Efficient Computation of Integer-constrained Cones for Conformal Parameterizations (2512.20904v1)

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$\times$ faster on average) compared to state-of-the-art approaches while maintaining comparable cone numbers and parameterization distortion.

• The paper presents an alternate optimization approach that reduces MIQP variables by focusing only on cone singularity angles and vertex positions.
• It achieves an order-of-magnitude computational speedup with comparable or fewer cone singularities while maintaining low parameterization distortion.
• Adaptive cone insertion and removal, guided by distortion gradients and branch-driven strategies, ensure robustness for both genus-zero and high-genus surfaces.

Efficient Computation of Integer-constrained Cones for Conformal Parameterizations

Introduction and Motivation

The paper "Efficient Computation of Integer-constrained Cones for Conformal Parameterizations" (2512.20904) addresses the discrete optimization challenge of generating sparse, integer-constrained cone singularities to facilitate rotationally seamless conformal parameterizations with low distortion on surface meshes. Conformal parameterization is fundamental for applications in texture mapping, remeshing, and physical simulation due to its angle-preserving nature, but it suffers from area distortion that can be mitigated by strategically introducing cone singularities.

Integer-constrained cones—where cone angles are quantized to integer multiples of $\frac{\pi}{2}$—are essential for globally seamless or rotationally seamless parameterizations. The optimization requires not only low distortion but also minimal cone count and computational efficiency, all while constrained to mesh vertices and discrete angle values. Existing methods either rely on costly mixed-integer programming, iterative rounding, or exhaustive enumeration, each presenting scalability issues and often sacrificing optimality in trade-offs among distortion, cone sparsity, and runtime.

Methodology and Formulation

The core technical contribution is a highly efficient alternate optimization strategy on three discrete variable groups: cone positions (vertex-constrained), angles (integer multiples of $\frac{\pi}{2}$), and count. The conformal parameterization problem adopts a discrete Yamabe equation formulation, with area distortion measured via the $\ell_2$ Hencky energy metric.

A key innovation is variable reduction for cone angle optimization. Instead of solving a full-scale MIQP with as many integer variables as mesh vertices, the method identifies that, for fixed cone positions, only the angles at cone locations are relevant, which reduces the MIQP to a scale nearly matching the cone count. Further, a matrix transformation and exploitation of sparsity allow analytic expressions for solutions to the discrete equations governing curvature assignment, yielding significant computational speedup, especially for high-genus surfaces.

Optimization proceeds by alternately:

1. Angle Optimization: Solve the reduced MIQP for cone angles using structural constraints, including the Gauss-Bonnet theorem for total curvature and holonomy constraints for non-contractible cycles (genus $g > 0$).
2. Cone Position Update: Move cone positions along mesh vertices guided by a newly derived distortion gradient (shape derivative), tailored for fixed angles.
3. Adaptive Cone Addition/Removal: Employ branch-driven strategies to insert cones where distortion is concentrated and prune oppositely signed cone pairs that fail to decrease energy, ensuring monotonic improvement and convergence.

For genus-zero and high-genus surfaces, explicit constructions of null spaces permit adaptable variable assignment after mesh cutting and handle/tunnel duplication, enabling compliance with additional holonomy constraints required for rotationally seamless parameterization.

Numerical Results and Evaluation

Extensive evaluations on a dataset of 3885 diverse models demonstrate order-of-magnitude computational speedup (average 30$\times$ faster) compared to state-of-the-art methods [Li2022int], [zhang2023practical], with comparable or fewer cone singularities and similar or lower parameterization distortion. The method is robust to mesh tessellation irregularities and scales effectively with mesh size. Cone number, distortion, and runtime distributions empirically confirm efficiency and effectiveness.

The parameterization pipeline is validated for both genus-zero and non-zero genus surfaces, with the algorithm producing seamless parameterizations with low inter-cut distortion. In practical remeshing (e.g., quad extraction), the conformal parametrizations derived via this method yield quad meshes with more uniform interior angles and competitive singularity counts, supporting downstream mesh processing tasks.

Implications and Theoretical Significance

The explicit reduction of MIQP variable cardinality fundamentally improves tractability in discrete geometric optimization. The variable reduction framework, combined with adaptive cone insertion/removal strategies and branch-driven approaches, suggests a scalable paradigm for combinatorial geometric problems. This technique is not only relevant for conformal parameterization but can be generalized to other integer-constrained mesh processing tasks, such as field/singularity assignment, seamless mapping, and quadrangulation.

Restricting cones to mesh vertices precludes strict global seamlessness in high-genus cases unless the Abel-Jacobi condition is met, typically requiring mesh refinement. Thus, the work highlights an underlying theoretical barrier in conformal parametrization with vertex-constrained cones, motivating future research in topological pre-processing or mixed alignment constraints for broader applicability.

Prospects for Future Developments

Possible directions include developing more sophisticated pruning strategies to address cases with very low target distortion, algorithmic improvements for CAD-like meshes with extensive regions of quantized curvature, and dynamic mesh refinement during parameterization to increase cone placement flexibility. Integrating the variable reduction paradigm in alignment-constrained or feature-aware parameterization frameworks is a promising avenue for bridging geometric optimizations and application-specific requirements in mesh generation, texture synthesis, and simulation.

Conclusion

This work introduces an explicit, efficient computational framework for integer-constrained cone construction in conformal parameterization, establishing new standards for speed, scalability, and solution quality in discrete mesh optimization. Its technical innovations in variable reduction and adaptive alternating optimization are broadly applicable within computer graphics and geometry processing, with significant impact on practical mesh generation pipelines and underlying theoretical understanding of conformal mapping constraints.