Geometric Adaptive Sampling (GAS)

• Geometric Adaptive Sampling (GAS) is a framework for generating maximal Poisson-disk sets by ensuring every point in a domain is within a prescribed radius using geometric gap analysis.
• It employs fast geometric data structures, power diagrams, and regular triangulations to detect, cluster, and fill gaps, achieving blue-noise sampling and optimal mesh quality.
• GAS guarantees coverage and mesh quality through iterative refinement and dynamic maintenance, making it scalable for both Euclidean and manifold sampling applications.

Geometric Adaptive Sampling (GAS) is a mathematically rigorous framework for generating maximal Poisson-disk sets with non-uniform radii in Euclidean domains and on manifolds. GAS leverages geometric structures—especially power diagrams and regular triangulations—to detect and fill uncovered regions ("gaps") so that every point in the domain lies within a prescribed radius of some sample. Its algorithmic pipeline combines fast geometric data structures, exact gap analysis, and iterative refinement, providing guarantees for coverage and mesh quality that underpin advanced applications in blue-noise sampling, mesh generation, and remeshing. This entry provides a comprehensive technical account of GAS, focusing on the underlying geometric theory, algorithmic structures, extensions to manifolds, implementations, and practical performance (Yan et al., 2012).

1. Geometric Theory of Gaps

A Poisson-disk sample in $\Omega \subset \mathbb{R}^d$ is defined as a set of disks $\{(c_i, r_i)\}$ with centers $c_i \in \Omega$ and radii $r_i>0$ such that no sample center lies within the disk of any other: $\forall i \ne j$, $\|c_i - c_j\| \geq \max(r_i, r_j)$. The union of all disks is $$\mathcal{D} = \bigcup_i \{x \in \Omega : \|x - c_i\| \leq r_i\}$$, and the set of uncovered ("gap") regions is $G = \Omega \setminus \mathcal{D}$.

GAS achieves maximality—meaning no disk can be added without violating the separation constraint—by geometrically characterizing gaps and iteratively eliminating them. Each disk $d_i$ is represented as a weighted point $(c_i, w_i = r_i^2)$. The power distance between $x$ and $d_i$ is $\pi(x, d_i) = \|x - c_i\|^2 - r_i^2$. The domain is partitioned using the power diagram, with each cell

$$\Omega_i = \{ x \in \Omega : \pi(x, d_i) \leq \pi(x, d_j),\ \forall j \ne i \},$$

and its dual, the regular triangulation $\mathrm{RT}$, of the weighted set $\{(c_i, w_i)\}$. Each $d$-simplex $t \in \mathrm{RT}$ admits a unique power-center $m_t$ where the power distance to all associated disks is equal: $$\pi(m_t, d_{i_0}) = \cdots = \pi(m_t, d_{i_d}) = \Pi(t)$$. The core criterion for gap existence is:

Gap-Existence Theorem: $G$ is nonempty iff $\exists\ t\in\mathrm{RT}$ such that $\Pi(t) > 0$.

Thus, checking for sampling maximality reduces to testing all simplices in $\mathrm{RT}$: the sampling is maximal iff $\forall t \in \mathrm{RT}$, $\Pi(t) \leq 0$.

On triangulated surfaces ($\Omega \subset \mathbb{R}^3$), one uses the restricted power diagram (RPD) and its dual restricted regular triangulation (RRT). The local geometric criterion for gaps and maximality holds analogously.

2. Algorithms and Data Structures for Dynamic Gap Maintenance

GAS maintains a dynamic regular triangulation to enable rapid detection and update of gaps as disks are inserted, deleted, moved, or their radii are changed. This is crucial for efficient maximal sampling and adaptive refinement.

The regular triangulation is stored (availability via packages such as CGAL), where each simplex $t$ keeps:

• pointers to its $d+1$ vertex disks;
• its power-center $m_t$ and power-value $\Pi(t)$.

Primitive operations for dynamic maintenance include:

• Insertion: Add $(c, r)$ as a weighted point, update affected simplices, compute new $\Pi(t)$, and update the set of gap-simplices $\Delta = \{ t \mid \Pi(t) > 0 \}$. Amortized $\mathcal{O}(\log n)$.
• Deletion: Remove weighted point, update stars of affected simplices, recompute $\Pi(t)$, update $\Delta$. Complexity $\mathcal{O}(k)$.
• Movement/Radius Change: For movement, use delete+insert; for radius update, locally update weights and flip edges as necessary, recompute $\Pi(t)$ in affected region.
• Batch Gap Processing: Once $\Delta$ is maintained, operations such as gap detection, gap clustering (by simplex connectivity), and primitive extraction (covering the local uncovered region in each gap-simplex) are $\mathcal{O}(|\Delta|)$.

These mechanisms permit efficient, scalable maintenance and querying of sampling gaps, which is central to GAS.

3. GAS Sampling and Remeshing Pipeline

The GAS methodology implements a two-phase sampling process for both Euclidean and surface domains:

1. Initial Sampling: Use grid-accelerated dart-throwing to populate the domain with non-overlapping disks up to near-maximality, rejecting darts whose centers are inside any existing disk.
2. Gap Filling:
   • Detect all gap-simplices ($\Delta$).
   • Cluster gaps into independent gap-sets (IGS), each representing a region that can be filled in parallel.
   • Within each IGS, extract convex gap-primitives, union them to cover the IGS, construct area-weighted sampling distributions, and draw new disks within uncovered regions.
   • Repeat until no gaps remain (in practice, typically 4–5 iterations suffice for full maximality).

Extension to Manifolds: On triangulated surfaces, GAP uses the restricted power diagram (RPD) and RRT. Each sample’s surface-polygon is obtained by clipping its RPD cell with its associated sphere; the MUSP operations (detect, cluster, extract, fill) are adapted accordingly, yielding maximal coverage on manifolds.

Surface Remeshing: The GAS sampling provides seeds for remeshing:

• Prescribed radii encode density function $\rho(x)$, yielding $r(x) \propto \rho(x)^{-1/2}$ sampling.
• Dual triangulation of the RPD is extracted to form initial mesh.
• Optional randomized local optimization enforces angular and valence bounds: e.g., angle-bounds $[\theta_{\min},\theta_{\max}]$ or discrete valence sets $\{5,6,7\}$.
• Edge-length optimization may be applied.

4. Performance and Empirical Results

GAS exhibits high efficiency and superior mesh regularity, as demonstrated by extensive experiments:

Application	Key Result Metrics	Timings
2D Uniform/Adaptive	1M samples in unit square: full maximality in ≈2s; gap-filling converges in 4–5 iterations	≈2s (sampling)
	Min angle $\theta_{\min} \geq 30^\circ$, valence ≈98% at 6, blue-noise spectrum confirmed
Surface Remeshing	Bunny (30k verts): $\theta_{\min} \geq 30^\circ,Q(t) \geq 0.3$; $>95\%$ vertices valence in {5,6,7}	Initial: 12s, gap: 6s, opt.: 5s
	Adaptive density: angle bounds $[32^\circ, 115^\circ]$, Hausdorff error ≈0.5% of bbox	23s total

In comparative experiments, GAS produces blue-noise, maximal samples and meshes with tight angle and valence guarantees at markedly lower computational cost than centroidal Voronoi tessellation (CVT) or capacity-constrained (CAP) approaches, which require 180–400s for equivalent quality (Yan et al., 2012).

5. Properties, Guarantees, and Quality Criteria

GAS delivers provable coverage and mesh-quality properties:

• Maximality is certified via the local power-center gap test ($\forall t \in \mathrm{RT},\ \Pi(t) \leq 0$).
• Angle and Valence Optimization: Uniform disk case yields $\theta_{\min} \geq 30^\circ$, $\theta_{\max} \leq 120^\circ$; local randomized optimization can drive adaptive angle bounds toward these uniform values. Surface remeshing outputs $>95\%$ of vertices with valence $\{5,6,7\}$.
• Blue-Noise Spectrum: Spectral analysis confirms GAS-sampled sets exhibit blue-noise properties, advantageous for reducing simulation bias in applications such as SPH and finite element analysis.
• Scalability and Efficiency: Primitive gap operations require only local updates ($\mathcal{O}(1)$ per simplex), leading to scaling to millions of samples in seconds.

A plausible implication is that GAS enables high-quality, adaptive sampling and remeshing pipelines in computational geometry, computer graphics, and simulation, where both spatial regularity and sampling maximality are critical.

6. Context, Extensions, and Significance

GAS builds upon and generalizes classical Poisson-disk sampling by incorporating exact gap detection and strictly enforcing maximality under varying radii. The geometric framework—grounded in power diagrams and regular triangulations—supports dynamic, fully adaptive operations, and admits extensibility from Euclidean to curved manifold domains.

Key algorithmic ingredients include:

• Unified geometric gap existence characterization,
• Dynamic maintenance of regular triangulations,
• Hierarchical gap-filling,
• Local randomized optimization for mesh metrics.

Applications of GAS span blue-noise sampling, surface remeshing, meshing for simulation, and any domain where maximal disk cover and controllable sample densities are central. Its guarantees and empirical results position GAS as a robust foundation for further developments in geometric sampling, coverage theory, and mesh optimization (Yan et al., 2012).