A Novel Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

Abstract

Numerous methods for computing conformal mesh paramterizations has been developed due to the vast applications in the field of geometry processing. Spectral conformal parameterization (SCP) is one of these methods to computing a quality conformal parameterization based on the spectral technique. SCP focus on a generalized eigenvalue problem (GEP) $L_{C}\mathbf{f} = λB\mathbf{f}$ whose eigenvector(s) associated with the smallest positive eigenvalue(s) will provide the parameterization result. This paper devotes to study a novel eigensolver for this GEP. Based on the structures of matrix pair $(L_{C},B)$, we show that this GEP can be transformed into a small-scaled compressed deating standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a skew-Hamiltonian isotropic Lanczos algorithm (SHILA) to solve the reducing problem. Numerical experiments show that our compressed deating skill remove the inuence of the kernel of $L_{C}$ and transform the original problem to a more robust system. The novel SHILA method can effective avoid the disturbance of duplicate eigenvalues. As a result, our numerical eigensolver can accurately and efficiently compute the conformal parameterization based on the spectral model of SCP.