Browsing by Author "Yau, Shing-Tung"

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A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds
(2019-01-01)
Yueh, Mei-Heng
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Li, Tiexiang
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Lin, Wen-Wei
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Yau, Shing-Tung
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應用數學系
Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle- and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.
A Novel Stretch Energy Minimization Algorithm for Equiareal Parameterizations
(2019-03-01)
Yueh, Mei-Heng
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Lin, Wen-Wei
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Wu, Chin-Tien
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Yau, Shing-Tung
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應用數學系
Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with very small area distortions and highly improved computational efficiencies. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the accuracy, effectiveness, and robustness of the proposed SEM algorithm outperform the other state-of-the-art algorithms. Applications of the SEM on surface remeshing, registration and morphing for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computation for these applications can be carried out efficiently and reliably.
A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations
(2014-12-01)
Huang, Wei-Qiang
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Gu, Xianfeng David
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Lin, Wen-Wei
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Yau, Shing-Tung
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應用數學系
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) 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 , 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 (HILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of and transform the original problem to a more robust system. The novel 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.
An Efficient Energy Minimization for Conformal Parameterizations
(2017-10-01)
Yueh, Mei-Heng
;
Lin, Wen-Wei
;
Wu, Chin-Tien
;
Yau, Shing-Tung
;
應用數學系
Surface parameterizations have been widely applied to digital geometry processing. In this paper, we propose an efficient conformal energy minimization (CEM) algorithm for computing conformal parameterizations of simply-connected open surfaces with a very small angular distortion and a highly improved computational efficiency. In addition, we generalize the proposed CEM algorithm to computing conformal parameterizations of multiply-connected surfaces. Furthermore, we prove the existence of a nontrivial accumulation point of the proposed CEM algorithm under some mild conditions. Several numerical results show the efficiency and robustness of the CEM algorithm comparing to the existing state-of-the-art algorithms. An application of the CEM on the surface morphing between simply-connected open surfaces is demonstrated thereafter. Thanks to the CEM algorithm, the whole computations for the surface morphing can be performed efficiently and robustly.
iSIRA: Integrated shift-invert residual Arnoldi method for graph Laplacian matrices from big data
(2019-01-15)
Huang, Wei-Qiang
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Lin, Wen-Wei
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Lu, Henry Horng-Shing
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Yau, Shing-Tung
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交大名義發表
The eigenvalue problem of a graph Laplacian matrix L arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated matrix L is symmetric, positive semi-definite, and is usually large and sparse. It is often of interest for finding some smallest positive eigenvalues and corresponding eigenvectors. However, the singularity of L makes the classical eigensolvers inefficient since we need to factorize L for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually prohibitive to factorize L generated by real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections. In this paper, we propose a trimming to cure the singularity of L according to its special property: zero row/column sum. This remedy technique leads us to solve a positive definite linear system reduced in one dimension and then recover the result to get a suitable solution of the original system involved in an eigensolver. Besides, we apply a deflating approach to exclude the influence of converged eigenvalues. We show how to apply the idea of trimming to the graph Laplacian eigenvalue problem together with a deflated term and a target shift. Accordingly, based on the inexact residual Arnoldi (Lee, 2007; Lee and Stewart, 2007) method, we propose an integrated eigensolver for this kind of L combining with the implicit remedy of the singularity, an effective deflation for convergent eigenvalues and the shift-invert enhancement. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation especially when the LU factorization is not available. (C) 2018 Elsevier B.V. All rights reserved.