A Filter for Minimizing Gaussian Curvature on 3D Triangular Meshes

Minimizing the Gaussian Curvature of triangular meshes can have important applications in 3D computer vision and graphics. However, traditional explicit methods require solving high-order partial differential equations which makes them computationally demanding and impractical in many applications. This paper presents a very fast and efficient adaptive filtering technique termed Gaussian Curvature Filtering (GCF) which optimizes the Gaussian curvature of the triangular meshes through exploiting the properties of developable surfaces. By moving a vertex along its normal direction such that one of its 1-ring neighbors falls onto the vertex's tangent plane, GCF minimizes Gaussian curvature without explicitly computing the Gaussian curvature. A novel multi tangent plane projection strategy is developed to adaptively determine a vertex's moving distance which enables the GCF to achieve Gaussian curvature minimization while preserving important geometric features. We present extensive experiments to demonstrate that GCF outperforms state of the art methods in Gaussian curvature minimization and shape-preserving model smoothing, and that it is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$7\sim 50$</tex> times faster than previous explicit optimization methods.