NDLI: Jacobians and Hessians of mean value coordinates for closed triangular meshes

Content Provider	Springer Nature Link
Author	Thiery, Jean Marc Tierny, Julien Boubekeur, Tamy
Copyright Year	2013
Abstract	Mean value coordinates provide an efficient mechanism for the interpolation of scalar functions defined on orientable domains with a nonconvex boundary. 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 (Ben-Chen, Weber, Gotsman, ACM Trans. Graph.:1–11, 2009) for interactive 3D model deformation.In this paper, we introduce the analytic expressions of the Jacobian and the Hessian of functions interpolated through mean value coordinates. We provide these expressions both for the 2D and 3D case. We also provide a thorough analysis of their degenerate configurations along with accurate approximations of the partial derivatives in these configurations. Extensive numerical experiments show the accuracy of our derivation. In particular, we illustrate the improvements of our formulae over a variety of finite differences 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 reused for further manipulations, which makes our framework directly compatible with existing software supporting mean value coordinates based deformations.
Starting Page	981
Ending Page	995
Page Count	15
File Format	PDF
ISSN	01782789
Journal	The Visual Computer
Volume Number	30
Issue Number	9
e-ISSN	14322315
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-10-17
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cage coordinates Mean value coordinates Constrained interpolation Implicit cage deformation Computer Graphics Computer Science Artificial Intelligence (incl. Robotics) Image Processing and Computer Vision
Content Type	Text
Resource Type	Article
Subject	Computer Graphics and Computer-Aided Design Computer Vision and Pattern Recognition Software

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