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26
A complex view of Barycentric mappings
, 2011
"... Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two-dimensional data in the plane, and in particular for holomorphic maps. We extend and genera ..."
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Cited by 8 (2 self)
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Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two-dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real-valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real-valued barycentric coordinates from a given weight function can be applied to generating complex-valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex-weighted combination of edge-to-edge similarity transformations, allowing the design of “home-made ” barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.
Optimizing voronoi diagrams for polygonal finite element computations
- In Proceedings of the 19th International Meshing Roundtable
, 2010
"... Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectiv ..."
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Cited by 8 (0 self)
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Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities—short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number. 1
On the Virtual Element Method for three-dimensional linear elasticity
, 2014
"... problems on arbitrary polyhedral meshes ..."
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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 gradient-domain image processing and shape-preserving geometric computation. We propose Poisson coordinat ..."
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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 gradient-domain image processing and shape-preserving 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 n-dimensional 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 coordinates-based interpolation and fast estimation of harmonic functions. Index Terms—Poisson integral formula, transfinite interpolation, barycentric coordinates, pseudoharmonic Ç 1
Poisson-based Weight Reduction of Animated Meshes
- COMPUTER GRAPHICS FORUM
, 2009
"... While animation using barycentric coordinates or other automatic weight assignment methods has become a popular method for shape deformation, the global nature of the weights limits their use for real-time applications. We present a method that reduces the number of control points influencing a vert ..."
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Cited by 6 (0 self)
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While animation using barycentric coordinates or other automatic weight assignment methods has become a popular method for shape deformation, the global nature of the weights limits their use for real-time applications. We present a method that reduces the number of control points influencing a vertex to a user-specified number such that the deformations created by the reduced weight set resemble that of the original deformation. To do so we show how to set up a Poisson minimization problem to solve for a reduced weight set and illustrate its advantages over other weight reduction methods. Not only does weight reduction lower the amount of storage space necessary to deform these models but also allows GPU acceleration of the resulting deformations. Our experiments show that we can achieve a factor of 100 increase in speed over CPU deformations using the full weight set, which makes real-time deformations of large models possible.
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 finite-difference 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 finite-difference 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 in-depth understand-ing of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node 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 semi-definite 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 poly-hedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based 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 post-processing of field variables and
Weighted averages on surfaces
- ACM Trans. Graph
, 2013
"... Figure 1: Interactive control for various geometry processing and modeling applications made possible with weighted averages on surfaces. From left to right: texture transfer, decal placement, semiregular remeshing and Laplacian smoothing, splines on surfaces. We consider the problem of generalizing ..."
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Cited by 5 (1 self)
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Figure 1: Interactive control for various geometry processing and modeling applications made possible with weighted averages on surfaces. From left to right: texture transfer, decal placement, semiregular remeshing and Laplacian smoothing, splines on surfaces. We consider the problem of generalizing affine combinations in Eu-clidean spaces to triangle meshes: computing weighted averages of points on surfaces. We address both the forward problem, namely computing an average of given anchor points on the mesh with given weights, and the inverse problem, which is computing the weights given anchor points and a target point. Solving the forward problem on a mesh enables applications such as splines on surfaces, Laplacian smoothing and remeshing. Combining the forward and inverse problems allows us to define a correspondence mapping be-tween two different meshes based on provided corresponding point pairs, enabling texture transfer, compatible remeshing, morphing and more. Our algorithm solves a single instance of a forward or an inverse problem in a few microseconds. We demonstrate that anchor points in the above applications can be added/removed and moved around on the meshes at interactive framerates, giving the user an immediate result as feedback.
Quadratic maximum-entropy serendipity shape functions
"... for arbitrary planar polygons ..."
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Bijective composite mean value mappings
- Computer Graphics Forum
, 2013
"... We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU-assisted implementation for ..."
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Cited by 4 (1 self)
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We introduce the novel concept of composite barycentric mappings and give theoretical conditions under which they are guaranteed to be bijective. We then focus on mean value mappings and derive a simple procedure for computing their Jacobians, leading to an efficient GPU-assisted implementation for interactively designing com-posite mean value mappings which are bijective up to pixel resolution. We provide a number of examples of 2D image deformation and an example of 3D shape deformation based on a natural extension of the concept to spatial mappings.
