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Symmetry Identification of a 3-D Object Represented by Octree
May 1993 (vol. 15 no. 5)
pp. 507-514

An algorithm for identifying symmetry of a 3-D object given by its octree is presented, and the symmetry degree (a measure of object symmetry) is proposed. The algorithm is based on traversals of the octree obtained by the principal axis transform of an input octree. An object can be in an arbitrary position and with arbitrary orientation within the octree space, and a wide range of symmetries represented by groups of proper and improper rotations can be identified. It is shown that the octree data structure supports these operations well, especially for objects whose symmetry types are simpler or equal in complexity with a fourfold rotational symmetry. The operation of the algorithm is illustrated using some synthetic test objects. The results, which are composed of identified symmetry types and the corresponding symmetry degrees, were satisfactory.

[1] Y. S. Abu-Mostafa and D. Psaltis, "Image normalization by complex moments,"IEEE Trans. Patt. Anal. Machine Intel., vol. PAMI-7, pp. 46-55, 1985.
[2] N. Alexandridis and A. Klinger, "Picture decomposition, tree data structures, and identifying directional symmetries as node combination,"Comput. Graphics Image Processing, vol. 8, pp. 43-77, 1978.
[3] M. D. Atkinson, "An optimal algorithm for geometrical congruence,"J. Algorithms, vol. 8, pp. 159-172, 1987.
[4] J. Bigun, "Local symmetry features in image processing," Ph.D. thesis, Linkoeping Univ., Linkoeping, Sweden, 1988, no. 179 (ISBN 91-7870- 334-4).
[5] J. Bigün, "A structure feature for image processing applications based on spiral functions,"Comput. Vision Graphics Image Processing, vol. 51, pp. 166-194, 1990.
[6] C.H. Chien and J.K. Aggarwal, "Identification of 3-D Objects from Multiple Silhouettes Using Quadtrees/Octrees,"Computer Vision, Graphics, and Image Processing, Nov./Dec. 1986, pp. 256-273.
[7] P. Eades, "Symmetry finding algorithms," inComputational Morphology(G. T. Toussaint, Ed.). Amsterdam: North-Holland, 1988, pp. 41-51.
[8] T. G. Evans, "A program for the solution of a class of geometric-analogy intelligence-test questions," inSemantic Information Processing(M. Minsky, Ed.). Cambridge, MA: MIT Press, 1968, pp. 271-353.
[9] S. A. Friedberg, "Finding axes of skewed symmetry,"Comput. Vision Graphics Image Processing, vol. 34, pp. 138-155, 1986.
[10] J. Gips, "A syntax-directed program that performs a three-dimensional task,"Patt. Recogn., vol. 6, pp. 189-199, 1974.
[11] R.C. Gonzalez and P. Wintz,Digital Image Processing, Addison-Wesley, Reading, Mass., 1987.
[12] J. J. Leou and W. H. Tsai, "Automatic rotational symmetry determination for shape analysis,"Patt. Recogn., vol. 20, pp. 571-582, 1987.
[13] C. H. Lo and H. S. Don, "3-D moment forms: their construction and application to object identification and positioning,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, pp. 1053-1064, 1989.
[14] G. Marola, "On the detection of the axes of symmetry of symmetric and almost symmetric planar images,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, pp. 104-108, 1989.
[15] P. Minovic, S. Ishikawa, and K. Kate, "Three-dimensional symmetry identification,"Memoirs Kyushu Inst. Tech. (Eng.), no. 21, pp. 1-38, 1992.
[16] H. Noborio, S. Fukuda, and S. Arimoto, "Construction of the octree approximating three dimensional objects by using multiple views,"IEEE Trans. Patt. Anal. Machine Intell., vol. 10, pp. 769-782, 1988.
[17] H. Samet,Applications of Spatial Data Structures, Addison-Wesley, Reading, Mass., 1989.
[18] S. Srivastava and N. Ahuja, "Octree generation from object silhouettes in perspective views,"Comput. Vision Graphics Image Processing, vol. 49, No. 1, pp. 68-84, 1990.
[19] K. Sugihara, "Annlognalgorithm for determining the congruity of polyhedra,"J. Comput., Syst. Sci., vol. 29, pp. 36-47, 1984.
[20] K. R. Symon,Mechanics. Reading, MA: Addison-Wesley, 1965.
[21] J. Weng and N. Ahuja, "Octrees of Objects in Arbitrary Motion: Representation and Efficiency,"Computer Vision, Graphics, and Image Processing, Aug. 1987, pp. 167-185.
[22] H. Weyl,Symmetry. Princeton, NJ: Princeton Univ. Press, 1952.
[23] J. D. Wolter, T. C. Woo, and R. A. Volz, "Optimal algorithms for symmetry detection in two and three dimensions,"Visual Comput., vol. 1, pp. 37-48, 1985.
[24] M. Yau, "Generating quadtrees of cross sections from octrees,"Comput. Vision Graphics Image Processing, vol. 27, pp. 211-238, 1984.
[25] E. Yodogawa, "Symmetropy, an entropy-like measure of visual symmetry,"Perception Psychophys., vol. 32, no. 3, pp. 230-240, 1982.
[26] L. Zusne, "Measures of symmetry,"Perception Psychophys., vol. 9, no. 38, pp. 363-366, 1971.

Index Terms:
3D object recognition; image recognition; symmetry identification; octree; principal axis transform; complexity; image recognition; trees (mathematics)
P. Minovic, S. Ishikawa, K. Kato, "Symmetry Identification of a 3-D Object Represented by Octree," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 5, pp. 507-514, May 1993, doi:10.1109/34.211472
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