triangulation methods

This paper addresses the high computational complexity of two-view triangulation in multiview geometry. We propose a reweighted projection error optimization framework that theoretically derives an optimal weighting strategy, reducing the conventional sextic polynomial minimization to a quadratic form amenable to closed-form analytical solution. To our knowledge, this is the first application of reweighted least squares to triangulation, achieving a favorable trade-off between efficiency and accuracy: geometric consistency is preserved while computation speed is significantly improved. We derive a theoretical upper bound on the approximation error, guaranteeing solution reliability. Extensive experiments on real-world datasets demonstrate that the proposed method achieves accuracy comparable to globally optimal solutions, with rigorously provable error control.

This work introduces, for the first time, a prediction-augmented algorithmic framework to computational geometry, specifically targeting the efficient computation of two-dimensional Delaunay triangulations (DT). Given a point set \( P \) and a predicted triangulation \( G \) that approximates the true DT, the paper proposes an adaptive correction algorithm based on an edge-difference metric \( D \), a violation measure \( d_{\text{vio}} \), and a randomized sampling probability \( \rho \). The main contributions include a deterministic algorithm running in \( O(n + D \log^3 n) \) time and an optimal randomized algorithm with expected time \( O(n + D \log n) \). Under a stochastic model combining edge inclusion and violation degrees, the approach further yields an almost-linear-time solution, which is also extended to related problems such as the Euclidean minimum spanning tree.

This paper addresses the CG:SHOP 2025 challenge: computing constrained non-obtuse triangulations (all interior angles ≤ 90°) of planar domains, subject to mandatory inclusion of specified vertices and edges, while minimizing the number of Steiner points. We propose a local search framework built upon dynamically constrained Delaunay triangulation, integrating geometric conflict detection, adaptive Steiner point relocation, and a synergistic optimization strategy—simultaneously considering removal, relocation, and insertion of Steiner points. To our knowledge, this is the first approach achieving a Pareto balance between solution size and quality for this problem. Our method guarantees 100% compliance with the non-obtuse angle constraint. On diverse, complex benchmark instances, it reduces the average number of Steiner points by 37% compared to prior methods—marking a substantial improvement. The approach secured first place in the CG:SHOP 2025 competition.

Craig and uniform interpolation lack theoretical guarantees and are computationally intractable in description logics and logic programming. Method: This paper systematically characterizes the existence boundaries of interpolation for prominent formalisms—including ALC, EL, and Answer Set Programming—by integrating model-theoretic and proof-theoretic criteria; it proposes a theoretically complete, polynomial-time interpolant construction framework. Contribution/Results: We establish the first interpolation property hierarchy across multiple sublogics, develop an extensible interpolant generator, and empirically validate its efficiency and practicality on standard ontologies and rule sets. The approach significantly advances key knowledge engineering tasks, including knowledge forgetting, modular reuse, and explainable reasoning.

Existing explicit 3D triangular mesh representations lack simultaneous topological flexibility and full differentiability. Method: We propose DMesh, a differentiable triangular mesh representation that jointly models geometry and connectivity. DMesh generates candidate faces via weighted Delaunay tetrahedralization and models face existence probabilistically through a differentiable parametric model, enabling arbitrary topology. Contribution/Results: DMesh enables the first end-to-end differentiable explicit mesh reconstruction, overcoming the dual limitations of traditional explicit methods (fixed topology) and implicit methods (non-differentiable or topologically ambiguous). Evaluated on point cloud and multi-view image inputs, DMesh achieves topology-adaptive, high-fidelity, and optimization-stable mesh reconstruction, significantly improving geometric accuracy and training robustness compared to prior approaches.

This study addresses the limitations of traditional two-dimensional nonparametric regression methods, which often ignore boundary geometry on non-rectangular domains and struggle to accommodate spatially heterogeneous smoothness. The authors propose a novel approach that constructs locally adaptive splines based on constrained Delaunay triangulation and introduces a carefully designed adaptive prior within a Bayesian framework. This method achieves optimal spatial adaptivity to unknown local smoothness over irregular domains while strictly respecting domain boundaries. It is applicable to any triangulation satisfying mild shape-regularity conditions. Theoretical analysis establishes that the posterior contraction rate attains the oracle-optimal rate, and simulations demonstrate superior estimation accuracy and lower model complexity compared to existing methods.

This study resolves the long-standing open question of whether six specific normal extensions of the modal logic S4 possess the Craig interpolation property. Building on Smoryński’s approach, the authors introduce Fine’s frame formulas for splitting clusters into interpolation analysis for the first time, integrating modal semantics, model-theoretic techniques, and classical interpolation proof strategies to develop a novel methodological framework. This work provides a complete solution to Maksimova’s classification problem concerning interpolation in S4 extensions, establishing that all six previously unresolved cases indeed enjoy the Craig interpolation property. The results not only settle a decades-old conjecture but also open new avenues for research in interpolation within modal logic.

This work proposes a template-driven local remeshing framework to address the limitations of conventional constrained Delaunay triangulation in the stable discretization of PDEs over image-derived domains—namely, global update requirements, poor parallelizability, and insufficient geometric fidelity. The method selectively retriangulates only those elements intersecting domain boundaries while preserving the underlying mesh structure. By leveraging discrete equivalence and symmetry classification, it constructs a finite symbolic lookup table that enables deterministic, conflict-free, and scalable template mapping. The resulting meshes are conforming, angle-bounded, compatible with standard finite element methods, and amenable to synchronization-free parallel execution. Experiments demonstrate significant reductions in sliver elements, improved triangle regularity, and enhanced geometric fidelity near complex boundaries, making the approach well-suited for real-time geometric analysis and physics-based simulation.

This study addresses the inferential bias arising from misspecification of a single causal model under model uncertainty. The authors propose a weighted triangulation framework that integrates identification functionals from multiple candidate causal models through a data-driven measure of model validity, enabling robust causal effect estimation without explicit model selection. This approach uniquely bridges testability in causal discovery with semiparametric inference, formalizing robustness under causal pluralism without requiring consensus among models or reliance on any single specification. Theoretically, the proposed functional is shown to converge to the true causal effect with high probability. Both simulation studies and empirical analyses demonstrate the method’s robustness and effectiveness.

This work addresses the ambiguity in simplex diameter shrinkage during iterative refinement of spherical Delaunay complexes, which arises from their non-nested structure. To resolve this issue, the authors introduce several Steiner point insertion strategies that recast the shrinkage analysis as a covering estimation problem for Euclidean simplices. By proposing a sharp, dimension-dependent variant of the approximate Carathéodory theorem and integrating a key sampling technique, they establish, for the first time, an explicit bound on diameter shrinkage in Delaunay refinement. Both theoretical analysis and numerical experiments demonstrate that the proposed approach achieves stronger and quantifiable simplex shrinkage compared to conventional subdivision strategies.