Similarity and Symmetry Measures for Convex Shapes Using Minkowski Addition

This paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. This means, in particular, that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they can be chosen to be invariant under rotations, multiplications, reflections, or the class of affine transformations. It is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle /spl theta/ is a piecewise concave function of /spl theta/. This and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures. Various results obtained in this paper are illustrated by experimental data. Although the paper deals exclusively with the two-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces.

[1]  Stuart A. Friedberg,et al.  Finding axes of skewed symmetry , 1986, Comput. Vis. Graph. Image Process..

[2]  Hiromitsu Yamada,et al.  Detection of partial symmetry using correlation with rotated-reflected images , 1993, Pattern Recognit..

[3]  Lyal B. Harris.  November , 1890, The Hospital.

[4]  H. Heijmans.  Morphological image operators , 1994 .

[5]  Tat-Jen Cham,et al.  Symmetry detection through local skewed symmetries , 1995, Image Vis. Comput..

[6]  P. J. Van Otterloo,et al.  A contour-oriented approach to digital shape analysis , 1988 .

[7]  H.J.A.M. Heijmans,et al.  Similarity and symmetry measures for convex sets based on Minkowski addition , 1996 .

[8]  Farzin Mokhtarian,et al.  Scale-Based Description and Recognition of Planar Curves and Two-Dimensional Shapes , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Hagit Hel-Or,et al.  Symmetry as a Continuous Feature , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Daniel P. Huttenlocher,et al.  Comparing Images Using the Hausdorff Distance , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Charles K. Chui,et al.  Measures of $N$-fold symmetry for convex sets , 1970 .

[12]  Ernst Snapper,et al.  Metric affine geometry , 1973 .

[13]  Pijush K. Ghosh,et al.  A unified computational framework for Minkowski operations , 1993, Comput. Graph..

[14]  B. Abel Devalcourt.  Measures of axial symmetry for ovals , 1966 .

[15]  King-Sun Fu,et al.  Shape Discrimination Using Fourier Descriptors , 1977, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Azriel Rosenfeld,et al.  Computer Vision , 1988, Adv. Comput..

[17]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[18]  金谷 健一.  Group-theoretical methods in image understanding , 1990 .

[19]  Michael Werman,et al.  Similarity and Affine Invariant Distances Between 2D Point Sets , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Phillip H. Schmidt,et al.  Computing volumes of polyhedra , 1986 .

[21]  Ralph Roskies,et al.  Fourier Descriptors for Plane Closed Curves , 1972, IEEE Transactions on Computers.

[22]  Esther M. Arkin,et al.  An efficiently computable metric for comparing polygonal shapes , 1991, SODA '90.

[23]  H. Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , 1957 .

[24]  Rama Chellappa,et al.  Classification of Partial 2-D Shapes Using Fourier Descriptors , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Charles R. Giardina,et al.  Elliptic Fourier features of a closed contour , 1982, Comput. Graph. Image Process..

[26]  Mikhail J. Atallah,et al.  On Symmetry Detection , 1985, IEEE Transactions on Computers.

[27]  Y. J. Tejwani,et al.  Robot vision , 1989, IEEE International Symposium on Circuits and Systems,.

[28]  T ZahnCharles,et al.  Fourier Descriptors for Plane Closed Curves , 1972 .

[29]  Soo-Chang Pei,et al.  Image normalization for pattern recognition , 1995, Image Vis. Comput..

[30]  E. Dubois,et al.  Digital picture processing , 1985, Proceedings of the IEEE.

[31]  B. Grünbaum.  Measures of symmetry for convex sets , 1963 .

[32]  Dmitry Chetverikov,et al.  A multiresolution algorithm for rotation-invariant matching of planar shapes , 1992, Pattern Recognit. Lett..

[33]  Micha Sharir,et al.  Identification of Partially Obscured Objects in Two and Three Dimensions by Matching Noisy Characteristic Curves , 1987 .

[34]  Wen-Hsiang Tsai,et al.  Detection of generalized principal axes in rotationally symmetric shapes , 1991, Pattern Recognit..

[35]  Giovanni Marola,et al.  On the Detection of the Axes of Symmetry of Symmetric and Almost Symmetric Planar Images , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[37]  Jiawei Hong,et al.  Recognize The Similarity Between Shapes Under Affine Transformation , 1988, [1988 Proceedings] Second International Conference on Computer Vision.

[38]  A. V. Shubnikov,et al.  Symmetry in Science and Art , 1974 .