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18
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.
Maximum Entropy Coordinates for Arbitrary Polytopes
, 2008
"... Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and ..."
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Cited by 26 (7 self)
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Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton’s method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higherdimensional polytopes.
On Transfinite Barycentric Coordinates
, 2006
"... A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinate ..."
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Cited by 20 (0 self)
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A general construction of transfinite barycentric coordinates is obtained as a simple and natural generalization of Floater's mean value coordinates [Flo03, JSW05b]. The GordonWixom interpolation scheme [GW74] and transfinite counterparts of discrete harmonic and WachspressWarren coordinates are studied as particular cases of that general construction. Motivated by finite element/volume applications, we study capabilities of transfinite barycentric interpolation schemes to approximate harmonic and quasiharmonic functions. Finally we establish and analyze links between transfinite barycentric coordinates and certain inverse problems of di#erential and convex geometry.
Higher order barycentric coordinates
 COMPUTER GRAPHICS FORUM (PROC. EUROGRAPHICS
, 2008
"... In recent years, a wide range of generalized barycentric coordinates has been suggested. However, all of them lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We int ..."
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Cited by 13 (0 self)
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In recent years, a wide range of generalized barycentric coordinates has been suggested. However, all of them lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We introduce a method to modify existing barycentric coordinates to higher order barycentric coordinates and demonstrate, using higher order mean value coordinates, that our method, although conceptually simple and easy to implement, can be used to give easy and intuitive control at interactive frame rates over local space deformations such as rotations.
Mesh Denoising via L0 Minimization
"... Figure 1: From left to right: initial surface, surface corrupted by Gaussian noise in random directions with standard deviationσ = 0.4le (le is the mean edge length), bilateral filtering [Fleishman et al. 2003], prescribed mean curvature flow [Hildebrandt and Polthier 2004], mean filtering [Yagou et ..."
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Cited by 6 (0 self)
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Figure 1: From left to right: initial surface, surface corrupted by Gaussian noise in random directions with standard deviationσ = 0.4le (le is the mean edge length), bilateral filtering [Fleishman et al. 2003], prescribed mean curvature flow [Hildebrandt and Polthier 2004], mean filtering [Yagou et al. 2002], bilateral normal filtering [Zheng et al. 2011], our method. The wireframe shows folded triangles as red edges. We present an algorithm for denoising triangulated models based on L0 minimization. Our method maximizes the flat regions of the model and gradually removes noise while preserving sharp features. As part of this process, we build a discrete differential operator for arbitrary triangle meshes that is robust with respect to degenerate triangulations. We compare our method versus other anisotropic denoising algorithms and demonstrate that our method is more robust and produces good results even in the presence of high noise.
Poisson Coordinates
"... Abstract—Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradientdomain image processing and shapepreserving geometric computation. We propose Poisson coordinat ..."
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Cited by 6 (0 self)
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Abstract—Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradientdomain image processing and shapepreserving geometric computation. We propose Poisson coordinates, a novel transfinite interpolation scheme based on the Poisson integral formula, as a rapid way to estimate a harmonic function on a certain domain with desired boundary values. Poisson coordinates are an extension of the Mean Value coordinates (MVCs) which inherit their linear precision, smoothness, and kernel positivity. We give explicit formulas for Poisson coordinates in both continuous and 2D discrete forms. Superior to MVCs, Poisson coordinates are proved to be pseudoharmonic (i.e., they reproduce harmonic functions on ndimensional balls). Our experimental results show that Poisson coordinates have lower Dirichlet energies than MVCs on a number of typical 2D domains (particularly convex domains). As well as presenting a formula, our approach provides useful insights for further studies on coordinatesbased interpolation and fast estimation of harmonic functions. Index Terms—Poisson integral formula, transfinite interpolation, barycentric coordinates, pseudoharmonic Ç 1
New perspectives on polygonal and polyhedral finite element methods
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2014
"... Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary ..."
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Cited by 5 (2 self)
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Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the Virtual Element Method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more indepth understanding of mimetic schemes, and also endows polygonalbased Galerkin methods with greater flexibility than threenode and fournode finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semidefinite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinatebased Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates postprocessing of field variables and
Positive GordonWixom coordinates
 Computer Aided Design
, 2011
"... We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in 2D. Our method extends weighted GordonWixom interpolation to nonconvex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with ..."
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Cited by 3 (0 self)
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We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in 2D. Our method extends weighted GordonWixom interpolation to nonconvex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with smooth boundaries. We achieve these properties by using the distance to lines tangent to the boundary curve to define a weight function that is positive and smooth. We derive closedform expressions for arbitrary polygons in 2D and compare the basis functions of our coordinates with several other types of barycentric coordinates. Key words: barycentric coordinates, transfinite, interpolant 1.