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    Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographical Elevation Data
    (Blackwell Science Ltd and the Eurographics Association, 1995) Takahashi, Shigeo; Ikeda, Tetsuya; Shinagawa, Yoshihisa; Kunii, Tosiyasu L.; Ueda, Minoru
    Researchers in the fields of computer graphics and geographical information systems (GISs) have extensively studied the methods of extracting terrain features such as peaks, pits, passes, ridges, and ravines from discrete elevation data. The existing techniques, however, do not guarantee the topological integrity of the extracted features because of their heuristic operations, which results in spurious features. Furthermore, there have been no algorithms for constructing topological graphs such as the surface network and the Reeb graph from the extracted peaks, pits, and passes. This paper presents new algorithms for extracting features and constructing the topological graphs using the features. Our algorithms enable us to extract correct terrain features; i.e., our method extracts the critical points that satisfy the Euler formula, which represents the topological invariant of smooth surfaces. This paper also provides an algorithm that converts the surface network to the Reeb graph for representing contour changes with respect to the height. The discrete elevation data used in this paper is a set of sample points on a terrain surface. Examples are presented to show that the algorithms also appeal to our visual cognition.
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    Function Representation of Solids Reconstructed from Scattered Surface Points and Contours
    (Blackwell Science Ltd and the Eurographics Association, 1995) Savchenko, Vladimir V.; Pasko, Alexander A.; Okunev, Oleg G.; Kunii, Tosiyasu L.
    This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ? 0. The volume spline is based on use of the Green s function for interpolation of scalar function values of a chosen"carrier" solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross-sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.