Local Barycentric Coordinates

ACM Transactions on Graphics (Proceedings of Siggraph Asia), 2014

Abstract

Barycentric coordinates yield a powerful and yet simple paradigm to interpolate data values on polyhedral domains. They represent interior points of the domain as an affine combination of a set of control points, defining an interpolation scheme for any function defined on a set of control points. Numerous barycentric coordinate schemes have been proposed satisfying a large variety of properties. However, they typically define interpolation as a combination of all control points. Thus a local change in the value at a single control point will create a global change by propagation into the whole domain. In this context, we present a family of local barycentric coordinates (LBC), which select for each interior point a small set of control points and satisfy common requirements on barycentric coordinates, such as linearity, non-negativity, and smoothness. LBC are achieved through a convex optimization based on total variation, and provide a compact representation that reduces memory footprint and allows for fast deformations. Our experiments show that LBC provide more local and finer control on shape deformation than previous approaches, and lead to more intuitive deformation results.



Results



Deformation of the cactus model with different control weight functions. The control points in red and in green are subject to rigid and non-rigid transformations, respectively. The color-coding shows the absolute sums of weight functions for the green and red control points, respectively. LBC preserve the shape of the hat, since it is only influenced by the red control points, and deformed by the same rigid transformation. Other coordinate schemes distort the hat shape, due to the influence from the green control points.



The gecko image is deformed using control points close to the tail (in red) and the feet (in green). The color-coding shows the absolute sum of control weight functions for the red control points. The elbow and the head remain fixed with LBC, while being deformed by other weight functions. A comparison of the deformations can be found in the supplementary video.




Video