Abstract
The aim of this paper is to describe a conceptual model for surface representation based on topological coding, which de.nes a sketch of a surface usable for classi.cation or compression purposes. Theoretical approaches based on di.erential topology and geometry have been used for surface coding, for example Morse theory and Reeb graphs. To use these approaches in discrete geometry, it is necessary to adapt concepts developed for smooth manifolds to discrete surface models, as for example piecewise linear approximations. A typical problem is represented by degenerate critical points, that is non-isolated critical points such as plateaux and .at areas of the surface. Methods proposed in literature either do not consider the problem or propose local adjustments of the surface, which solve the theoretical problem but may lead to a wrong interpretation of the shape by introducing artefacts, which do not correspond to any shape feature. In this paper, an Extended Reeb Graph representation (ERG) is proposed, which can handle degenerate critical points, and an algorithm is presented for its construction.
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References
B. Falcidieno, M. Spagnuolo. Shape Abstraction Paradigm for Modelling Geometry and Semantics. In Proceedings of Computer Graphics International, pp. 646–656, Hannover, June 1998. 185, 186
W. D’Arcy Thompson. On growth and form. MA: University Press, Cambridge, second edition, 1942 185
P. Pentland. Perceptual organization and representation of natural form. Artificial Intelligence, Vol.28, pp. 293–331, 1986. 185
L. R. Nackman. Two-dimensional Critical Point Configuration Graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-6, No. 4, p. 442–450, 1984. 186
Y. Shinagawa, T. L. Kunii, Y. L. Kergosien. Surface Coding Based on Morse Theory. IEEE Computer Graphics & Applications, pp 66–78, September 1991. 186, 187, 189
J. L. Pfaltz. Surface Networks. Geographical Analysis, Vol. 8, pp. 77–93, 1990. 186
C. Bajaj, D. R. Schikore. Topology preserving data simplification with error bounds. Computer & Graphics, 22(1), pp. 3–12, 1998. 186, 189
Y. Shinagawa, T. L. Kunii. Constructing a Reeb graph automatically from cross sections. IEEE Computer Graphics and Applications, 11(6), pp 44–51, 1991. 186, 189
S. Takahashi, T. Ikeda, Y. Shinagawa, T. L. Kunii, M. Ueda. Algorithms for Extracting Correct Critical Points and Construction Topological Graphs from Discrete geographical Elevation Data. Eurographics’ 95, Vol. 14, Number 3, 1995. 186, 189
A. Fomenko. Visual Geometry and Topology. Springer-Verlag, 1994. 186
V. Guillemin, A. Pollack. Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974. 186
J. Milnor. Morse Theory. Princeton University Press, New Jersey, 1963. 186
G. Reeb. Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numérique. Comptes Rendus Acad. Sciences, Paris, 222:847–849, 1946. 187
S. Biasotti. Rappresentazione di superfici naturali mediante grafi di Reeb. Thesis for the Laurea Degree, Department of Mathematics, University of Genova, September 1998. 189, 191
G. Aumann, H. Ebner, L. Tang. Automatic derivation of skeleton lines from digitized contours. ISPRS Journal of Photogrammetry and Remote Sensing, 46, pp. 259–268, 1991. 190, 192
M. De Martino, M. Ferrino. An example of automated shape analysis to solve human perception problems in anthropology. International Journal of Shape Modeling, Vol. 2 No. 1, pp 69–84, 1996. 190, 192
C. Pizzi, M. Spagnuolo. Individuazione di elementi morfologici da curve di livello. Technical Report IMA, No. 1/98, 1998. 190, 191, 192
S. Biasotti, M. Mortara, M. Spagnuolo. Surface Compression and Reconstruction using Reeb graphs and Shape Analysis. Spring Conference on Computer Graphics, Bratislava, May 2000. 194, 196
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Biasotti, S., Falcidieno, B., Spagnuolo, M. (2000). Extended Reeb Graphs for Surface Understanding and Description. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds) Discrete Geometry for Computer Imagery. DGCI 2000. Lecture Notes in Computer Science, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44438-6_16
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