In this paper, we present a novel method on surface partition from the perspective of approximation theory. Different from previous shape proxies, the ellipsoidal variance proxy is proposed to penalize the partition results falling into disconnected parts. On its support, the Principle Component Analysis (PCA) based energy is developed for asymptotic cluster aspect ratio and size control. We provide the theoretical explanation on how the minimization of the PCA-based energy leads to the optimal asymptotic behavior for approximation. Moreover, we show the partitions on densely sampled triangular meshes converge to the theoretic expectations. To evaluate the effectiveness of surface approximation, polygonal/triangular surface remeshing results are generated. The experimental results demonstrate the high approximation quality of our method.

Abstract Superpixels have been widely used as a preprocessing step in various computer vision tasks. Spatial compactness and color homogeneity are the two key factors determining the quality of the superpixel representation. In this paper, these two objectives are considered separately and anisotropic superpixels are generated to better adapt to local image content. We develop a unimodular Gaussian generative model to guide the color homogeneity within a superpixel by learning local pixel color variations. It turns out maximizing the log‐likelihood of our generative model is equivalent to solving a Centroidal Voronoi Tessellation (CVT) problem. Moreover, we provide the theoretical guarantee that the CVT result is invariant to affine illumination change, which makes our anisotropic superpixel generation algorithm well suited for image/video analysis in varying illumination environment. The effectiveness of our method in image/video superpixel generation is demonstrated through the comparison with other state‐of‐the‐art methods.