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13
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 ..."
Abstract

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.
Complex Barycentric Coordinates with Applications to Planar Shape Deformation
, 2009
"... Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real and positive. In this paper we relax this requirement, allowing t ..."
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Cited by 17 (4 self)
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Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real and positive. In this paper we relax this requirement, allowing the barycentric coordinates to be complex numbers. This allows us to generate new families of barycentric coordinates, which have some powerful advantages over traditional ones. Applying complex barycentric coordinates to data which is itself complexvalued allows to manipulate functions from the complex plane to itself, which may be interpreted as planar mappings. These mappings are useful in shape and image deformation applications. We use Cauchy’s theorem from complex analysis to construct complex barycentric coordinates on (not necessarily convex) polygons, which are shown to be equivalent to planar Green coordinates. These generate conformal mappings from a given source region to a given target region, such that the image of the source region is close to the target region. We then show how to improve the Green coordinates in two ways. The first provides a much better fit to the polygonal target region, and the second allows to generate deformations based on positional constraints, which provide a more intuitive user interface than the conventional cagebased approach. These define two new types of complex barycentric coordinates, which are shown to be very effective in interactive deformation and animation scenarios.
Fast automatic skinning transformations
 ACM TRANS. GRAPH
"... Skinning transformations are a popular way to articulate shapes and characters. However, traditional animation interfaces require all of the skinning transformations to be specified explicitly, typically using a control structure (a rig). We propose a system where the user specifies only a subset ..."
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Cited by 15 (2 self)
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Skinning transformations are a popular way to articulate shapes and characters. However, traditional animation interfaces require all of the skinning transformations to be specified explicitly, typically using a control structure (a rig). We propose a system where the user specifies only a subset of the degrees of freedom and the rest are automatically inferred using nonlinear, rigidity energies. By utilizing a loworder model and reformulating our energy functions accordingly, our algorithm runs orders of magnitude faster than previous methods without compromising quality. In addition to the immediate boosts in performance for existing modeling and real time animation tools, our approach also opens the door to new modes of control: disconnected skeletons combined with shapeaware inverse kinematics. With automatically generated skinning weights, our method can also be used for fast variational shape modeling.
Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates
, 2011
"... We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex ngon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrangelike fashion to each v ..."
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Cited by 9 (3 self)
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We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex ngon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrangelike fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the socalled ‘serendipity’ elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.
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
Cubic Mean Value Coordinates
"... We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closedform interpolant that produces naturallooking functions while allowing flexible control of boundary constraints. Our ..."
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Cited by 3 (0 self)
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We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closedform interpolant that produces naturallooking functions while allowing flexible control of boundary constraints. Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant. We rederive the interpolant from the mean value property of biharmonic functions, and prove that the interpolant indeed matches the gradient constraints when the boundary is piecewise linear. We then give closedform formula (as generalized barycentric coordinates) for boundary constraints represented as polynomials up to degree 3 (for values) and 1 (for normal derivatives) over each polygon edge. We demonstrate the flexibility and efficiency of our coordinates in two novel applications, smooth image deformation using curved cage networks and adaptive simplification of gradient meshes.
Pointwise radial . . . arbitrary domains
, 2008
"... In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivative ..."
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In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely.
Linear Subspace Design for RealTime Shape Deformation
"... Figure 1: Our linear subspaces are very fast to compute. This enables the users to add (or remove) control handles very quickly, allowing them to realize their creative intent in a single interactive session. We propose a method to design linear deformation subspaces, unifying linear blend skinning ..."
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Figure 1: Our linear subspaces are very fast to compute. This enables the users to add (or remove) control handles very quickly, allowing them to realize their creative intent in a single interactive session. We propose a method to design linear deformation subspaces, unifying linear blend skinning and generalized barycentric coordinates. Deformation subspaces cut down the time complexity of variational shape deformation methods and physicsbased animation (reducedorder physics). Our subspaces feature many desirable properties: interpolation, smoothness, shapeawareness, locality, and both constant and linear precision. We achieve these by minimizing a quadratic deformation energy, built via a discrete Laplacian inducing linear precision on the domain boundary. Our main advantage is speed: subspace bases are solutions to a sparse linear system, computed interactively even for generously tessellated domains. Users may seamlessly switch between applying transformations at handles and editing the subspace by adding, removing or relocating control handles. The combination of fast computation and good properties means that designing the right subspace is now just as creative as manipulating handles. This paradigm shift in handlebased deformation opens new opportunities to explore the space of shape deformations.