GeloVec Module for Adaptive Sampling

• GeloVec Module is an adaptive sampling framework that leverages geometric gap detection via power diagrams and regular triangulation.
• It employs dynamic regular triangulation updates and efficient gap-filling routines to ensure provably maximal Poisson-disk sets with blue-noise distribution.
• The framework supports advanced applications in mesh generation and remeshing by ensuring high-quality sample distributions with robust performance guarantees.

Geometric Adaptive Sampling (GAS) refers to a mathematically principled framework for achieving maximal Poisson-disk sets under spatially varying radii, subject to geometric constraints, with particular emphasis on the efficient detection and resolution of "gaps"—uncovered regions—during the dynamic evolution of the sample set. GAS provides a unified geometric criterion for the characterization of coverage completeness, leveraging structures such as the power diagram and regular triangulation to detect, maintain, and fill gaps arising in both Euclidean domains and manifold surfaces. Fundamentally, GAS enables provably maximal, blue-noise sample distributions suitable for demanding applications in mesh generation and numerical simulation (Yan et al., 2012).

1. Geometric Foundations: Gaps, Power Diagram, and Regular Triangulation

GAS is grounded in the analysis of gaps in a set of disks $d_i = (c_i, r_i)$, where $c_i$ denotes disk centers and $r_i$ their radii in a compact domain $\Omega \subset \mathbb R^d$. No disk center lies inside another disk: $\forall i \ne j,\; \|c_i-c_j\| \geq \max(r_i, r_j)$. The union $$\mathcal D = \cup_i \{x \in \Omega : \|x-c_i\| \leq r_i\}$$, defines the covered region, and the set of gaps is $G = \Omega \setminus \mathcal D$.

Each disk $d_i$ is viewed as a weighted point $w_i = (c_i, w_i)$; weights are $w_i = r_i^2$. The power distance from $x$ to $d_i$ is $\pi(x, d_i) := \|x-c_i\|^2 - r_i^2$, leading to the partition of $\Omega$ into convex sets $$\Omega_i = \{x \in \Omega : \pi(x, d_i) \leq \pi(x, d_j),\; \forall j\}$$, which form the power diagram. The dual structure, the regular triangulation (RT), consists of $d$-simplices, each with a unique power-center $m_t$ satisfying $\pi(m_t, d_{i_0}) = ... = \pi(m_t, d_{i_d})$. The value $\Pi(t) = \pi(m_t, d_{i_0})$ identifies whether simplex $t$ contributes a gap.

The key result, termed the Gap-Existence Theorem, states that $G$ is nonempty if and only if $\exists t \in RT : \Pi(t) > 0$. In other words, a gap arises precisely when the power-center of a simplex lies outside all its disks. On triangulated surfaces embedded in $\mathbb R^3$, the restricted power diagram and restricted regular triangulation provide analogous definitions, enabling gap analysis under geometric constraints and facilitating adaptive sampling on manifolds (Yan et al., 2012).

2. Dynamic Data Structures and Gap Processing Algorithms

GAS relies on the dynamic maintenance of the regular triangulation (RT) to efficiently track coverage changes due to disk insertion, deletion, movement, or radius modification. RTs store explicit pointers to disk centers/radii and the corresponding power-center and gap-value for each simplex.

The atomic update operations are as follows:

• Insertion: Adding a disk requires inserting its weighted point into RT (average $O(\log n)$) and re-evaluating all new simplices; those with $\Pi(t) > 0$ are marked as gap contributors.
• Deletion: Removing a disk involves updating all simplices in its local star, recomputing their gap status ($O(k)$ per Devillers ’06).
• Movement/Radius Change: Disk movement is handled as sequential delete and insert; local radius changes propagate via edge flips until regularity is restored, incurring local $O(k)$ complexity.

Higher-level operations include explicit gap detection (by scanning RT), gap clustering (to identify independent gap components), and primitive extraction (locally carving out convex polygons that exactly define uncovered regions for subsequent filling). Primitive extraction is $O(1)$ per simplex; detection and clustering are $O(|RT|)$ and $O(|\Delta|)$ respectively, where $\Delta$ is the set of gap simplices (Yan et al., 2012).

3. Adaptive Sampling Protocols: Euclidean and Manifold Domains

GAS employs a two-stage adaptive sampling workflow:

1. Grid-Accelerated Dart Throwing: Rapidly seeds disks to achieve near-maximal coverage, with rejection of samples violating the pairwise center-distance constraint.
2. Gap-Filling: Detects gap-simplices, clusters them into independent gap-sets, and fills each set by sampling uniformly (or under local density) within convex gap primitives until all gaps are eliminated. For adaptive Poisson-disk sampling, disk radii may vary as $r = c \rho(x)^{-1/2}$.

For manifold sampling, the restricted power diagram and regular triangulation are localized to the mesh surface. Gap primitives become surface polygons arising from the intersection of spheres and mesh triangles; gap detection, clustering, and primitive extraction proceed analogously, yielding robust surface sampling suitable for remeshing algorithms (Yan et al., 2012).

4. Applications in Mesh Generation and Remeshing

GAS is directly applicable to surface remeshing and quality blue-noise sample generation. The algorithmic pipeline includes:

• Sampling seeds with prescribed density and radii within bounds.
• Topological correctness via $\epsilon$-sampling or local post-checks.
• Extraction of the dual triangulation of the (restricted) power diagram.
• Optional randomized local optimization enforcing constraints such as angle bounds ($\theta_{\min}, \theta_{\max}$), valence (number of incident edges per vertex), and edge lengths.

Empirical results demonstrate:

• For $1$M samples in a unit square: maximality reached in $\approx 2$ s, gap-filling converges in $4$–$5$ iterations vs $>10$ for previous methods.
• Angle bounds after maximal insertion: $\theta_{\min} \geq 30^\circ$, $\theta_{\max} \leq 120^\circ$. Adaptive cases achieve $\theta_{\min} \geq 35^\circ$, $\theta_{\max} \leq 105^\circ$.
• Valence distribution: $>95\%$ of vertices have degree $6$ (periodic), $1\%$ degree $5$ or $7$.
• Surface remeshing (Bunny, 30k verts): Initial sampling $\approx 12$ s, gap-filling $\approx 6$ s, optimization $\approx 5$ s ($23$ s total), compared to $180$–$400$ s for comparable CVT/CAP methods (Yan et al., 2012).

Blue-noise sampling reduces bias in particle simulations and finite element methods; adaptive angle and valence optimization yield meshes with high-quality statistics and low geometric error.

5. Coverage Guarantees and Quality Properties

A sampling is maximal if and only if all $\Pi(t) \leq 0$ across RT. This equivalence provides an efficient maximality test: scan simplices and evaluate gap-values. Uniform maximality guarantees local angle bounds, e.g., $\theta_{\min} \geq 30^\circ$, and provable blue-noise spectral properties.

Local randomized optimization further improves adaptive meshes toward uniform guarantees, extending to constraints relevant in computational simulation, mesh generation, and geometric modeling.

6. Summary and Key Features

GAS integrates geometric principles and algorithmic efficiency through:

• Geometric gap characterization via power-diagrams and regular triangulations.
• Dynamic RT maintenance for disk modification operations.
• Efficient gap detect–cluster–extract–fill routines in $O(1)$ per affected simplex.
• Two-stage sampling for provably maximal Poisson-disk sets in both Euclidean and manifold domains.
• Optional randomized local optimization for mesh quality enforcement.

The framework leads to fast, high-quality sample sets and remeshing outputs with predictable coverage and mesh-quality properties, supported by rigorous geometric analysis and efficient algorithms (Yan et al., 2012).