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Irregular Shape Symmetry Analysis: Theory and Application to Quantitative Galaxy Classification

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Irregular Shape Symmetry Analysis: Theory and Application to Quantitative Galaxy Classification 1730 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 Irregular Shape Symmetry Analysis: Theory and Application to Quantitative Galaxy Classification Qi Guo, Falei Guo, and Jiaqing Shao Abstract—This paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the “amount” of symmetry of arbitrary shapes. The definitions of both bilateral symmetricity and rotational symmetricity give new insight into analyzing the geometrical property of a shape and enable characterizing arbitrary shapes in a new way. We developed a set of criteria for quantitative galaxy classification using our proposed irregular shape symmetry measures. Our study has demonstrated the effectiveness of the proposed method for the characterization of the shape of the celestial bodies. The concepts described in the paper are applicable to many fields, such as mathematics, artificial intelligence, digital image processing, robotics, biomedicine, etc. Index Terms—Bilateral and rotational symmetry, irregularity, symmetry measure, galaxy classification. Ç 1INTRODUCTION YMMETRY is one of the basic features of shapes and objects and accident, the one formal rigidity and constraint, the S[1]. The traditional viewpoint of symmetry is often a other life, play and freedom.” Mathematician Weyl [2] gave binary concept: Either an object is symmetric or it is not at the definition of symmetry as: “symmetry = harmony of all. However, the exact mathematical definition of symme- proportions.” “... symmetric means something like well- try [2], [3] is inadequate to describe and quantify the proportioned, well-balanced, and symmetry denotes that symmetries found in real objects and images. Most objects sort of concordance of several parts by which they integrate have only approximate or imperfect symmetries. To into a whole.” describe the imperfect symmetry quantitatively, one has We argue that one of the basic regularity features of to deal with approximate symmetries. shapes is symmetry. No matter how complicated a In this paper, we propose a set of geometric symmetry geometric shape is, the basic regularity property of the measures which can be used to quantitatively measure the shape in the transition from irregular shape to regular “amount” of imperfect symmetry of arbitrary shapes. This method provides a new way to analyze and characterize the shape is its increasing “amount” of symmetry. Fractals are irregularity of arbitrary shapes. We develop a new mathematical sets with a high degree of geometric complex- quantitative scheme for galaxy classification using our ity. However, many fractals with highly complicated proposed irregular shape symmetry measures. structures in fact possess a small “amount” of regularity, that is, symmetry. Fig. 1 illustrates some examples [5] of fractals. The regularity of these fractals is reflected by one of 2REGULARITY AND SYMMETRY the properties of these shapes, which is bilateral symmetry. Regularity means binding, order, and harmony. For a Basic symmetries are bilateral symmetry (reflection), rota- regular shape, all points in its contour should be arranged tional symmetry, and translational symmetry. For a simple following certain mathematical rules in order that the motif, different combinations of reflection, rotation, and system formed by these points achieves order and harmony translation can form very complicated symmetric shapes of proportions instead of disorder and disruption. Symme- and mathematical groups. try is one of the basic properties of nature. Frey [4] said, A variety of methods for dealing with the symmetry of “symmetry signifies rest and binding, asymmetry motion shape have been developed over the years. In an early and loosening, the one order and law, the other arbitrariness work, Blum and Nagel [6] developed medial axis transform (MAT) for shape description. Brady and Asada [7] intro- duced a representation of planar shape called smoothed . Q. Guo is with the Strangeways Research Laboratory, University of local symmetries. Other medial axis-based methods have Cambridge, Worts Causeway, Cambridge CB1 8RN, UK. E-mail: [email protected]. been developed for shape representation, reconstruction, . F. Guo, PO Box 081, No 216, Luoyang, Henan 471003, China. and shape matching, such as [41], [42], [43], [44], [45], [46]. E-mail: [email protected]. Zabrodsky et al. [1] proposed a symmetry measure for . J. Shao is with the Department of Electronics, University of Kent, Canterbury, Kent CT2 7NT, UK. E-mail: [email protected]. shapes that quantifies the closeness of a given shape and its Manuscript received 17 Mar. 2009; revised 16 Aug. 2009; accepted 21 Sept. symmetric approximation. Heijmans and Tuzikov [8] 2009; published online 6 Jan. 2010. defined symmetry measures for convex sets using Min- Recommended for acceptance by K. Siddiqi. kowski addition and the Brunn-Minkowski inequality. For information on obtaining reprints of this article, please send e-mail to: Reisfeld et al. [10] proposed a symmetrical metric and the [email protected], and reference IEEECS Log Number TPAMI-2009-03-0174. symmetry transform was used as a context-free attention Digital Object Identifier no. 10.1109/TPAMI.2010.13. operator. Studies of symmetry can also be found in medical 0162-8828/10/$26.00 ß 2010 IEEE Published by the IEEE Computer Society GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1731 Fig. 2. Examples of symmetries. (a) C2-symmetry. (b) C3-symmetry. (c) D1-symmetry. (d) D3-symmetry. the measurement of the areas of the intersection of the Fig. 1. (a) Koch curve. (b) Sierpinski gasket. (c) Cantor set [5]. original shape and its transformed shapes during the rotations. The proposed rotational central symmetry degree (RCSD) and rotational symmetricity can be computed applications [22], color images [23], and 3D objects [9], [46]. R from these areas of intersection. Similarly, when we rotate the However, most of the studies are generally either restricted leaf shape n times about its centroid, but, after each rotation, to certain types of symmetry, e.g., [11], [12], [14], [15], [16], we perform a reflection about a mirror line and calculate the [17], [18], [19], [20], or to certain shapes, e.g., [8], [11], [13], area of the intersection of rotated shape and reflected shape, [14], [15]. Some studies focused on finding the symmetry then the leaf shape is reflected again and continues the next axis only, e.g., [11], [12], [21]. Furthermore, symmetry has rotation. The series of rotations followed by two reflection rarely been studied as a regularity attribute of the shapes motions constitute a group which we call H. The proposed themselves, quantitatively and systematically. Our study bilateral central symmetry degree (BCSD) and bilateral focuses on both the bilateral and rotational symmetry as symmetricity B can be computed from these areas of well as on finding the symmetry axis for any arbitrary intersection. Groups H and Cn provide the most suitable shape. Our approach to the symmetry measure is also and rigorous mathematical tool to describe the relationships different from other shape representation methods, e.g., [6], of all of the geometric operations performed. They can also be [7], [41], [42], [43], [44], [45], [46]. Following our earlier work regarded as a kind of “coordinate system” for specifying the [47], [48], we proposed a generalized and systematic position of the individual motion during the transformation method for characterizing the shape irregularity and process. Furthermore, apart from parameters B and R, our quantitatively classifying arbitrary shapes. Using our proposed measures also include the number of bilateral and proposed method, we are able to quantitatively classify rotational symmetry axes Nb and Nr.Itisrelatively different galaxy shapes and define new type of galaxies. straightforward to define the position and the number of The proposed scheme for galaxy classification utilized the bilateral symmetry axes. But defining the rotational symme- information derived from bilateral symmetry, rotational try axis is a difficult problem in the original shape. However, by studying the Cayley diagram of the group C , we are able symmetry, and the number of symmetry axes of the shape. n to define both the position and the number of the rotational symmetry axes. 3SHAPE SYMMETRY ANALYSIS AND CONTINUOUS 3.1 Continuous Bilateral Symmetry Measure MEASURE 3.1.1 Construction of the Abelian Group H For perfect-symmetric shapes, there are two types of rotation-based symmetries, given by Leonardo’s table (see For a given shape M, the origin of the Cartesian coordinate n r [2]) as follows: system is set at its centroid. Let be a positive number, be a counterclockwise rotation of M through 2/n about the C1;C2;C3; ...; z-axis passing through the origin O and perpendicular to ð1Þ the x-O-y plane. Then, I; r; r2; ...;rnÀ1 ðrn ¼ IÞ represents D2;D2;D3; ... n-fold counterclockwise rotations of M about the centroid Fig. 2 shows a few examples of these two types of (origin O) through i ¼ 2i=n, i ¼ 0; 1; 2; ...;nÀ 1, respec- symmetries which correspond to cyclic group (C2 and C3) tively, each of which leaves the system unchanged in form; and dihedral group (D1 and D3). It is clear that a group can be I represents the identity operation. The symbol fII denotes a used to describe the perfect symmetry of a given shape. In pair of mirror reflection operations f of M about the x-axis 2 this paper, we use the concept of group as a mathematic tool as mirror line. Thus, fII ¼ I ð¼ f Þ. We combine every to describe the series of rotation and/or reflection (geo- rotational operation ri of shape M with a pair of reflection metric) motions. The main idea of our work is to derive a set operations fII to form a set H: of imperfect symmetry measures for an arbitrary shape based 2 ðnÀ1Þ n on these geometric transformations.
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