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Symmetry as a Continuous Feature
December 1995 (vol. 17 no. 12)
pp. 1154-1166

Abstract—Symmetry is treated as a continuous feature and a Continuous Measure of Distance from Symmetry in shapes is defined. The Symmetry Distance (SD) of a shape is defined to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape. This general definition of a symmetry measure enables a comparison of the “amount” of symmetry of different shapes and the “amount” of different symmetries of a single shape. This measure is applicable to any type of symmetry in any dimension. The Symmetry Distance gives rise to a method of reconstructing symmetry of occluded shapes. We extend the method to deal with symmetries of noisy and fuzzy data. Finally, we consider grayscale images as 3D shapes, and use the Symmetry Distance to find the orientation of symmetric objects from their images, and to find locally symmetric regions in images.

[1] H. Alt,K. Mehlhorn,H. Wagener,, and E. Welzl,“Congruence, similarity, and symmetries of geometric objects,” ACM J. Computing, vol. 4, pp. 308-315, 1987.
[2] J.L. Amoros,M.J. Buerger,, and M. Canut de Amoros,The Laue Method.New York: Academic Press, 1975.
[3] M. Atallah,“On symmetry detection,” IEEE Trans. Computers, vol. 34, no. 7, pp. 663-666, 1985.
[4] F. Attneave,“Symmetry information and memory for patterns,” Am. J. Psychology, vol. 68, pp. 209-222, 1955
[5] D. Avnir and A.Y. Meyer,“Quantifying the degree of molecular shapedeformation. A chirality measure,” J. Molecular Structure (Theochem), vol. 94, pp. 211-222, 1991.
[6] J. Bigün,“Recognition of local symmetries in gray value images by harmonicfunctions,” Proc. Int’l Conf. Pattern Recognition, pp. 345-347, 1988.
[7] A. Blake,M. Taylor,, and A. Cox,“Grasping visual symmetry,” Proc. Int’l Conf. Pattern Recognition,Berlin, pp. 724-733, May 1993.
[8] H. Blum and R.N. Nagel,“Shape description using weighted symmetric axisfeatures,” Pattern Recognition, vol. 10, pp. 167-180, 1978.
[9] Y. Bonneh,D. Reisfeld,, and Y. Yeshurun,“Texture discrimination by localgeneralized symmetry,” Proc. Int’l Conf. Pattern Recognition, pp. 461-465,Berlin, May, 1993.
[10] M. Brady and H. Asada,“Smoothed local symmetries and theirimplementation,” Int’l J. Robotics Research, vol. 3, no. 3, pp. 36-61, 1984.
[11] P.J. Burt and E.H. Adelson, “The Laplacian Pyramid as a Compact Image Code,” IEEE Trans. Comm., vol. 31, no. 4, pp. 532-540, 1983.
[12] M.H. DeGroot,“Probability and statistics.”Reading, Mass.: Addison-Wesley, 1975.
[13] G. Gilat,“Chiral coefficient(A measure of the amount of structuralchirality,” J. Phys. A: Math. Gen., vol. 22, p. 545, 1989.
[14] A.D. Gross and T.E. Boult,“Analyzing skewed symmetry,” Int’l J. Computer Vision, vol. 13, no. 1, pp. 91-111, 1994.
[15] B. Grünbaum,“Measures of symmetry for convex sets,” Proc. Symp. Pure Math:Am. Mathematical Soc., vol. 7, pp. 233-270, 1963.
[16] Y. Hel-Or,S. Peleg.,, and D. Avnir,“Characterization of right handed andleft handed shapes,” Computer Vision, Graphics, and Image Processing, vol. 53, no. 2, 1991.
[17] M.K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE Trans. Information Theory, vol. 8, pp. 179-187, 1962.
[18] T. Kanade,“Recovery of the three-dimensional shape of an object from asingle view,” Artificial Intelligence, vol. 17, pp. 409-460, 1981.
[19] M. Kass,A. Witkin,, and D. Terzopoulos,“Snakes: Active contour models,” Int’l J. Computer Vision, vol. 1, pp. 322-332, 1988.
[20] M. Kirby and L. Sirovich,“Application of Karhunen-Loève procedure for the characterization of human faces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 1, pp. 103-108, Jan. 1990.
[21] M. Leyton,Symmetry, Causality, Mind.Cambridge, Mass.: MIT Press, 1992.
[22] G. Marola, "On the Detection of the Axes of Symmetry of Symmetric and Almost Symmetric Planar Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 104-108, 1989.
[23] W. Miller,Symmetry Groups And Their Applications.”London: Academic Press, 1972.
[24] H. Mitsumoto,S. Tamura,K. Okazaki,N. Kajimi,, and Y. Fukui,“3D reconstruction using mirror images based on a plane symmetry recoveringmethod,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 9, pp. 941-946, 1992.
[25] F. Mokhtarian and A.K. Mackworth, “A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 8, pp. 789-805, Aug. 1992.
[26] V.S. Nalwa, “Line-Drawing Interpretation: Bilateral Symmetry,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 10, pp. 1117-1120, Oct. 1989.
[27] W.G. Oh,M. Asada,, and S. Tsuji, “Model-based matching using skewed symmetryinformation,” Proc. Int’l Conf. Pattern Recognition, pp. 1,043-1,045, 1988.
[28] J. Ponce,“On characterizing ribbons and finding skewed symmetries,” Computer Vision, Graphics, and Image Processing, vol. 52, pp. 328-340, 1990.
[29] D. Reisfeld,H. Wolfson,, and Y. Yeshurun, “Robust detection of facialfeatures by generalized symmetry,” Proc. Int’l Conf. Pattern Recognition,Champaign, Ill., pp. 117-120, June 1992.
[30] H. Samet,“The quadtree and related hierarchical data structures,” ACM Computing Surveys, vol. 16, no. 2, pp. 187-260, June 1984.
[31] D. Terzopoulos,A. Witkin,, and M. Kass, “Symmetry seeking models and objectreconstruction,” Int’l J. Computer Vision, vol. 1, pp. 211-221, 1987.
[32] H. Weyl,Symmetry.Princeton, N.J.: Princeton Univ. Press, 1952.
[33] E. Yodogawa,“Symmetropy, An entropy-like measure of visual symmetry,” Perception and Psychophysics, vol. 32, no. 3, pp. 230-240, 1982.
[34] H. Zabrodsky,“Computational aspects of pattern characterization—Continuoussymmetry,” PhD thesis, Hebrew Univ., Jerusalem, Israel, 1993.
[35] H. Zabrodsky and D. Avnir,“Continuous symmetry measures, IV: Chirality,” J. Am. Chemical Soc., vol. 117, pp. 462-473, 1995.
[36] H. Zabrodsky and S. Peleg,“Attentive transmission,” J. Visual Comm. and Image Representation, vol. 1, no. 2, pp. 189-198, Nov. 1990.
[37] H. Zabrodsky,S. Peleg,, and D. Avnir,“Continuous symmetry measures II:Symmetry groups and the tetrahedron,” J. Am. Chemical Soc., vol. 115, pp. 8,278-8,298, 1993.
[38] H. Zabrodsky,S. Peleg,, and D. Avnir,“Symmetry of fuzzy data,” Proc. Int’l Conf. Pattern Recognition,Tel-Aviv, Israel, pp. 499-504, Oct. 1994.
[39] H. Zabrodsky and D. Weinshall,“3D symmetry from 2D data,” Proc. European Conf. Computer Vision,Stockholm, Sweden, May 1994.

Index Terms:
Symmetry, local symmetry, symmetry distance, similarity measure, occlusion, fuzzy shapes, face orientation.
Hagit Zabrodsky, Shmuel Peleg, David Avnir, "Symmetry as a Continuous Feature," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 12, pp. 1154-1166, Dec. 1995, doi:10.1109/34.476508
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