Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

We present, in details, a generic tool to estimate differential geometric quantities on digital shapes, Shapedigital which are subsets of ℤd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}^{d}$$ \end{document}. This tool, called digital integral invariant, Invariantdigital integral simply places a ball at the point of interest, and then examines the intersection of this ball with input data to infer local geometric information. Just counting the number of input points within the intersection provides curvature estimation in 2D and mean curvature estimation in 3D. The covariance matrix of the same point set allows to recover principal curvatures, principal directions and normal direction estimates in 3D. We show the multigrid convergence of all these estimators, which means that their estimations tend toward the exact geometric quantities on—smooth enough—Euclidean shapes digitized with finer and finer gridsteps. During the course of the chapter, we establish several multigrid convergence results of digital volume and moments estimators in arbitrary dimensions. Afterwards, we show how to set automatically the radius parameter while keeping multigrid convergence properties. Our estimators are then demonstrated to be accurate in practice, with extensive comparisons with state-of-the-art methods. To conclude the exposition, we discuss their robustness to perturbations and noise in input data and we show how such estimators can detect features using scale-space arguments.