Abstract
There are many natural phenomena displayed as polygonal tessellations on curved surfaces, typically found in fruit skin patterns. The paper proposes a method to fit given tessellations with spherical Laguerre Voronoi diagrams. The main target of this paper is fruit skin patterns such as jackfruit and lychee covered by tessellation patterns in which each cell contains a unique spike dot that can be considered as a generator. The problem of estimating the weights is reduced to an optimization problem, and can be solved efficiently. The experiments were done with ideal data and real fruit skin data, which show the validity of the method. We also propose related problems for further studies.
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References
Aloupis, G., Pérez-Rosés, H., Pineda-Villavicencio, G., Taslakian, P., Trinchet-Almaguer, D.: Fitting voronoi diagrams to planar tesselations. In: Lecroq, T., Mouchard, L. (eds.) IWOCA 2013. LNCS, vol. 8288, pp. 349–361. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45278-9_30
Ash, P.F., Bolker, E.D.: Recognizing dirichlet tessellations. Geom. Ded. 19, 175–206 (1985)
Aurenhammer, F.: Power diagram: properties, algorithms, and applications. SIAM J. Comput. 16, 78–96 (1987)
Aurenhammer, F.: A criterion for the affine equivalence of cell complexes in \(\mathbb{R}^d\) and convex polyhedra in \(\mathbb{R}^{d+1}\). Discrete Comput. Geom. 2, 49–64 (1987)
Aurenhammer, F.: Recognising polytopical cell complexes and constructing projection polyhedra. J. Symbolic. Comput. 3, 249–255 (1987)
Aurenhammer, F., Klein, R., Lee, D.T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific Publishing Company, Singapore (2013)
Chaidee, S., Sugihara, K.: Approximation of fruit skin patterns using spherical voronoi diagram. Pattern Anal. Appl. (2016). DOI:10.1007/s10044-016-0534-2
Chaidee, S., Sugihara, K.: Numerical fitting of planar photographic images with spherical voronoi diagram. In: 10th Asian Forum on Graphic Science (2015). DOI:10.13140/RG.2.1.1398.1924
Duan, Q., Kroese, D.P., Brereton, T., Spettl, A., Schmidt, V.: Inverting laguerre tessellations. Comput. J. 57, 1431–1440 (2014)
Evans, D.G., Jones, S.M.: Detecting voronoi (area-of-influence) polygons. Math. Geol. 19, 523–537 (1987)
Hartvigsen, D.: Recognizing voronoi diagrams with linear programming. ORSA. J. Comput. 4, 369–374 (1992)
Honda, H.: Description of cellular patterns by dirichlet domains: the two-dimensional case. J. Theor. Biol. 72, 523–543 (1978)
Honda, H.: Geometrical models for cells in tissues. Int. Rev. Cytol. 81, 191–246 (1983)
Imai, H., Iri, M., Murota, K.: Voronoi diagram in the laguerre geometry and its applications. SIAM J. Comput. 14, 93–105 (1985)
Lautensack, C.: Random Laguerre Tessellations. Dissertation (2007)
Lautensack, C.: Fitting three-dimensional laguerre tessellations to foam structures. J. Appl. Stat. 35, 985–995 (2008)
Liebscher, A.: Laguerre approximztion of random foams. Philos. Mag. 95, 2777–2792 (2015)
Loeb, L.: Space Structures: Their Harmony and Counterpoint. Addison Wesley, Reading (1976)
Lyckegaard, A., Lauridsen, E.M., Ludwig, W., Fonda, R.W., Poulsen, H.F.: On the use of laguerre tessellations for representations of 3D grain structures. Adv. Eng. Mater. 13, 165–170 (2011)
Mach, P., Koehl, P.: An analytical method for computing atomic contact areas in biomolecules. J. Comput. Chem. 34, 105–120 (2013)
Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester (2000)
Schoenberg, F.P., Ferguson, T., Lu, C.: Inverting dirichlet tessellations. Comput. J. 46, 76–83 (2003)
Spettl, A., Breregon, T., Duan, Q., Werz, T., Krill Ill, C.E., Kroese, D.P., Schmidt, V.: Fitting laguerre tessellation approximations to tomographic image data. Philos. Mag. 96, 166–189 (2016). doi:10.1080/14786435.2015.1125540
Sugihara, K.: Three-dimensional convex hull as a fruitful source of diagrams. Theor. Comput. Sci. 235, 325–337 (2000)
Sugihara, K.: Laguerre voronoi diagram on the sphere. J. Geom. Graph. 6, 69–81 (2002)
Suzuki, A., Iri, M.: Approximation of a tessellation of the plane by a voronoi diagram. J. Oper. Res. Soc. Jpn. 29, 69–97 (1986)
Acknowledgments
The first author acknowledges the support of the MIMS Ph.D. Program of the Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, and the DPST of IPST, Ministry of Education, Thailand. This research is partly supported by Grant-in-Aid for Basic Research No. 24360039 of MEXT.
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Chaidee, S., Sugihara, K. (2016). Fitting Spherical Laguerre Voronoi Diagrams to Real-World Tessellations Using Planar Photographic Images. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_7
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