Anisotropic Green Coordinates

2025-12-23

AI Summary

To address insufficient modeling capability for anisotropic deformations in cage-based deformation, this paper proposes Anisotropic Green Coordinates (AGC). Methodologically, we introduce the anisotropic Laplace equation into coordinate construction for the first time, establishing a variational deformation framework that simultaneously ensures linear reproduction and translation invariance; we derive a closed-form analytical solution under joint vertex and normal constraints, and achieve efficient numerical evaluation via boundary integral formulation and simplex-cage discretization. Our contribution lies in endowing deformations with directional sensitivity, enabling high-fidelity hybrid deformations combining rigidity-preserving and soft behaviors. This significantly enhances diversity, controllability, and geometric intuitiveness in artistic deformation tasks. Experiments demonstrate that AGC outperforms state-of-the-art methods in stability, computational efficiency, and visual quality.

Technology Category

• Computer Vision: Computational Photography, Image & Video Synthesis
• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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 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 $ ablacdot(mathbf{A} abla 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. The deformation satisfies basic properties such as linear reproduction and translation invariance, and has closed-form expressions for both 2D and 3D scenarios. We also offer intuitive geometric interpretations of this method. Furthermore, our approach computes the gradient and Hessian of the deformation coordinates and employs 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.

Problem

• Introduces anisotropic Green coordinates for cage-based deformation
• Derives coordinates from anisotropic Laplacian for 2D and 3D deformations
• Enables variational shape deformation with as-rigid-as-possible manipulation

Innovation

• Anisotropic Green Coordinates derived from anisotropic Laplacian equation
• Boundary integral formulation discretized for cage-based deformation
• Local-global optimization enables as-rigid-as-possible variational shape deformation