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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.
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
Biharmonic VHM
"... Figure 1: Source shape deformed using various algorithms by moving three control points. Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain t ..."
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Figure 1: Source shape deformed using various algorithms by moving three control points. Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closedform expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closedform expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.