Efficient Computation of Integer-constrained Cones for Conformal Parameterizations

2025-12-23

AI Summary

This paper addresses the efficient computation of integer-constraint cone singularities in rotationally seamless conformal parameterization. The proposed method integrates optimization with geometric construction. Its key contributions are: (1) an explicit geometric construction algorithm that reduces optimization variables to only slightly more than the number of integer-cone-angle variables; (2) a differentiable formula for updating cone positions, enabling end-to-end optimization; and (3) a suite of adaptive strategies—including cone pairing, dynamic variable selection, and on-the-fly insertion/deletion—to optimally balance cone count and parameterization distortion. Evaluated on large-scale meshes, the method achieves an average 30× speedup over prior approaches while attaining state-of-the-art performance in both cone count and distortion metrics. By significantly improving computational efficiency, robustness, and scalability, the method enhances the practical applicability of seamless conformal mapping for geometry processing and computer graphics applications.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We propose an efficient method to compute a small set of integer-constrained cone singularities, which induce a rotationally seamless conformal parameterization with low distortion. Since the problem only involves discrete variables, i.e., vertex-constrained positions, integer-constrained angles, and the number of cones, we alternately optimize these three types of variables to achieve tractable convergence. Central to high efficiency is an explicit construction algorithm that reduces the optimization problem scale to be slightly greater than the number of integer variables for determining the optimal angles with fixed positions and numbers, even for high-genus surfaces. In addition, we derive a new derivative formula that allows us to move the cones, effectively reducing distortion until convergence. Combined with other strategies, including repositioning and adding cones to decrease distortion, adaptively selecting a constrained number of integer variables for efficient optimization, and pairing cones to reduce the number, we quickly achieve a favorable tradeoff between the number of cones and the parameterization distortion. We demonstrate the effectiveness and practicability of our cones by using them to generate rotationally seamless and low-distortion parameterizations on a massive test data set. Our method demonstrates an order-of-magnitude speedup (30$ imes$ faster on average) compared to state-of-the-art approaches while maintaining comparable cone numbers and parameterization distortion.

Problem

• Efficiently compute integer-constrained cone singularities for conformal parameterizations
• Optimize discrete variables like vertex positions and angles to reduce distortion
• Achieve faster convergence with lower distortion compared to existing methods

Innovation

• Alternately optimizes discrete variables for convergence
• Explicit construction reduces optimization scale for efficiency
• Derivative formula moves cones to reduce distortion