Abstract
This paper deals with the problem of reconstructing shapes from an unorganized set of sample points (called S). First, we give an intuitive notion for gathering sample points in order to reconstruct a shape. Then, we introduce a variant of α-shape [1] which takes into account that the density of the sample points varies from place to place, depending on the required amount of details. The Locally-Density-Adaptive-α-hull (LDA-α-hull) is formally defined and some nice properties are proven. It generates a monotone family of hulls for α ranging from 0 to 1. Afterwards, from LDA-α-hull, we formally define the LDA-α-shape, describing the boundaries of the reconstructed shape, and the LDA-α-complex, describing the shape and its interior. Both describe a monotone family of subgraphs of the Delaunay triangulation of S (called Del(S)). That is, for α varying from 0 to 1, LDA-α-shape (resp. LDA-α-complex) goes from the convex hull of S (resp. Del(S)) to S. These definitions lead to a very simple and efficient algorithm to compute LDA-α-shape and LDA-α-complex in O(n log(n)). Finally, a few meaningful examples show how a shape is reconstructed and underline the stability of the reconstruction in a wide range of α even if the density of the sample points varies from place to place.
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References
Edelsbrunner, H., Kirkpatrick, D., Seidel, R.: On the shape of a set of points in the plane. IEEE Transactions on Information Theory 29(4), 551–559 (1983)
Urquhart, R.: Graph theoretical clustering based on limited neighborhood sets. Pattern Recognition, 173–187 (1992)
Ahuja, N., Tuceryan, M.: Extraction of early perceptual structure in dot patterns: Integrating region, boundary, and component gestalt. Computer Vision, Graphics, and Image Processing 48(3), 304–356 (1989)
Jarvis, R.: Computing the shape hull of points in the plane. In: Pattern Recognition and Image Processing, pp. 231–241 (1977)
Melkemi, M.: A-shapes of a finite point set. In: SCG 1997: Proceedings of the Thirteenth Annual Symposium on Computational Geometry, pp. 367–369. ACM, New York (1997)
Teichmann, M., Capps, M.: Surface reconstruction with anisotropic density-scaled alpha shapes. In: Proceedings of IEEE Visualization, pp. 67–72. IEEE, Los Alamitos (1998)
Dey, T.K., Kumar, P.: A simple provable algorithm for curve reconstruction. In: SGP 2007: Proceedings of the Fifth Eurographics Symposium on Geometry Processing, pp. 893–894 (1999)
Amenta, N., Choi, S., Kolluri, R.K.: The power crust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications 19, 127–153 (2001)
Amenta, N., Bern, M.: Surface reconstruction by voronoi filtering. Discrete and Computational Geometry 22, 481–504 (1999)
Boissonnat, J.D., Oudot, S.: Provably good surface sampling and approximation. In: SGP 2003: Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pp. 9–18. Eurographics Association. Aire-la-Ville (2003)
Chazal, F., Lieutier, A.: Weak feature size and persistent homology: computing homology of solids in rn from noisy data samples. In: SCG 2005: Proceedings of the Twenty-First Annual Symposium on Computational Geometry, pp. 255–262. ACM, New York (2005)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: SCG 2005: Proceedings of the Twenty-First Annual Symposium on Computational Geometry, pp. 263–271. ACM, New York (2005)
Dey, T.K., Li, K., Ramos, E.A., Wenger, R.: Isotopic reconstruction of surfaces with boundaries. Comput. Graph. Forum 28(5), 1371–1382 (2009)
Boissonnat, J.D., Geiger, B.: Three-dimensional reconstruction of complex shapes based on the delaunay triangulation. Research Report 1697, INRIA (1992)
Barequet, G., Goodrich, M.T., Levi-Steiner, A., Steiner, D.: Contour interpolation by straight skeletons. Graph. Models 66(4), 245–260 (2004)
Boissonnat, J.D., Memari, P.: Shape reconstruction from unorganized cross-sections. In: SGP 2007: Proceedings of the fifth Eurographics Symposium on Geometry Processing, pp. 89–98, Eurographics Association, Aire-la-Ville (2007)
Edelsbrunner, H.: Weighted alpha shapes. Technical report, Champaign, IL, USA (1992)
Cazals, F., Giesen, J., Pauly, M., Zomorodian, A.: The conformal alpha shape filtration. The Visual Computer 22(8), 531–540 (2006)
Dey, T.K., Goswami, S.: Provable surface reconstruction from noisy samples. Comput. Geom. Theory Appl. 35(1), 124–141 (2006)
Edelsbrunner, H., Shah, N.R.: Triangulating topological spaces. In: SCG 1994: Proceedings of the tenth annual symposium on Computational geometry, pp. 285–292. ACM, New York (1994)
Boissonnat, J.D., Oudot, S.: Provably good sampling and meshing of lipschitz surfaces. In: SCG 2006: Proceedings of the Twenty-Second Symposium on Computational Geometry, pp. 337–346. ACM, New York (2006)
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Maillot, Y., Adam, B., Melkemi, M. (2010). Shape Reconstruction from Unorganized Set of Points. In: Campilho, A., Kamel, M. (eds) Image Analysis and Recognition. ICIAR 2010. Lecture Notes in Computer Science, vol 6111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13772-3_28
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DOI: https://doi.org/10.1007/978-3-642-13772-3_28
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