DMCA
Mean Value Coordinates based deformations.
BibTeX
@MISC{Thiery_meanvalue,
author = {Jean-marc Thiery and Julien Tierny and Tamy Boubekeur and J. Tierny and T. Boubekeur},
title = {Mean Value Coordinates based deformations.},
year = {}
}
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Abstract
Abstract Mean Value Coordinates provide an efficient mechanism for the interpolation of scalar functions de-fined on orientable domains with non-convex bound-ary. They present several interesting features, including the simplicity and speed that yield from their closed-form expression. In several applications though, it is desirable to enforce additional constraints involving the partial derivatives of the interpolated function, as done in the case of the Green Coordinates approximation scheme [2] for interactive 3D model deformation. In this paper, we introduce the analytic expressions of the Jacobian and the Hessian of functions interpo-lated throughMean Value Coordinates.We provide these expressions both for the 2D and 3D case. We also pro-vide a thorough analysis of their degenerate configura-tions along with accurate approximations of the partial derivatives in these configurations. Extensive numeri-cal experiments show the accuracy of our derivation. In particular, we illustrate the improvements of our formu-lae over a variety of Finite Difference schemes in terms of precision and usability. We demonstrate the utility of this derivation in several applications, including cage-based implicit 3D model deformations (i.e. Variational MVC deformations). This technique allows for easy and interactive model deformations with sparse positional, rotational and smoothness constraints. Moreover, the cages produced by the algorithm can be directly re-used
Keyphrases
mean value coordinate several application partial derivative model deformation efficient mechanism present several interesting feature interpolated function finite difference scheme sparse positional scalar function orientable domain variational mvc deformation extensive numeri-cal experiment degenerate configura-tions interactive model deformation closed-form expression smoothness constraint analytic expression accurate approximation function interpo-lated throughmean value coordinate green coordinate approximation scheme abstract mean value coordinate thorough analysis additional constraint
