Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants


In this chapter, we present in details a generic tool to estimate differential geometric quantities on digital shapes, which are subsets of Zd . This tool, called digital integral invariant, simply places a ball at the point of interest, and then examines the inter- section 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 direc- tion 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 dig- ital 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 ex- tensive 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.

Modern Approaches to Discrete Curvature

Caption: Integral invariant based differential estimators on a digital surface (256^3 OCTAFLOWER shape). From left to right, mean curvature, Gaussian curvature, first principal direction, second principal direction and normal vector field

      author = {Lachaud, Jacques-Olivier and Coeurjolly, David and Levallois, Jérémy},
      booktitle = {Modern Approaches to Discrete Curvature},
      editor = {Laurent Najman, Pascal Romon},
      hal_id = {hal-01576020},
      hal_version = {v1},
      keywords = {Digital geometry ; Curvature Estimation ; digital
      pdf = { copie.pdf},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Mathematics},
      title = {Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants},
      url = {},
      volume = {2184},
      year = {2017}