Numerical methods for spectral problems: theory and applications

P000010

A Structured Eigensolver for Spectral Conformal Parameterizations

*Wei-Qiang Huang (General Education, National Formosa University, Taiwan)
Xianfeng David Gu (Department of Computer Science, Stony Brook University)
Wen-Wei Lin (Department of Applied Mathematics, National Chiao Tung University)
Yau Shing-Tung (Mathematics Department, Harvard University)

Mesh parameterization is one of the fundamental operations in computer graphics. Its wide application includes texture mapping, surface fitting, surface remeshing and so on. Spectral conformal parameterization (SCP) is one of these methods for computing a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP), $\mathcal{A}\mathbf{x} = \lambda \mathcal{B}\mathbf{x}$, whose eigenvector associated with the smallest positive eigenvalue provides the parameterization result.
In this talk, we will show that this GEP can be transformed into a standard eigenvalue problem, $\mathcal{H}\mathbf{s} = \mu \mathbf{s}$, with a symmetric positive definite skew-Hamiltonian operator $\mathcal{H}$. Furthermore, we develop an eigensolver, based on this particular structure, to solve the resulting problem. Numerical experiments show that this proposed algorithm can compute the conformal parameterization accurately and efficiently.