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Anisotropic Green Coordinates

Dong Xiao, Renjie Chen, Bailin Deng

TL;DR

This paper presents anisotropic Green coordinates by embedding a constant positive-definite matrix $\mathbf{A}$ into the anisotropic Laplacian $\nabla\cdot(\mathbf{A}\nabla u)=0$ and deriving a boundary-integral formulation on a cage. The method yields closed-form, linear-reproducing, translation-invariant coordinates on both 2D and 3D cages and offers a geometric interpretation as a quasi-conformal mapping, with gradients and Hessians computed to support variational shape deformation in an ARAP framework. Key contributions include the derivation of 2D/3D closed-form expressions, a detailed discussion of properties (notably quasi-conformality bounded by $\text{cond}(\mathbf{A})$), and extensive experiments demonstrating a broader repertoire of deformation styles beyond isotropic Green coordinates. The practical impact lies in providing artists and engineers with flexible, anisotropy-aware deformation tools that integrate cage-based and variational paradigms while enabling efficient computation through closed-form expressions. A notable limitation is the fixed global $\mathbf{A}$, suggesting future work to learn or optimize $\mathbf{A}$ from deformation constraints to further enhance realism and control.

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.

Anisotropic Green Coordinates

TL;DR

This paper presents anisotropic Green coordinates by embedding a constant positive-definite matrix into the anisotropic Laplacian and deriving a boundary-integral formulation on a cage. The method yields closed-form, linear-reproducing, translation-invariant coordinates on both 2D and 3D cages and offers a geometric interpretation as a quasi-conformal mapping, with gradients and Hessians computed to support variational shape deformation in an ARAP framework. Key contributions include the derivation of 2D/3D closed-form expressions, a detailed discussion of properties (notably quasi-conformality bounded by ), and extensive experiments demonstrating a broader repertoire of deformation styles beyond isotropic Green coordinates. The practical impact lies in providing artists and engineers with flexible, anisotropy-aware deformation tools that integrate cage-based and variational paradigms while enabling efficient computation through closed-form expressions. A notable limitation is the fixed global , suggesting future work to learn or optimize from deformation constraints to further enhance realism and control.

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 , where 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 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.
Paper Structure (18 sections, 2 theorems, 43 equations, 11 figures)

This paper contains 18 sections, 2 theorems, 43 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Cage-based Deformation
  4. Variational Shape Deformation
  5. Anisotropy in Geometry Processing
  6. Anisotropic Green Coordinates
  7. Anisotropic Laplacian Equation
  8. Construction of Anisotropic Green Coordinates
  9. Properties
  10. Derivation of Closed-form Anisotropic Green Coordinates
  11. Closed-form Expressions for 2D Scenario
  12. Closed-form Expressions for 3D Scenario
  13. Discussion
  14. Variational Shape Deformation of Anisotropic Green Coordinates
  15. Experiments
  16. ...and 3 more sections

Key Result

Theorem 1

The fundamental solution of the anisotropic Laplacian Equation $\nabla_{\xi}\cdot(\mathbf{A}\nabla_{\xi} G_{\mathbf{A}}(\xi, \eta))=\delta(\xi-\eta)$ is: where $\delta(\mathbf{x})$ is the Dirac delta distribution satisfying $\delta(\mathbf{x})=0$ for $\mathbf{x} \neq \mathbf{0}$ and $\int_{\mathbb{R}^{d}}{\delta(\mathbf{x})=1}$, and $\omega_{d}={2\pi}^{d/2}/\Gamma(d/2)$ represents the surface are

Figures (11)

  • Figure 1: Illustration of the hat function in 2D and 3D.
  • Figure 2: The illustration of a 2D cage and its mathematical notations.
  • Figure 3: A tetrahedron formed by a triangular face $t_j=\triangle \mathbf{v}_{j1}\mathbf{v}_{j2}\mathbf{v}_{j3}$ of a cage and the query point $\eta$.
  • Figure 4: Cage-based deformation of anisotropic Green coordinate in 2D. Different rotation matrices lead to varying "stretching" effects along different directions.
  • Figure 5: Cage-based deformation results of an image containing circles and cubes. Anisotropic Green coordinates brings diverse deformation effects.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2