Anisotropic Green Coordinates

TL;DR

This paper introduces anisotropic Green coordinates derived from anisotropic Laplacian equations, enabling versatile, quasi-conformal, and as-rigid-as-possible shape deformations in 2D and 3D for improved spatial manipulation.

Contribution

The paper develops anisotropic Green coordinates with closed-form expressions and boundary integral formulation, extending deformation techniques to anisotropic settings with a geometric interpretation.

Findings

Provides versatile deformation options for 2D and 3D shapes.

Achieves as-rigid-as-possible shape deformation using local-global optimization.

Demonstrates enhanced flexibility for artists in shape manipulation.

Abstract

We live in a world filled with anisotropy, a ubiquitous characteristic of both natural and engineered systems. In this study, we concentrate on space deformation and introduce \textit{anisotropic Green coordinates}, which provide versatile effects for cage-based and variational deformations in both two and three dimensions. The anisotropic Green coordinates are derived from the anisotropic Laplacian equation $\nabla\cdot(\mathbf{A}\nabla u)=0$, where $\mathbf{A}$ is a symmetric positive definite matrix. This equation belongs to the class of constant-coefficient second-order elliptic equations, exhibiting properties analogous to the Laplacian equation but incorporating the matrix $\mathbf{A}$ to characterize anisotropic behavior. Based on this equation, we establish the boundary integral formulation, which is subsequently discretized to derive anisotropic Green coordinates defined on the vertices and normals of oriented simplicial cages. Our method satisfies basic properties such as linear reproduction and translation invariance, and possesses closed-form expressions for both 2D and 3D scenarios. We also give an intuitive geometric interpretation of the approach, demonstrating that our method generates a quasi-conformal mapping. Furthermore, we derive the gradients and Hessians of the deformation coordinates and employ the local-global optimization framework to facilitate variational shape deformation, enabling flexible shape manipulation while achieving as-rigid-as-possible shape deformation. Experimental results demonstrate that anisotropic Green coordinates offer versatile and diverse deformation options, providing artists with enhanced flexibility and introducing a novel perspective on spatial deformation.