The design and analysis of spatial data structures
The design and analysis of spatial data structures
The quickhull algorithm for convex hulls
ACM Transactions on Mathematical Software (TOMS)
C++ Toolbox for Verified Scientific Computing - Theory, Algorithms and Programs: Basic Numerical Problems
Geometric computations with interval and new robust methods: applications in computer graphics, GIS and computational geometry
Verified convex hull and distance computation for octree-encoded objects
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
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In this paper, we discuss approaches to constructing convex polyhedral enclosures of interval-based hierarchical structures. Hierarchical object representations are the data structures most frequently used for reconstructing real scenes. This object modelling does not depend on the nature of a real solid but only on the chosen maximum level of the hierarchical structure. This is a useful property for objects with complex shapes that are difficult to describe via exact mathematical formulas. We focus on reliable object modeling using an interval-based octree data structure. To obtain a convex polyhedral enclosure of an octree, we seek feasible ways to limit the number of considered points. For this purpose, we use the concept of extreme vertices of the tree nodes. Accurate algorithms for constructing the convex hull of these vertices yield a convex polyhedron as an adaptive and reliable object enclosure at each level of the tree.