Abstract
We study the computation of rotation-invariant similarity measures of convex polyhedra, based on Minkowski’s theory of mixed volumes. To compute the similarity measure, a (mixed) volume functional has to be minimized over a number of critical orientations of these polyhedra. These critical orientations are those relative con.gurations where faces and edges of the two polyhedra are as much as possible parallel. Two types of critical orientations exist for two polyhedra A and B. Type-1 critical orientations are those relative orientations where a face of B is parallel to a face of A, and an edge of B is parallel to a face of A, or vice versa. Type-2 critical orientations correspond to the case that three edges of A are parallel to three faces of B, or vice versa. It has been conjectured that to perform minimization of the volume functional, it is suficient to consider Type-1 critical orientations only. Here we present experimental proof showing this conjecture to be false.
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Bekker, H., and Roerdink, J. B. T. M. Calculating critical orientations of polyhedra for similarity measure evaluation. In Proc. 2nd Annual IASTED International Conference on Computer Graphics and Imaging, Palm Springs, California USA, Oct. 25–27(1999), pp. 106–111.
Heijmans, H. J. A. M., and Tuzikov, A. Similarity and symmetry measures for convex shapes using Minkowski addition. IEEE Trans. Patt. Anal. Mach. Intell. 20, 9 (1998), 980–993.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. Numerical Recipes, the Art of Scientific Computing. Cambr. Univ. Press, New York, 1986.
Schneider, R. Convex Bodies. The Brunn-Minkowski Theory. Cambridge University Press, 1993.
Tuzikov, A. V., Roerdink, J. B. T. M., and Heijmans, H. J. A. M. Similarity measures for convex polyhedra based on Minkowski addition. Pattern Recognition 33, 6 (2000), 979–995.
Tuzikov, A. V., and Sheynin, S. A. Minkowski sum volume minimization for convex polyhedra. In Mathematical Morphology and its Applications to Image and Signal Processing, J. Goutsias, L. Vincent, and D. S. Bloomberg, Eds. Kluwer Acad. Publ., Dordrecht, 2000, pp. 33–40.
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Roerdink, J.B.T.M., Bekker, H. (2001). Similarity Measure Computation of Convex Polyhedra Revisited. In: Bertrand, G., Imiya, A., Klette, R. (eds) Digital and Image Geometry. Lecture Notes in Computer Science, vol 2243. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45576-0_23
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DOI: https://doi.org/10.1007/3-540-45576-0_23
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