Convergent geometric estimators with digital volume and surface integrals

Abstract

This paper presents several methods to estimate geometric quantities on subsets of the digital space ℤd. We take an interest both on global geometric quantities like volume and area, and on local geometric quantities like normal and curvatures. All presented methods have the common property to be multigrid convergent, i.e. the estimated quantities tend to their Euclidean counterpart on finer and finer digitizations of (smooth enough) Euclidean shapes. Furthermore, all methods rely on digital integrals, which approach either volume integrals or surface integrals along shape boundary. With such tools, we achieve multigrid convergent estimators of volume, moments and area in ℤd, of normals, curvature and curvature tensor in ℤ2 and ℤ3, and of covariance measure and normals in ℤd even with Hausdorff noise.