Results 1  10
of
64
GPUassisted Positive Mean Value Coordinates for Mesh Deformations
, 2007
"... In this paper we introduce positive mean value coordinates (PMVC) for mesh deformation. Following the observations of Joshi et al. [JMD ∗ 07] we show the advantage of having positive coordinates. The control points of the deformation are the vertices of a "cage " enclosing the defo ..."
Abstract

Cited by 40 (3 self)
 Add to MetaCart
In this paper we introduce positive mean value coordinates (PMVC) for mesh deformation. Following the observations of Joshi et al. [JMD ∗ 07] we show the advantage of having positive coordinates. The control points of the deformation are the vertices of a &quot;cage &quot; enclosing the deformed mesh. To define positive mean value coordinates for a given vertex, the visible portion of the cage is integrated over a sphere. Unlike MVC [JSW05], PMVC are computed numerically. We show how the PMVC integral can be efficiently computed with graphics hardware. While the properties of PMVC are similar to those of Harmonic coordinates [JMD ∗ 07], the setup time of the PMVC is only of a few seconds for typical meshes with 30K vertices. This speedup renders the new coordinates practical and easy to use.
Barycentric rational interpolation with no poles and high rates of approximation
 MATH
, 2006
"... It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that ha ..."
Abstract

Cited by 39 (7 self)
 Add to MetaCart
(Show Context)
It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no poles and arbitrarily high approximation orders, regardless of the distribution of the points. The family includes a construction of Berrut as a special case.
A Finite Element Method on Convex Polyhedra
 EUROGRAPHICS 2007 / D. COHENOR AND P. SLAVÍK (GUEST EDITORS)
, 2007
"... We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis funct ..."
Abstract

Cited by 27 (4 self)
 Add to MetaCart
We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.
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 ..."
Abstract

Cited by 26 (7 self)
 Add to MetaCart
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.
Spherical Barycentric Coordinates
, 2006
"... We develop spherical barycentric coordinates. Analogous to classical, planar barycentric coordinates that describe the positions of points in a plane with respect to the vertices of a given planar polygon, spherical barycentric coordinates describe the positions of points on a sphere with respect ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
We develop spherical barycentric coordinates. Analogous to classical, planar barycentric coordinates that describe the positions of points in a plane with respect to the vertices of a given planar polygon, spherical barycentric coordinates describe the positions of points on a sphere with respect to the vertices of a given spherical polygon. In particular,
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 ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
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.
On the derivation of green coordinates
, 2008
"... Coordinates in 3D admits a quasiconformal deformation. In (f) the result using Mean Value Coordinates is presented. Note how Green Coordinates nicely preserve the shape of the Ogre’s head. We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green’s third int ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
Coordinates in 3D admits a quasiconformal deformation. In (f) the result using Mean Value Coordinates is presented. Note how Green Coordinates nicely preserve the shape of the Ogre’s head. We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green’s third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shapepreserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasiconformal mappings in 3D. In both cases we derive closedform expressions for the coordinates, yielding a simple and fast algorithm for cagebased space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shapepreserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages. 1
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 ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
(Show Context)
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.
Transfinite mean value interpolation
, 2007
"... Transfinite mean value interpolation has recently emerged as a simple and robust way to interpolate a function f defined on the boundary of a planar domain. In this paper we study basic properties of the interpolant, including sufficient conditions on the boundary of the domain to guarantee interpol ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
(Show Context)
Transfinite mean value interpolation has recently emerged as a simple and robust way to interpolate a function f defined on the boundary of a planar domain. In this paper we study basic properties of the interpolant, including sufficient conditions on the boundary of the domain to guarantee interpolation when f is continuous. Then, by deriving the normal derivative of the interpolant and of a mean value weight function, we construct a transfinite Hermite interpolant and discuss various applications.
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 ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
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.