Discrete Geometry for Computer Imagery

A digital discrete hyperplane in Z is defined by a normal vector v, a shift μ, and a thickness θ. The set of thicknesses θ for which the hyperplane is connected is a right unbounded interval of R. Its lower bound, called the connecting thickness of v with shift μ, may be computed by means of the fully subtractive algorithm. A careful study of the behaviour of this algorithm allows us to give exhaustive results about the connectedness of the hyperplane at the connecting thickness in the case μ = 0. We show that it is connected if and only if the sequence of vectors computed by the algorithm reaches in finite time a specific set of vectors which has been shown to be Lebesgue negligible by Kraaikamp & Meester.

[1]  Chandler Davis.  The rotation of eigenvectors by a perturbation , 1963 .

[2]  H. Weyl.  On the Volume of Tubes , 1939 .

[3]  Pierre Alliez,et al.  Eurographics Symposium on Geometry Processing (2007) Voronoi-based Variational Reconstruction of Unoriented Point Sets , 2022 .

[4]  Jacques-Olivier Lachaud,et al.  Voronoi-Based Geometry Estimator for 3D Digital Surfaces , 2014, DGCI.

[5]  Marc Pouget,et al.  Estimating differential quantities using polynomial fitting of osculating jets , 2003, Comput. Aided Geom. Des..

[6]  François de Vieilleville,et al.  Comparison and improvement of tangent estimators on digital curves , 2009, Pattern Recognit..

[7]  Yan Gérard,et al.  Estimation of the Derivatives of a Digital Function with a Convergent Bounded Error , 2011, DGCI.

[8]  Jacques-Olivier Lachaud,et al.  Integral Based Curvature Estimators in Digital Geometry , 2013, DGCI.

[9]  Jacques-Olivier Lachaud,et al.  Curvature estimation along noisy digital contours by approximate global optimization , 2009, Pattern Recognit..

[10]  Jean-Marie Morvan,et al.  Approximation of the Normal Vector Field and the Area of a Smooth Surface , 2004, Discret. Comput. Geom..

[11]  Rémy Malgouyres,et al.  Normals and Curvature Estimation for Digital Surfaces Based on Convolutions , 2008, DGCI.

[12]  Leonidas J. Guibas,et al.  Voronoi-Based Curvature and Feature Estimation from Point Clouds , 2011, IEEE Transactions on Visualization and Computer Graphics.

[13]  Quentin Mérigot,et al.  Détection de structure géométrique dans les nuages de points. (Geometric structure detection in point clouds) , 2009 .

[14]  Jacques-Olivier Lachaud,et al.  Accurate Curvature Estimation along Digital Contours with Maximal Digital Circular Arcs , 2011, IWCIA.