Generalized Balanced Power Diagrams

Updated 3 July 2026

Generalized Balanced Power Diagrams are geometric models that extend classical tessellations by incorporating anisotropic metrics and balancing weights.
They unify Voronoi, Laguerre, and anisotropic diagrams and are applied in polycrystal modeling, image segmentation, and spatial resource allocation.
Optimization techniques like linear programming, gradient descent, and stochastic methods enable precise volume constraints and efficient cell boundary computation.

Generalized Balanced Power Diagrams (GBPDs) constitute a geometric and optimization-based extension of classical tessellation models, unifying Voronoi, Laguerre (power), and anisotropic diagrams under a common framework. GBPDs are widely applied in the modeling of microstructures (e.g., polycrystals), image segmentation, clustering, and spatial resource allocation, where control over partition cell size, anisotropy, and boundary curvature are essential. Key features include flexible metric tensors, additive weights (balancing parameters), nonlinear cell boundaries, and the capacity to enforce or optimize for prescribed cell measures such as volumes, areas, or counts.

1. Mathematical Foundations and Definition

Let $\psi = \{ (s_i, M_i, w_i) : s_i \in \mathbb{R}^d, M_i \succ 0, w_i \in \mathbb{R} \}_{i=1}^n$ denote a finite set of generators, where $s_i$ is the site (center), $M_i$ is a symmetric positive-definite metric tensor, and $w_i$ is the balancing weight. For any $x \in \mathbb{R}^d$ , the generalized (anisotropic power) distance to generator $i$ is defined as

$\operatorname{dist}_G(x; s_i, M_i, w_i) = (x-s_i)^\top M_i (x-s_i) - w_i.$

The $i$ -th GBPD cell is the region

$C_i = \left\{ x \in \mathbb{R}^d : \operatorname{dist}_G(x; s_i, M_i, w_i) \leq \operatorname{dist}_G(x; s_j, M_j, w_j),\ \forall j \right\}.$

Special cases:

$M_i \equiv I,\ w_i \equiv 0$ yields the ordinary Voronoi diagram.
$s_i$ 0 and general $s_i$ 1 gives the Laguerre (power) diagram.
$s_i$ 2 equal and general $s_i$ 3 yields the anisotropic Voronoi diagram (Ballani, 26 Jan 2026, Fiedler et al., 2020, Alpers et al., 2014, Jung et al., 2023).

The boundaries between GBPD cells correspond to quadric surfaces; in 2D, the bisector between two generators is a conic (ellipse, hyperbola, or degenerate line) and in higher dimensions, a quadric hypersurface (Jung et al., 2023, Ballani, 26 Jan 2026). The defining equations guarantee that any uniform shift in weights or common scaling of $s_i$ 4 leaves the tessellation unchanged.

2. Optimization Formulations and Balanced Assignments

A central property of GBPDs is their relation to balanced least-squares assignments. Let $s_i$ 5 be a set of points (e.g., voxels, residents, pixels). Given $s_i$ 6 sites and quota vector $s_i$ 7 (cell sizes or measures), the classic optimization is

$s_i$ 8

Alternatively, for pixel/voxel assignments: $s_i$ 9 are indicator variables, each point assigned to one site, each site receives prescribed measure, and

$M_i$ 0

with constraints $M_i$ 1 (unique assignment), $M_i$ 2 (balancing), $M_i$ 3 (Fiedler et al., 2020, Alpers et al., 18 Jul 2025, Alpers et al., 2014, Cohen-Addad et al., 2017). The Lagrange dual provides the weights $M_i$ 4 enforcing these constraints.

In spatial allocation, such as electoral districting, the GBPD constructs districts whose populations differ by at most one, each defined as the intersection of a polygonal (power cell) region with a geographic domain (Cohen-Addad et al., 2017).

For clustering and outlier detection, the GBPD framework—via LPs with margin slacks—can handle soft assignments and quantify departures from balanced assignments, providing optimality certificates on the number of margin errors and support vectors (Borgwardt, 2013).

3. Analytical and Algorithmic Aspects

Boundary Structure

Between two generators $M_i$ 5, $M_i$ 6, the cell boundary is the locus

$M_i$ 7

which is a conic in 2D (general quadric in higher dimensions) (Jung et al., 2023, Ballani, 26 Jan 2026). In the isotropic case or when $M_i$ 8, this reduces to planar boundaries; otherwise, faces can be curved.

Affine Invariance and Sections

GBPDs are closed under invertible affine maps. For $M_i$ 9, the map is realized as $w_i$ 0, $w_i$ 1, $w_i$ 2 unchanged; other special cases (translation, rotation, scaling) are subsumed (Ballani, 26 Jan 2026). Flat sections (intersections with affine subspaces) of GBPDs yield lower-dimensional GBPDs with modified parameters.

Algorithmic Construction

Discrete GBPDs (over gridded domains) can be constructed by minimizing the assignment or clustering LP, with strong duality ensuring optimal balancing (Fiedler et al., 2020, Alpers et al., 2014, Cohen-Addad et al., 2017). Efficient algorithms include:

Capacitated Lloyd–style iteration: alternating balanced assignment (min-cost flow/LP) and centroid update, guaranteeing monotonic decrease of the k-means cost and finite termination (Cohen-Addad et al., 2017).
Improved rendering algorithm: two-stage assignment over large grids achieves $w_i$ 3 complexity under stochastic generator configurations (Ballani, 26 Jan 2026).
Analytical construction of 2D diagrams: full vertex-edge incidence and topology via explicit conic intersection routines (Jung et al., 2023).

4. Fitting GBPDs to Data and Methodological Trade-Offs

GBPDs are systematically fitted to high-dimensional structural data (e.g., segmented 3D images of polycrystals) by optimizing generator parameters to maximize agreement with ground-truth grain partitions. Canonical methods include:

Linear Programming: fixing $w_i$ 4 (e.g., by PCA or first and second moments) and solving for $w_i$ 5 subject to exact or bounded volume constraints, yielding globally optimal assignments and integral solutions when the LP matrix is totally unimodular (Alpers et al., 18 Jul 2025, Alpers et al., 2014).
Gradient Descent (Softmax Relaxation): simultaneously optimizing all generator parameters over smooth surrogate losses; efficient with GPU acceleration, sensitive to initialization (Alpers et al., 18 Jul 2025).
Stochastic Optimization (Cross-Entropy): treating generator parameters as random variables and minimizing interface-based discrepancy; robust to local minima but computationally costly (Alpers et al., 18 Jul 2025).
Derivative-Free Local Optimization: e.g., Subplex, Praxis methods directly fitting generator parameters by minimizing point-to-boundary discrepancies (Alpers et al., 18 Jul 2025).

The choice of method depends on balancing model and algorithmic complexity, convergence guarantees, fidelity to volume/area/topology, and computational resources.

A consolidated table of main fitting methods (Alpers et al., 18 Jul 2025):

Method	Variables	Optimality	Constraints/Focus
LP (weighted assign.)	$w_i$ 6 (weights)	Global	Exact/bounded volumes
GD (softmax)	$w_i$ 7	Local	Volume or shape loss
Cross-entropy (CE)	$w_i$ 8 or all	Heuristic	Interface placement
Neper/DFO	$w_i$ 9	Local	Boundary agreement

5. Analytical Representation and Cell Topology

Recent advances include analytic representations of 2D GBPDs, allowing for exact computation of cell boundaries, vertices, and adjacency relations without pixelization (Jung et al., 2023). The bisector between two sites is a general conic $x \in \mathbb{R}^d$ 0, classified by the inertia of the associated matrix. Vertices are obtained as solutions to pairs of such quadratic equations; edge arcs are assembled by sorting intersection parameters and validating local cell inclusions. This framework enables area and perimeter evaluation via closed-form line integrals, accurate topology analysis, and robust handling of degenerate cases such as lenses and unique-neighbor cells.

6. Applications and Empirical Performance

Polycrystal Modeling

GBPDs efficiently reconstruct polycrystalline grain maps from minimal statistics (center-of-mass, volume, second moments), achieving high accuracy in voxel labeling and topological neighborhood reconstruction, even under anisotropic conditions. In practical tests, GBPDs consistently outperform heuristic Laguerre tessellations and power diagrams on labeling rates and neighborhood accuracy, with only a handful of parameters per grain needed (Alpers et al., 2014, Alpers et al., 18 Jul 2025).

Superpixel Image Segmentation

Power-SLIC applies GBPDs to superpixel segmentation, generating piecewise-quadratic cell boundaries with exact area balancing, competitive boundary recall, undersegmentation error, and robustness to noise. Continuous diagram representations ensure resolution-independence and efficient compression (Fiedler et al., 2020).

Redistricting and Spatial Allocation

GBPD-based algorithms ensure perfectly balanced, compact, and contiguous partitioning of geographic domains, with cells given by truncations of convex polygons (or more general quadric cells) to the domain boundary. Iterative capacitated Lloyd–style algorithms provide local minima with provable compactness and side count bounds (Cohen-Addad et al., 2017).

Clustering and Outlier Detection

GBPD formulations enable efficient linear programming algorithms for clustering with balanced assignments, margin maximization, and explicit control over misclassification and support vector counts, facilitating robust outlier detection (Borgwardt, 2013).

7. Theoretical Properties and Invariance

GBPDs inherit several crucial invariance properties:

Affine Covariance: Closed under invertible affine maps; directions and anisotropies transform appropriately, preserving diagram topology (Ballani, 26 Jan 2026).
Sectional Stability: Slices through GBPDs yield lower-dimensional GBPDs, enabling detailed study of microstructure cross-sections or embedding problems (Ballani, 26 Jan 2026).
Parameter Redundancy: Uniform weight shifts and common scaling of $x \in \mathbb{R}^d$ 1 leave the diagram unchanged, implying redundancy in parameter specification (Ballani, 26 Jan 2026).
Compactness and Cell Topology: The set-theoretic and geometrical structure of GBPDs restricts the average number of sides (in planar domains) and maintains centroidality between generator positions and resulting cell mass centers when optimization is used (Cohen-Addad et al., 2017, Alpers et al., 2014).

Robust theoretical guarantees exist in special cases (e.g., volume-matching LPs, support vector counts in soft-margins), while in general, practical optimization is subject to complexity and local minima considerations (Alpers et al., 18 Jul 2025, Borgwardt, 2013).

References:

(Cohen-Addad et al., 2017, Fiedler et al., 2020, Alpers et al., 2014, Borgwardt, 2013, Ballani, 26 Jan 2026, Alpers et al., 18 Jul 2025, Jung et al., 2023)