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Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 54 (5 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
Conformal Flattening by Curvature Prescription and Metric Scaling
, 2008
"... We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topo ..."
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Cited by 51 (2 self)
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We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasiconformal distortion.
3D Face Matching and Registration Based on Hyperbolic Ricci Flow
"... 3D surface matching is fundamental for shape analysis. As a powerful method in geometric analysis, Ricci flow can flexibly design metrics by prescribed target curvature. In this paper we describe a novel approach for matching surfaces with complicated topologies based on hyperbolic Ricci flow. For s ..."
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Cited by 4 (4 self)
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3D surface matching is fundamental for shape analysis. As a powerful method in geometric analysis, Ricci flow can flexibly design metrics by prescribed target curvature. In this paper we describe a novel approach for matching surfaces with complicated topologies based on hyperbolic Ricci flow. For surfaces with negative Euler characteristics, such as a human face with holes (eye contours), the canonical hyperbolic metric is conformal to the original and can be efficiently computed. Then the surface can be canonically decomposed to hyperbolic hexagons. By matching the corresponding hyperbolic hexagons, the matching between surfaces can be easily established. Compared to existing methods, hyperbolic Ricci flow induces diffeomorphisms between surfaces with complicated topologies with negative Euler characteristics, while avoiding singularities. Furthermore, all the boundaries are intrinsically mapped to hyperbolic lines as alignment constraints. Finally, we demonstrate the applicability of this intrinsic shape representation for 3D face matching and registration. local isometric mapping [28], summation invariants [21], landmarksliding [7], physicsbased deformable models [30], FreeForm Deformation (FFD) [14], and LevelSet based methods [23]. However, many surface representations that use local geometric invariants can not guarantee a global convergence and might suffer from local minima in the presence of nonrigid deformations. To address this issue, many global parameterization methods have been developed recently based on conformal geometric maps
Slit map: conformal parameterization for multiply connected surfaces
 In Proc. Geometric Modeling and Processing
, 2008
"... Abstract. Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply connected disks. This work introduces a novel method to handle multiply connected surfaces based on holomorphic oneforms. The method maps genus zero surfaces with ar ..."
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Cited by 3 (1 self)
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Abstract. Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply connected disks. This work introduces a novel method to handle multiply connected surfaces based on holomorphic oneforms. The method maps genus zero surfaces with arbitrary number of boundaries to an annulus with concentric circular slits. Any two boundaries can be chosen to map to the inner circle and the outer circle, the other boundaries to slits. Equivalently, the surfaces can be mapped to a rectangle with horizontal slits. Compared to existing linear methods that require surface partition, this method is more intrinsic and automatic. Compared to the existing holomorphic oneform method that requires double covering, it is more efficient and has better control over singularities. Compared to the existing Ricci flow method, this one is linear and simpler. The proposed method has many merits. The images of boundaries are parallel line segments. This regularity not only helps improve the accuracy for surface matching with boundaries, but also makes quadremeshing or meshspline conversion conversion convenient. The whole rectangle in texture domain is fully occupied without any gap or overlapping; this improves the packing efficiency for texture mapping. The positions of the slits are completely determined by the surface geometry, which can be treated as the finger print of the surface to classify surfaces by conformal equivalence. The algorithm is thoroughly explained in detail. Experimental results are demonstrated to show the usefulness of the algorithm for multiply connected domains. Key words: conformal parameterization, slits, holomorphic oneform 1
TO APPEAR IN IEEE TVCG 1 Metric Driven RoSy Field Design and
"... Abstract—Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic NRoSy field design on arbitrary surfaces with user defined field topologies. The user has full control of th ..."
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Abstract—Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic NRoSy field design on arbitrary surfaces with user defined field topologies. The user has full control of the number, positions and indices of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate NRoSy field construction as designing a Riemannian metric, such that the holonomy along any loop is compatible with the local symmetry of NRoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of NRoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility, such that, it can design NRoSy fields with single singularity, and mixedRoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to penandink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.
Volume xx (200y), Number z, pp. 1–10 Fast MeanCurvature Flow via FiniteElements Tracking
"... In this paper, we present a novel approach for efficiently evolving meshes using meancurvature flow. We use a finiteelements hierarchy that supports an efficient multigrid solver for performing the semiimplicit timestepping. Though expensive to compute, we show that it is possible to track this ..."
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In this paper, we present a novel approach for efficiently evolving meshes using meancurvature flow. We use a finiteelements hierarchy that supports an efficient multigrid solver for performing the semiimplicit timestepping. Though expensive to compute, we show that it is possible to track this hierarchy through the process of surface evolution. As a result, we provide a way to efficiently flow the surface through the evolution, without requiring a costly initialization at the beginning of each timestep. Using our approach, we demonstrate a factor of nearly sevenfold improvement over the nontracking implementation, supporting the evolution of surfaces consisting of 1M triangles at a rate of just a few seconds per update.
Geometric Sampling of Images, Vector Quantization and Zador’s Theorem
"... We present several consequences of the geometric approach to image sampling and reconstruction we have previously introduced. We single out the relevance of the geometric method to the vector quantization of images and, more important, we give a concrete and candidate for the optimal embedding dimen ..."
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We present several consequences of the geometric approach to image sampling and reconstruction we have previously introduced. We single out the relevance of the geometric method to the vector quantization of images and, more important, we give a concrete and candidate for the optimal embedding dimension in Zador’s Thorem. An additional advantage of our approach is that that this provides a constructive proof of the aforementioned theorem, at least in the case of images. Further applications are also briefly discussed. 1.
HIGH RESOLUTION CARDIAC SHAPE REGISTRATION USING RICCI FLOW
"... Current CT techniques are able to produce isotropic high resolution CT images (0.5mm). Recent research has revealed that the interior of the left ventricle has complex structures and topology, which has potentially valuable information. However, this makes the matching between models much more chall ..."
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Current CT techniques are able to produce isotropic high resolution CT images (0.5mm). Recent research has revealed that the interior of the left ventricle has complex structures and topology, which has potentially valuable information. However, this makes the matching between models much more challenging. In this paper, we propose a novel method to match two models with nontrivial topology. 3D mesh models are flattened onto a 2D planar surfaces using discrete hyperbolic Ricci flow. Therefore, the 3D matching problem is converted to a much simpler 2D matching problem. We show the performance on the registration of high resolution left ventricle models. Index Terms — high resolution CT, shape registration, Ricci flow 1.
Author manuscript, published in "SAMPTA'09, Marseille: France (2009)" Geometric Wavelets for Image Processing: Metric Curvature of Wavelets
, 2010
"... We introduce a semidiscrete version of the FinslerHaantjes metric curvature to define curvature for wavelets and show that scale and curvature play similar roles with respect to image presentation and analysis. More precisely, we show that there is an inverse relationship between local scale and l ..."
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We introduce a semidiscrete version of the FinslerHaantjes metric curvature to define curvature for wavelets and show that scale and curvature play similar roles with respect to image presentation and analysis. More precisely, we show that there is an inverse relationship between local scale and local curvature in images. This allows us to use curvature as a geometrically motivated automatic scale selection in signal and image processing, this being an incipient bridging of the gap between the methods employed in Computer Graphics and Image Processing. A natural extension to ridgelets and curvelets is also given. Further directions of study, in particular the development of a curvature transform and the study of its link with wavelet and the scale transforms are also suggested. 1.