Stable Discrete Minimization of Conformal Energy for Disk Conformal Parameterization

TL;DR

This paper introduces SDMCE, a stable and efficient algorithm for conformal parameterization of simply connected surfaces, ensuring one-to-one mappings and robustness against poor initializations.

Contribution

The paper proposes a novel stable algorithm, SDMCE, that guarantees conformal parameterization stability, avoids degeneracy, and corrects poor initializations efficiently.

Findings

SDMCE guarantees one-to-one and onto conformal mappings.

SDMCE is folding free and avoids solution degeneration.

Numerical experiments show SDMCE is stable and efficient.

Abstract

Conformal energy minimization is an efficient approach to compute conformal parameterization. In this paper, we develop a stable algorithm to compute conformal parameterization of simply connected open surface, termed Stable Discrete Minimization of Conformal Energy (SDMCE). The stability of SDMCE is reflected in the guarantee of one-to-one and on-to property of computed parameterization and the insensitivity on the initial value. On one hand, SDMCE can avoid degeneration and overlap of solution, also, SDMCE is folding free. On the other hand, even if given poor initial value, it can still correct it in very little computational time. The numerical experiments indicate SDMCE is stable and competitive with state-of-the-art algorithms in efficiency.