Bounded Power Diagrams

• Bounded power diagrams are convex partitions defined by weighted generators that ensure all cells lie within a specified finite domain.
• They extend Voronoi diagrams by incorporating scalar weights to handle capacity constraints in optimal transport, adaptive binning, and neural rendering.
• Efficient algorithms like PowerBin and GPU-based pipelines yield near-linear scaling, enabling robust performance over large-scale, heterogeneous datasets.

Bounded power diagrams are convex polyhedral or polygonal partitions of a bounded subset of $\mathbb{R}^d$ into regions defined by both positional generators and associated scalar weights. These structures generalize Voronoi diagrams by allowing weighted generators, and the boundedness guarantees that all constructed cells lie within a prescribed finite domain. Bounded power diagrams offer robust solutions to capacity-constrained partitioning tasks, combining geometric optimality with computational scalability in large and heterogeneous datasets. They have critical relevance in optimal transport, adaptive data binning, mesh-based neural rendering, and related fields (Cappellari, 8 Sep 2025, Taveira et al., 7 May 2026).

1. Formal and Mathematical Framework

Let $X \subset \mathbb{R}^d$ be a bounded domain, with sites (generators) $g_j \in \mathbb{R}^d$ and scalar weights $w_j \in \mathbb{R}$. The power (Laguerre) distance of a point $x$ to generator $g_j$ with weight $w_j$ is

$d_w(x, g_j) = \|x - g_j\|^2 - w_j.$

The $j$th power cell—subject to the bounded domain constraint—is given by

$$V_j = \{ x \in X \mid d_w(x, g_j) \leq d_w(x, g_k),\ \forall k \neq j \}.$$

Each $X \subset \mathbb{R}^d$0 is a convex polyhedron (in 3D) or polygon (in 2D), as all pairwise cell bisectors are linear in $X \subset \mathbb{R}^d$1. For bounded power diagrams, every cell $X \subset \mathbb{R}^d$2 is further restricted to remain within a global bounding region, typically enforced by initializing each cell as the global axis-aligned bounding box (AABB) of $X \subset \mathbb{R}^d$3 and iteratively clipping it with half-spaces defined by power bisectors (Cappellari, 8 Sep 2025, Taveira et al., 7 May 2026).

The measure or capacity of a cell is given, in the continuous case, by $X \subset \mathbb{R}^d$4 for density $X \subset \mathbb{R}^d$5 (often discrete in implementations). In applications such as optimal transport, the partition seeks prescribed capacities $X \subset \mathbb{R}^d$6 per cell.

2. Optimal Transport and Capacity Constraints

Bounded power diagrams appear as solutions to the semi-discrete Monge optimal transport problem with quadratic cost, under equal-mass constraints per cell. The primal functional is

$X \subset \mathbb{R}^d$7

subject to $X \subset \mathbb{R}^d$8 for all $X \subset \mathbb{R}^d$9. Introducing Lagrangian multipliers $g_j \in \mathbb{R}^d$0 leads to a dual functional

$g_j \in \mathbb{R}^d$1

where $g_j \in \mathbb{R}^d$2 is the power cell under weights $g_j \in \mathbb{R}^d$3. The stationarity conditions

$g_j \in \mathbb{R}^d$4

require each generator coincide with its cell centroid $g_j \in \mathbb{R}^d$5 and each cell meet its target capacity. Such a configuration is a Centroidal Power Diagram (CPD) and optimal for the transport cost (Cappellari, 8 Sep 2025).

3. Algorithms for Bounded Power Diagram Construction

Scalable construction of bounded power diagrams requires efficient geometric and combinatorial operations. In 2D, the PowerBin algorithm employs a two-stage process:

• Bin-accretion initialization: Fast $g_j \in \mathbb{R}^d$6 seed-and-grow procedure based on Delaunay adjacency and heap-based priority selection to cluster points into compact, near-convex regions, suitable for generator initialization.
• Iterative regularization: Employs heuristic weight updates inspired by the physics of packed soap bubbles. The algorithm treats $g_j \in \mathbb{R}^d$7 as the squared radius $g_j \in \mathbb{R}^d$8, with the goal that each convex cell area matches the prescribed capacity via

$g_j \in \mathbb{R}^d$9

where $w_j \in \mathbb{R}$0 is the current area and $w_j \in \mathbb{R}$1 its estimated mass. Generator positions are updated to the geometric centroids at each step. Under-relaxation and clamping strategies ensure stability and convergence (Cappellari, 8 Sep 2025).

In 3D, highly parallel approaches are critical:

• GPU-parallel clipping pipeline: Each convex power cell is constructed by progressively clipping an initial AABB against bisector half-spaces induced by neighboring generators.
• Hierarchical best-first BVH traversal: A bounding volume hierarchy (BVH) over generator sites, augmented with per-node maximum weight, coordinates fast culling and neighbor identification. Directional geometric bounds from AABB corners are leveraged to efficiently reduce candidate neighbor sets.
• CUDA-specific optimizations: Coalesced data loads, global-memory cell buffers, fusion of neighbor search and clipping steps, and per-thread buffer management are exploited for speed and memory efficiency (Taveira et al., 7 May 2026).

4. Computational Complexity and Performance

PowerBin's 2D pipeline achieves $w_j \in \mathbb{R}$2 complexity in both initialization and regularization stages, a two-orders-of-magnitude speedup compared to classic VorBin and MWV-based methods, which empirically scale as $w_j \in \mathbb{R}$3. On million-pixel astronomical datasets, PowerBin reduces typical runtimes from hours to minutes and produces more uniform and convex bins, with S/N RMS scatter reduced from 7.3% (VorBin WVT) to 6.5% (Cappellari, 8 Sep 2025).

In 3D, GPU-parallel bounded power diagram construction achieves near-linear scaling up to 20 million sites. Quantitative results indicate that, for 2–4.2 million point scenes, the method delivers $w_j \in \mathbb{R}$4–$w_j \in \mathbb{R}$5 speedup over gDel3D and substantially outperforms Radiant Foam’s Delaunay-based pipeline (up to $w_j \in \mathbb{R}$6 faster at scale). The performance is robust to heterogeneity in point distributions and generalizes to arbitrary weights with no additional computational or memory overhead (Taveira et al., 7 May 2026).

5. Applications and Robustness

Bounded power diagrams support a range of domain-specific objectives:

• Adaptive binning for astronomy: By formulating binning as an optimal transport partition, bounded power diagrams ensure S/N uniformity and convexity of bins in spectral imaging, with practical robustness to Poisson and correlated noise as well as background-limited scenarios (Cappellari, 8 Sep 2025).
• Mesh-based neural rendering: In large-scale 3D settings, power diagrams underpin data structures for neural scene representations, stippling, and point-based rendering with computational performance enabling real-time or near real-time pipelines (Taveira et al., 7 May 2026).
• Capacity-constrained tessellations: The formulation accommodates arbitrary non-additive cell objectives (e.g., S/N with pixel covariance) and general black-box measures, enhancing applicability in heterogeneous scientific datasets.

Empirical tests demonstrate that bounded power diagrams reliably maintain cell convexity, uniform capacity targets, and resilience against cell fragmentation or non-convexity otherwise common in additively-weighted Voronoi schemes.

6. Limitations and Future Directions

Main limitations include the lack of formal global optimality guarantees when employing non-gradients-based, heuristic regularization loops, such as the soap-bubble update. In cases of highly irregular or elongated initial binning, more iterations or stronger under-relaxation may be required. Formal gradient-based CPD solvers become unstable when dealing with non-additive cell capacities. For extremely sparse or discontinuous signals, pre-selection of regions of interest may be necessary to avoid grouping pure noise pixels into large bins. GPU pipelines for 3D power diagrams exhibit a graceful degradation in performance when the generator weight range spans orders of magnitude, with an increased empty-cell fraction in extreme cases (Cappellari, 8 Sep 2025, Taveira et al., 7 May 2026).

7. Implementation and Reference Algorithms

A public Python implementation of the 2D PowerBin algorithm for capacity-constrained bounded power diagrams is provided by M. Cappellari (Cappellari, 8 Sep 2025). The full code and usage examples are available at https://pypi.org/project/powerbin/. The GPU-native 3D pipeline developed in (Taveira et al., 7 May 2026) leverages cuBQL for BVH construction and demonstrates state-of-the-art performance, with practical integration into neural rendering research at Zenseact.

Algorithm/Method	Complexity	Dimensionality	Unique Features
PowerBin (CPU, 2D)	$w_j \in \mathbb{R}$7	2D	Soap-bubble, bin accretion
BVH-GPU (3D)	$w_j \in \mathbb{R}$8	3D	Parallel clipping, BVH

The table above summarizes key computational characteristics and features of current scalable bounded power diagram algorithms appearing in recent literature.

Bounded power diagrams, as formalized and accelerated in recent work, provide foundations for convex partitioning in scientific, geometric, and computational applications where robustness, scalability, and adherence to prescribed region capacities are essential.