research-article

Symmetry as a Continuous Feature

Authors:

Hagit Zabrodsky,

Shmuel Peleg, and

David AvnirAuthors Info & Claims

IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume 17, Issue 12

Pages 1154 - 1166

https://doi.org/10.1109/34.476508

Published: 01 December 1995 Publication History

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.

References

[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.

Digital Library

[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. Burt and E.H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Comm., vol. 31, 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.

Digital Library

[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.

Digital Library

[17]

M-K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory, vol. 20, pp. 179-187, Feb. 1962.

[18]

T. Kanade, “Recovery of the three-dimensional shape of an object from asingle view,” Artificial Intelligence, vol. 17, pp. 409-460, 1981.

Digital Library

[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 the Karhunen-Loeve procedure forthe characterization of human faces,” IEEE Trans. Pattern Analysis andMachine Intelligence, vol. 12, no. 1, pp. 103-108, 1990.

Digital Library

[21]

M. Leyton, Symmetry, Causality, Mind. Cambridge, Mass.: MIT Press, 1992.

[22]

G. Marola, “On the detection of the axes of symmetry of symmetric andalmost symmetric planar images,” IEEE Trans. Pattern Analysis and MachineIntelligence, vol. 11, no. 1, pp. 104-108, 1989.

Digital Library

[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.

Digital Library

[25]

F. Mokhatarian and A. Mackworth, “A theory of multiscale, curvature-basedshape representation for planar curves,” IEEE Trans. Pattern Analysis andMachine Intelligence, vol. 14, pp. 789-805, 1992.

Digital Library

[26]

V.S. Nalwa, “Line-drawing interpretation: Bilateral symmetry,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 10, pp. 1,117-1,120, 1989.

Digital Library

[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.

Digital Library

[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.

Digital Library

[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.

Cited By

Hruda LKolingerová IVáša L(2022)Robust, fast and flexible symmetry plane detection based on differentiable symmetry measureThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-020-02034-w38:2(555-571)Online publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1007/s00371-020-02034-w
Moeck P(2019)On Classification Approaches for Crystallographic Symmetries of Noisy 2D Periodic PatternsIEEE Transactions on Nanotechnology10.1109/TNANO.2019.294659718(1166-1173)Online publication date: 18-Nov-2019
https://dl.acm.org/doi/10.1109/TNANO.2019.2946597
Sipiran IVeltkamp RTelea ATheoharis T(2018)Completion of cultural heritage objects with rotational symmetryProceedings of the 11th Eurographics Workshop on 3D Object Retrieval10.5555/3290638.3290653(87-93)Online publication date: 16-Apr-2018
https://dl.acm.org/doi/10.5555/3290638.3290653
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Index Terms

Symmetry as a Continuous Feature
1. Computing methodologies
  1. Computer graphics
    1. Image manipulation
    2. Shape modeling

Index terms have been assigned to the content through auto-classification.

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Published In

cover image IEEE Transactions on Pattern Analysis and Machine Intelligence

IEEE Transactions on Pattern Analysis and Machine Intelligence Volume 17, Issue 12

December 1995

122 pages

ISSN:0162-8828

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Copyright © Copyright © 1995 IEEE. All Rights Reserved.

Publisher

IEEE Computer Society

United States

Publication History

Published: 01 December 1995

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Cited By

Hruda LKolingerová IVáša L(2022)Robust, fast and flexible symmetry plane detection based on differentiable symmetry measureThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-020-02034-w38:2(555-571)Online publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1007/s00371-020-02034-w
Moeck P(2019)On Classification Approaches for Crystallographic Symmetries of Noisy 2D Periodic PatternsIEEE Transactions on Nanotechnology10.1109/TNANO.2019.294659718(1166-1173)Online publication date: 18-Nov-2019
https://dl.acm.org/doi/10.1109/TNANO.2019.2946597
Sipiran IVeltkamp RTelea ATheoharis T(2018)Completion of cultural heritage objects with rotational symmetryProceedings of the 11th Eurographics Workshop on 3D Object Retrieval10.5555/3290638.3290653(87-93)Online publication date: 16-Apr-2018
https://dl.acm.org/doi/10.5555/3290638.3290653
Saha S(2018)Enhancing point symmetry-based distance for data clusteringSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-016-2477-322:2(409-436)Online publication date: 1-Jan-2018
https://dl.acm.org/doi/10.1007/s00500-016-2477-3
Milczarski P(2018)Symmetry of Hue Distribution in the ImagesArtificial Intelligence and Soft Computing10.1007/978-3-319-91262-2_5(48-61)Online publication date: 3-Jun-2018
https://dl.acm.org/doi/10.1007/978-3-319-91262-2_5
Li BJohan HYe YLu Y(2016)Efficient 3D reflection symmetry detectionGraphical Models10.1016/j.gmod.2015.09.00383:C(2-14)Online publication date: 1-Jan-2016
https://dl.acm.org/doi/10.1016/j.gmod.2015.09.003
Adanova VTari S(2016)Beyond symmetry groupsGraphical Models10.1016/j.gmod.2015.09.00183:C(15-27)Online publication date: 1-Jan-2016
https://dl.acm.org/doi/10.1016/j.gmod.2015.09.001
Misztal KTabor J(2016)Ellipticity and Circularity Measuring via Kullback---Leibler DivergenceJournal of Mathematical Imaging and Vision10.1007/s10851-015-0618-455:1(136-150)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1007/s10851-015-0618-4
Vijendra SLaxman S(2015)Symmetry based automatic evolution of clustersComputational Intelligence and Neuroscience10.1155/2015/7962762015(13-13)Online publication date: 1-Jan-2015
https://dl.acm.org/doi/10.1155/2015/796276
Mavridis PSipiran IAndreadis APapaioannou G(2015)Object Completion using k-Sparse OptimizationComputer Graphics Forum10.1111/cgf.1274134:7(13-21)Online publication date: 1-Oct-2015
https://dl.acm.org/doi/10.1111/cgf.12741
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