Efficient Computation of Integer-constrained Cones for Conformal Parameterizations

TL;DR

This paper introduces an efficient method for computing integer-constrained cone singularities to produce low-distortion, seamless conformal parameterizations, significantly speeding up the process while maintaining quality.

Contribution

The authors develop a novel optimization algorithm that reduces computational complexity and improves speed for generating conformal parameterizations with integer-constrained cones.

Findings

Achieves 30x faster computation on average compared to existing methods.

Maintains comparable low distortion and cone numbers.

Effectively handles high-genus surfaces with improved efficiency.

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.