Convergence of barycentric coordinates to barycentric kernels

Kosinka, Jiří; Bartoň, Michael

doi:10.1016/j.cagd.2016.02.003

Cited by 6 publications

(3 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• The value of w † i in the interior of Γ is computed by interpolating the w * i along ∂Γ using a transfinite barycentric kernel Belyaev (2006); Kosinka and Bartoň (2016).…”

Section: Appendix a Intuition For Conjecturementioning

confidence: 99%

A fast numerical solver for local barycentric coordinates

Tao

Deng

Zhang

2019

Computer Aided Geometric Design

View full text Add to dashboard Cite

The local barycentric coordinates (LBC), proposed in Zhang et al. (2014), demonstrate good locality and can be used for local control on function value interpolation and shape deformation. However, it has no closedform expression and must be computed by solving an optimization problem, which can be time-consuming especially for high-resolution models. In this paper, we propose a new technique to compute LBC efficiently. The new solver is developed based on two key insights. First, we prove that the non-negativity constraints in the original LBC formulation is not necessary, and can be removed without affecting the solution of the optimization problem. Furthermore, the removal of this constraint allows us to reformulate the computation of LBC as a convex constrained optimization for its gradients, followed by a fast integration to recover the coordinate values. The reformulated gradient optimization problem can be solved using ADMM, where each step is trivially parallelizable and does not involve global linear system solving, making it much more scalable and efficient than the original LBC solver. Numerical experiments verify the effectiveness of our technique on a large variety of models.

show abstract

“…• The value of w † i in the interior of Γ is computed by interpolating the w * i along ∂Γ using a transfinite barycentric kernel Belyaev (2006); Kosinka and Bartoň (2016).…”

Section: Appendix a Intuition For Conjecturementioning

confidence: 99%

A fast numerical solver for local barycentric coordinates

Tao

Deng

Zhang

2019

Computer Aided Geometric Design

View full text Add to dashboard Cite

show abstract

“…Many methods [MBLD02; Das03; JSWD05] have been proposed to produce Wachspress' coordinates inside a convex n‐gon using different approaches. [WSHD07; DW08a; KB15; KB16] extended Wachpress method from a polygon to a convex closed curved bounded domain. However, the main limitation of Wachspress's method is the restriction to convex shapes.…”

Section: Related Workmentioning

confidence: 99%

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

Yan

Schaefer

2019

Computer Graphics Forum

View full text Add to dashboard Cite

We construct a family of barycentric coordinates for 2D shapes including non‐convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater‐Hormann‐Kós family of barycentric coordinates for 2D convex shapes. We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co‐dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well‐defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.

show abstract

“…We evaluate numerically our interpolation scheme (19) and compare it to the fundamental solution f (x) given by (20) for different values of p on a concave shape shown in Fig. 12.…”

Section: Let Us Test Interpolation Scheme (19) On the Fundamental Solmentioning

confidence: 99%

On modified Gordon-Wixom interpolation schemes and their applications to nonlinear and exterior domain problems

Belyaev

Fayolle

2017

Numer Algor

View full text Add to dashboard Cite

We introduce and study extensions and modifications of the GordonWixom transfinite barycentric interpolation scheme (Gordon and Wixom, SIAM J. Numer. Anal. 11(5), 1974). We demonstrate that the modified GordonWixom scheme proposed in Belyaev and Fayolle (Comput. Graph. 51, [74][75][76][77][78][79][80] 2015) reproduces harmonic quadratic polynomials in convex domains. We adapt the scheme for dealing with the exterior of a bounded domain and for the exterior of a disk, where we demonstrate that our interpolation formula reproduces harmonic functions. Finally, we show how to adapt the Gordon-Wixom approach for approximating p-harmonic functions and to derive computationally efficient approximations of the solutions to boundary value problems involving the p-Laplacian.

show abstract

Convergence of barycentric coordinates to barycentric kernels

Cited by 6 publications

References 23 publications

A fast numerical solver for local barycentric coordinates

A fast numerical solver for local barycentric coordinates

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

On modified Gordon-Wixom interpolation schemes and their applications to nonlinear and exterior domain problems

Contact Info

Product

Resources

About