Dual complexes of cubical subdivisions of ℝn

article Dual complexes of cubical subdivisions of ℝn published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Michael Kerber author 36E4574A-F248-11E8-B48F-1D18A9856A870000-0002-8030-9299 HeEd department We use a distortion to define the dual complex of a cubical subdivision of ℝ n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝ n. https://research-explorer.ista.ac.at/download/3256/4675/IST-2016-543-v1+1_2012-J-08-HierarchyCubeComplex.pdf application/pdfno Springer2012 eng Discrete & Computational Geometry 00029905720001010.1007/s00454-011-9382-4 472393 - 414 Edelsbrunner, Herbert, and Michael Kerber. “Dual Complexes of Cubical Subdivisions of ℝn.” Discrete & Computational Geometry. Springer, 2012. <a href="https://doi.org/10.1007/s00454-011-9382-4">https://doi.org/10.1007/s00454-011-9382-4</a>. Edelsbrunner H, Kerber M. Dual complexes of cubical subdivisions of ℝn. Discrete & Computational Geometry. 2012;47(2):393-414. doi:<a href="https://doi.org/10.1007/s00454-011-9382-4">10.1007/s00454-011-9382-4</a> Edelsbrunner H, Kerber M. 2012. Dual complexes of cubical subdivisions of ℝn. Discrete & Computational Geometry. 47(2), 393–414. Edelsbrunner, Herbert, and Michael Kerber. “Dual Complexes of Cubical Subdivisions of ℝn.” Discrete & Computational Geometry, vol. 47, no. 2, Springer, 2012, pp. 393–414, doi:<a href="https://doi.org/10.1007/s00454-011-9382-4">10.1007/s00454-011-9382-4</a>. Edelsbrunner, H., & Kerber, M. (2012). Dual complexes of cubical subdivisions of ℝn. Discrete & Computational Geometry. Springer. <a href="https://doi.org/10.1007/s00454-011-9382-4">https://doi.org/10.1007/s00454-011-9382-4</a> H. Edelsbrunner, M. Kerber, Discrete & Computational Geometry 47 (2012) 393–414. H. Edelsbrunner and M. Kerber, “Dual complexes of cubical subdivisions of ℝn,” Discrete & Computational Geometry, vol. 47, no. 2. Springer, pp. 393–414, 2012. 32562018-12-11T12:02:17Z2025-09-30T07:43:46Z