## Closed Object Boundaries from Scattered PointsThis monograph is devoted to computational morphology, particularly to the construction of a two-dimensional or a three-dimensional closed object boundary through a set of points in arbitrary position. By applying techniques from computational geometry and CAGD, new results are developed in four stages of the construction process: (a) the gamma-neighborhood graph for describing the structure of a set of points; (b) an algorithm for constructing a polygonal or polyhedral boundary (based on (a)); (c) the flintstone scheme as a hierarchy for polygonal and polyhedral approximation and localization; (d) and a Bezier-triangle based scheme for the construction of a smooth piecewise cubic boundary. |

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### Contents

I | 1 |

II | 2 |

III | 6 |

V | 9 |

VII | 11 |

VIII | 19 |

IX | 23 |

XII | 26 |

XXXIV | 67 |

XXXV | 71 |

XXXVI | 72 |

XXXVII | 73 |

XL | 81 |

XLI | 89 |

XLII | 90 |

XLIII | 93 |

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### Common terms and phrases

7-Graph 7-indicator 7-neighborhood graph approximation and localization approximation error ball touching Boissonnat boundary construction boundary edge boundary faces boundary triangle bounding area bounding volumes Chapter complexity Computational Geometry constriction algorithm contains control points Convex Hull CQ,CI cubic Bezier curvature curves and surfaces defined definition Delaunay Triangulation deleted denotes disc empty ball Equation example Farin flintstone scheme flintstone tree G1-continuous Gabriel Graph geometric continuity geometric graphs Hamilton cycle hierarchical approximation hyper-graph interpolation intersection iteration l)-simplices Lemma line segments linear macro triangle methods micro triangle edge Minimum Spanning Tree Nearest Neighbors Graph normal vector O(Nv object boundary open polyhedron parameter parameterization parametric continuity piecewise polyhedral polyhedron polyline polynomial problem Q(Nv radius result Section set of vertices smallest sphere spline split surface normal tangent plane continuity tangent vector tetrahedron tion vertex vertices in kD vertices lie Voronoi Diagram Voronoi Skeleton