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

Wei Qiang Huang*, Xianfeng David Gu, Wen-Wei Lin, Shing Tung Yau

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Scopus citations


In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP ’08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487–1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) LCf = ?Bf whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair (LC, B), we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm (SHILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of LC and transform the original problem to a more robust system. The novel SHILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.

Original languageEnglish
Pages (from-to)558-583
Number of pages26
JournalJournal of Scientific Computing
Issue number3
StatePublished - 14 Oct 2014


  • Conformal parameterization
  • Isotropic Lanczos
  • Nonequivalence deflation
  • Null-space free
  • Symmetric skew-Hamiltonian

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