Abstract
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) \(L_{C}{\mathbf {f}} = \lambda B{\mathbf {f}}\) 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 \((L_{C},B)\), 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 (\({\mathbb {S}}\)HILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of \(L_{C}\) and transform the original problem to a more robust system. The novel \({\mathbb {S}}\)HILA 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.
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Notes
For mesh with single boundary, we have \(B_\mathtt{b} = I_{2n_\mathtt{b}}\).
A \(2n \times 2n\) matrix \(G\) is said to be symplectic if \(G^{\top }JG = J\).
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Acknowledgments
Some of the mesh models are available on the websites CCGL (Susan and Sophie); Unwrapped meshes (Bunny, Camel, Dino, Fandisk, Hand, Foot and Isis), and Project page of ARAP (Beetle and Gorgyle). The remaining mesh models are courtesy of AIM@SHAPE Shape Repository. The first and third authors would like to acknowledge the support from the National Science Council and the National Centre for Theoretical Sciences in Taiwan. They also like to thank the ST Yau Centre at the National Chiao Tung University for the support.
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Huang, WQ., Gu, X.D., Lin, WW. et al. A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations. J Sci Comput 61, 558–583 (2014). https://doi.org/10.1007/s10915-014-9840-2
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DOI: https://doi.org/10.1007/s10915-014-9840-2
Keywords
- Conformal parameterization
- Nonequivalence deflation
- Null-space free
- Symmetric skew-Hamiltonian
- Isotropic Lanczos