Energy-Based Distortion-Balancing Parameterization for Open Surfaces

TL;DR

This paper introduces an iterative algorithm for surface parameterization that optimally balances angle and area distortions, ensuring bijective mappings and practical applications in geometry image representations.

Contribution

A novel iterative algorithm with proven global convergence for open surface parameterization balancing distortions, improving surface representation accuracy.

Findings

Algorithm converges globally

Mappings are bijective

Effectively balances distortions across meshes

Abstract

Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the systematic representation and manipulation of surfaces of complicated shapes by simplifying them into a manageable planar domain. In this paper, we propose a new iterative algorithm for computing the parameterization of simply connected open surfaces that achieves an optimal balance between angle and area distortions. We rigorously prove that the iteration in our algorithm converges globally, and numerical results demonstrate that the resulting mappings are bijective and effectively balance angular and area accuracy across various triangular meshes. Additionally, we present the practical usefulness of the proposed algorithm by applying it to represent surfaces as geometry images.