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INFOCOM 2011. 30th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 10-15 April 2011, Shanghai, China; 01/2011
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Proceedings of the 12th ACM Interational Symposium on Mobile Ad Hoc Networking and Computing, MobiHoc 2011, Paris, France, May 16-20, 2011; 01/2011
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Computer Aided Geometric Design. 01/2011; 28:475-496.
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ACM Symposium on Solid and Physical Modeling, Proceedings of the 14th ACM Symposium on Solid and Physical Modeling, SPM 2010, Haifa, Israel, September 1-3, 2010; 01/2010
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IEEE Trans. Vis. Comput. Graph. 01/2010; 16:95-108.
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ABSTRACT: Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the efficiency and efficacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison.
Shape Modeling and Applications, 2009. SMI 2009. IEEE International Conference on; 07/2009
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ABSTRACT: Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the potential value of the method for geometric database indexing, shape comparison and classification.
Shape Modeling and Applications, 2009. SMI 2009. IEEE International Conference on; 07/2009
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ABSTRACT: Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.
IEEE Transactions on Visualization and Computer Graphics 07/2009; · 2.21 Impact Factor
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IEEE Trans. Vis. Comput. Graph. 01/2009; 15:504-517.
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07/2008;
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Advances in Visual Computing, 4th International Symposium, ISVC 2008, Las Vegas, NV, USA, December 1-3, 2008. Proceedings, Part I; 01/2008
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Computer-Aided Design. 01/2008; 40:676-690.
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Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, Stony Brook, New York, USA, June 2-4, 2008; 01/2008
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International Journal of Shape Modeling. 01/2008; 14:169-188.
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IEEE Trans. Vis. Comput. Graph. 01/2008; 14:1030-1043.
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Advances in Visual Computing, 4th International Symposium, ISVC 2008, Las Vegas, NV, USA, December 1-3, 2008. Proceedings, Part I; 01/2008
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Emerging Trends in Visual Computing, LIX Fall Colloquium, ETVC 2008, Palaiseau, France, November 18-20, 2008. Revised Invited Papers; 01/2008
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ABSTRACT: Summary form only given. In this paper we generalize the shortest path algorithm to the shortest cycles in each homotopy class on a surface with arbitrary topology, utilizing the universal covering space (UCS) in algebraic topology. In order to store and handle the UCS, we propose a two-level data structure which is efficient for storage and easy to process. We also pointed several practical applications for our shortest cycle algorithms and the UCS data structure.
Computer-Aided Design and Computer Graphics, 2007 10th IEEE International Conference on; 11/2007
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ABSTRACT: Three-dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition, and classification. However, shape matching with noise, occlusion, and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps including harmonic maps, conformal maps, and least-squares conformal maps with regards to 3D shape matching. As a result, we propose a novel and computationally efficient shape matching framework by using least-squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise. Therefore, highly accurate and efficient 3D shape matching algorithms can be achieved by using the above three parametric maps. Finally, the robustness of least-squares conformal maps is evaluated and analyzed comprehensively in 3D shape matching with occlusion, noise, and resolution variation. In order to further demonstrate the performance of our proposed method, we also conduct a series of experiments on two computer vision applications, i.e., 3D face recognition and 3D nonrigid surface alignment and stitching.
IEEE Transactions on Pattern Analysis and Machine Intelligence 08/2007; 29(7):1209-20. · 4.91 Impact Factor
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ABSTRACT: Three-dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition, and classification. However, shape matching with noise, occlusion, and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps including harmonic maps, conformal maps, and least-squares conformal maps with regards to 3D shape matching. As a result, we propose a novel and computationally efficient shape matching framework by using least-squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise. Therefore, highly accurate and efficient 3D shape matching algorithms can be achieved by using the above three parametric maps. Finally, the robustness of least-squares conformal maps is evaluated and analyzed comprehensively in 3D shape matching with occlusion, noise, and resolution variation. In order to further demonstrate the performance of our proposed method, we also conduct a series of experiments on two computer vision applications, i.e., 3D face recognition and 3D nonrigid surface alignment and stitching.
IEEE Transactions on Pattern Analysis and Machine Intelligence 08/2007; 29(7):1209-1220. · 4.91 Impact Factor
