffer, Leo Foord-Kelcey, Nicholas Vining, Oliver Oxford, Suzuran Takikawa.

**Figure 1.** Figure 1: Ray-shooting based occupancy surface sampling [Ling et al view at source ↗

**Figure 2.** Figure 2: Random [Cline et al. 2009] (a) and uniformly random [Ling et al. 2025] (b) ray-casting provide provably random sampling of occupancy surfaces but generate no connectivity information. Grid based isosurfacing methods such as Marching Cubes [Lorensen and Cline 1987] (c) or Occupancy dual contouring [Hwang and Sung 2024] (d) output fully connected but grid-biased samplings. ADS generates pseudo-random samples… view at source ↗

**Figure 3.** Figure 3: 2D illustration of our method: Delaunay scaffold with vertices con view at source ↗

**Figure 4.** Figure 4: ADS overview (in 2D). Left to right: (a) we sample a coarse, well-spaced initial point set within the bounding domain using Poisson disc sampling, then view at source ↗

**Figure 5.** Figure 5: Impact of barrier test: (a) 2D illustration: naive midpoint sampling view at source ↗

**Figure 6.** Figure 6: Refinement: (a) initial sampling and iso-surface; (b) sampling and view at source ↗

**Figure 7.** Figure 7: Additional visual comparisons of our ADS method to uniform ray casting [Ling et al view at source ↗

**Figure 8.** Figure 8: A gallery of our samplings and isosurface meshes using different sampling resolutions and input sources. Skateboard and motorcycle [Zhang et al view at source ↗

**Figure 9.** Figure 9: Optimizing our iso-surface meshes by collapsing short edges and flipping edges to improve smoothness and aspect ratio: (ac) initial iso-surface meshes. view at source ↗

**Figure 10.** Figure 10: Curvature adaptive sampling: (ac) Standard ADS outputs; (bd) Strongly curvature adapted outputs with more samples in higher curvature areas. view at source ↗

**Figure 11.** Figure 11: ADS outputs (a) can be directly filtered (rejection sampling plus edge collapse) using different rejection sampling thresholds: (b) 10% filtering further view at source ↗

**Figure 12.** Figure 12: When the input OF depicts very thin shapes, using our default method and default initial scaffold may fail to capture them even at fine mesh resolution view at source ↗

read the original abstract

Dense random sampling and surfacing of shapes encoded via implicit occupancy functions (OFs) are critical elements of many applications. Existing methods largely provide either one or the other of random sampling or mesh surfaces: ray shooting approaches deliver random samples with no connectivity, and grid-based methods deliver mesh surfaces but their sampling is highly biased. We propose a new method which delivers both pseudo-random OF surface samples and an isosurface mesh connecting them. Our method achieves these goals while requiring an order of magnitude fewer function evaluations than prior approaches. Key to our Adaptive Delaunay Sampling (ADS) approach is a progressively computed Delaunay tetrahedralization of points in 3D space, which we use as a sampling and surfacing scaffold. Starting from an initial coarse Delaunay scaffold, we repeatedly refine crossing edges, ones whose end vertices lie on opposite sides of the surface, augmenting the scaffold with points closer and closer to the surface. Each refinement step uses the Delaunay criterion to incorporate the newly added vertices into the scaffold, introducing new crossing edges. We use the intersections of fine crossing edges with the OF surface as the output samples, and use the marching tetrahedra method to surface these samples. We subsequently use normal estimation to densify the sampling near fine features and in areas of high surface curvature. We validate ADS by sampling 150 inputs at different resolutions, and provide extensive comparisons to existing alternatives. Our experiments demonstrate significant improvement in accuracy/function evaluation count trade-off, and showcase downstream applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

ADS integrates Delaunay refinement for joint sampling and meshing of occupancy functions but needs better checks on sample uniformity.

read the letter

The main thing to know is that this paper presents an adaptive Delaunay scaffolding method that jointly samples occupancy function surfaces and builds a connecting mesh by refining crossing edges in a tetrahedralization. It claims an order of magnitude fewer function evaluations than prior work. What stands out as new is the way the Delaunay criterion is used to propagate refinements and generate both the samples and the mesh in one process, plus the normal-based densification step. This avoids the bias of grid methods and the lack of connectivity in ray shooting. The experiments on 150 inputs showing better accuracy-evaluation trade-offs are a good practical check. The soft spot is whether the samples are truly pseudo-random and unbiased. Refining crossing edges and adding points based on normals might cluster samples in high-curvature areas or along specific directions, and the paper does not appear to include quantitative checks like sample discrepancy or nearest-neighbor statistics against uniform ground truth. Without that, the efficiency claim is interesting but the quality for downstream use is not fully demonstrated. This work is aimed at computer graphics and geometry processing folks who need efficient surface sampling from implicits. Someone working on modeling or simulation pipelines could get value from the integrated approach. It deserves a serious referee because the core algorithm is distinct and the validation is there, even if more analysis on distribution is needed. I recommend sending it for peer review with a request for uniformity metrics in the revision.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces Adaptive Delaunay Sampling (ADS), which builds a progressively refined Delaunay tetrahedralization scaffold starting from a coarse set of points. Crossing edges (those with endpoints on opposite sides of the implicit occupancy surface) are repeatedly refined by adding points closer to the surface; intersections of the finest crossing edges with the occupancy function provide the output samples. Marching tetrahedra produces a mesh connecting the samples, and normal estimation is used to densify sampling near high-curvature features. The central claims are that ADS simultaneously delivers pseudo-random surface samples and a mesh while requiring an order of magnitude fewer occupancy evaluations than ray-shooting or grid baselines, supported by experiments on 150 inputs showing improved accuracy-versus-evaluation trade-offs and downstream applications.

Significance. If the efficiency and unbiased-sampling claims hold, ADS would constitute a meaningful advance for graphics pipelines that require both dense random sampling and explicit meshing of implicit surfaces. The scale of the empirical validation (150 inputs with direct accuracy-versus-evaluation comparisons) is a clear strength and provides a reproducible basis for assessing practical utility.

major comments (1)

The validation experiments (described in the abstract and results) report accuracy-versus-evaluation comparisons and visual results across 150 inputs but supply no quantitative uniformity or bias metrics (e.g., star discrepancy, nearest-neighbor distance histograms, or direct comparison against ground-truth uniform samples on a sphere). This is load-bearing for the central claim because the pseudo-random guarantee and the asserted lack of systematic artifacts both rest on the untested assumption that repeated Delaunay edge refinement plus normal-based densification produces an unbiased distribution; without such a check the order-of-magnitude efficiency advantage cannot be fully separated from possible directional or curvature-induced bias.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the scale of our empirical validation. We address the single major comment below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: The validation experiments (described in the abstract and results) report accuracy-versus-evaluation comparisons and visual results across 150 inputs but supply no quantitative uniformity or bias metrics (e.g., star discrepancy, nearest-neighbor distance histograms, or direct comparison against ground-truth uniform samples on a sphere). This is load-bearing for the central claim because the pseudo-random guarantee and the asserted lack of systematic artifacts both rest on the untested assumption that repeated Delaunay edge refinement plus normal-based densification produces an unbiased distribution; without such a check the order-of-magnitude efficiency advantage cannot be fully separated from possible directional or curvature-induced bias.

Authors: We agree that quantitative uniformity metrics would provide stronger, more direct support for the pseudo-random sampling claim. Our current experiments emphasize practical accuracy-versus-evaluation trade-offs and visual inspection over 150 diverse inputs, which show no obvious directional or curvature-induced artifacts. Nevertheless, we acknowledge that explicit checks (star discrepancy, nearest-neighbor histograms, or comparison to ground-truth uniform samples on a sphere) are valuable to separate efficiency gains from any latent bias. In the revised version we will add these metrics on canonical shapes (including a unit sphere) and report the results alongside the existing comparisons. The Delaunay refinement criterion is intended to maintain a locally uniform scaffold, but we accept that this property requires quantitative verification rather than relying solely on design and visuals. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic construction is self-contained

full rationale

The paper describes an algorithmic procedure (Delaunay tetrahedralization scaffold, repeated crossing-edge refinement, intersection sampling, marching tetrahedra, and normal-based densification) that is presented as a novel construction rather than a derivation from equations. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. Claims of improved accuracy/evaluation trade-off rest on empirical comparisons to external baselines across 150 inputs, not on reductions to the method's own inputs by construction. The absence of any quoted equations or uniqueness theorems that collapse to tautology supports a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard properties of Delaunay tetrahedralizations and marching tetrahedra from computational geometry; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption Delaunay tetrahedralization maintains useful geometric properties when vertices are inserted along crossing edges
Invoked as the mechanism that incorporates new points and generates new crossing edges at each refinement step.
standard math Marching tetrahedra produces a valid isosurface mesh from the refined samples
Used to convert the final set of samples into the output mesh.

pith-pipeline@v0.9.0 · 5578 in / 1385 out tokens · 63773 ms · 2026-05-07T12:39:20.260976+00:00 · methodology

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discussion (0)

Reference graph

Works this paper leans on

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2025], marching cubes (MC) [Lorensen and Cline 1987], and occupancy dual contouring (ODC) [Hwang and Sung 2024]

Additional visual comparisons of our ADS method to uniform ray casting [Ling et al . 2025], marching cubes (MC) [Lorensen and Cline 1987], and occupancy dual contouring (ODC) [Hwang and Sung 2024]. We achieve higher accuracy and higher sample count with fewer occupancy function evaluations and less time. Contrary to [Ling et al. 2025], we generate both sa...

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Skateboard and motorcycle [Zhang et al

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Box delineates the OF domain

When the input OF depicts very thin shapes, using our default method and default initial scaffold may fail to capture them even at fine mesh resolution (a,b); this can be mitigated by using a significantly higher number of refinement iterations (c) or a finer initial scaffold (d). Box delineates the OF domain. SIGGRAPH Conference Papers’26, July 19–23, 20...

2026