Bias-Variance Tradeoff for Adaptive Surface Meshes

Abstract

This paper presents a novel statistical methodology for exerting control over adaptive surface meshes. The work builds on a recently reported adaptive mesh which uses split and merge operations to control the distribution of planar or quadric surface patches. Hitherto, we have used the target variance of the patch fit residuals as a control criterion. The novelty of the work reported in this paper is to focus on the variance-bias tradeoff that exists between the size of the fitted patches and their associated parameter variances. In particular, we provide an analysis which shows that there is an optimal patch area which minimises the variance in the fitted patch parameters. This area offers the best compromise between the noise-variance, which decreases with increasing area, and the model-bias, which increases in a polynomial manner with area. The computed optimal areas of the local surface patches are used to exert control over the facets of the adaptive mesh. We use a series of split and merge operations to distribute the faces of the mesh so that each resembles as closely as possible its optimal area. In this way the mesh automatically selects its own model-order by adjusting the number of control-points or nodes. We provide experiments on both real and synthetic data. This experimentation demonstrates that our mesh is capable of efficiently representing high curvature surface detail.