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; ...;rn 1 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 2 i=n, i ¼ 0; 1; 2; ...;n 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 ðn 1Þ n on these geometric transformations. Let’s take an example: H : I;rfII;r fII; ...;r fII ðr fII ¼ IÞ: ð2Þ for a leaf shape , when we rotate it 120 degrees counter- clockwise three times about its bottom point, these three We have proven that the set H is a finite Abelian group of rotation motions constitute a cyclic group C3. But, when we order n under the binary operation “succession” (see the rotate the leaf shape 360=n degrees counterclockwise n times Appendix). about its centroid instead of the bottom end point, these A Cayley diagram is an effective graph to visualize some of n times rotational motions constitute a cyclic group Cn. the structural properties of groups [24]. Fig. 3 illustrates a Note that we are not interested in what kind of shape these Cayley diagram of constructed group H when n ¼ 12.We 12 operations will form in the end. What we are interested in is have 12 vertices and the defining relation is r fII ¼ I. The 1732 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010
Fig. 4. Transformation of shape M. (a) Original shape M. 0 (b) Transformed shape (denoted Mi) after the ith rotation. 00 0 (c) Transformed shape (denoted Mi ) after one reflection from Mi. 0 (d) Transformed shape (denoted Mi) after another reflection.
Fig. 3. Cayley diagram of group H when n ¼ 12. symmetricity is obtained at word wi is defined as the bilateral symmetry axis of the shape M. The number of bilateral vertex I has been selected arbitrarily. Each word representing symmetry axes is denoted by Nb. an element in the group H can be interpreted as a path or a 00 specific sequence of directed segments of the Cayley diagram. When the transformed shape Mi is in congruence with 0 the transformed shape Mi after ith rotation operations, the 3.1.2 Bilateral Central Symmetry Degree and Bilateral shape M becomes perfectly bilaterally symmetric; therefore, Symmetricity the bilateral symmetry axis becomes a perfectly bilateral We introduce the general definitions of bilateral central symmetry axis ( B ¼ 1). NB is used to denote the number of symmetry degree, bilateral symmetricity ( B), and the perfectly bilateral symmetry axes. The number of bilateral N N number of bilateral symmetry axes (Nb). symmetry axes b ( B) of a given shape is counted as the half of the number of the peaks of BCSD curves. Definition 1 (Bilateral central symmetry degree). Let M be a shape in the euclidean space R2. The group H with generators r Remarks. For an irregular shape, bilateral symmetricity B can be used to quantitatively measure the degree of and fII is denoted by gpfr; fIIg. Let W:wi, i ¼ 0; 1; 2; ...;n 0 imperfect symmetry. If the shape is strictly symmetric, 1 be a set of words on all elements of group H. Let Mi be a transformed shape from the original shape M after the ith then B ¼ 1, which corresponds to symmetry of a 0 i 00 dihedral group. Bilateral symmetricity B quantitatively rotation operations, which corresponds to word wi ¼ r . Let Mi 0 characterizes the “amount” of bilateral symmetry pos- be a transformed shape from the shape Mi after one reflection sessed by the arbitrary shape. The combination of B, Nb, operation, which correspondspffiffiffiffiffi to a sequence of operations, 00 i i and NB results in different levels of symmetry: denoted by wi ¼ r fII ¼ r f. Let A be an area. We define the bilateral central symmetry degree, BCSDðiÞ, about the x-axis as 1. Asymmetric: The bilateral symmetricity B ap- being the ratio of the area of the intersection of shape M0 and its i proaches the minimum value, B ¼ð BÞ min .We M00 M reflected shape i to the area of , that is, also have N > 0 ðN ¼ 0Þ. b B 0 00 2. Intermediate: ð BÞmin < B < 1;Nb > 0 ðNB ¼ 0Þ, AðM \ M Þ H ¼ gpfr; fIIg BCSDðiÞ¼ i i ð3Þ the irregular shape is imperfectly symmetric. The AðMÞ word wi;i¼ 0; 1; 2; ...;n 1: symmetry property of these shapes is intermediate between asymmetry and perfect symmetry. Fig. 4 shows an example of the transformations of a leaf 3. Symmetric: ¼ 1; 1 N < 1. The majority of shape to give a pictorial representation of the set of motions B B the regular shapes are within this range. performed in Definition 1. 4. Most perfectly symmetric: The shape is both Definition 2 (Bilateral symmetricity). Let W:wi, i ¼ bilaterally symmetric B ¼ 1 and rotationally 0; 1; 2; ...;n 1 be a set of all words on all elements of symmetric NB !1. Only one type of shape group H. We define the bilateral symmetricity B of a shape M satisfies these two conditions, which is a circle relative to the group H as being the maximum value of the in a plane or a sphere in space. bilateral central symmetry degree with the word wi, Fig. 5 illustrates the relationships between different shapes and their corresponding values of B, Nb, and H ¼ gpfr; fIIg B ¼ maxfBCSDðiÞg ð4Þ NB. For an irregular shape, its bilateral symmetry is word wi;i¼ 0; 1; 2; ...;n 1: mainly characterized by B. Nb plays a less important role in describing the irregularity of the shape. With the The value of the bilateral symmetricity B ranges from 0 increasing value of B, a shape is in the transition from to 1. When the value of B approaches the maximum value of 1, the shape M becomes perfectly bilaterally symmetric. completely irregular shape to the lowest possible regularity ( B ¼ 1). For a regular shape, with the increas- When the value of B approaches the minimum value, the shape M has the lowest bilateral symmetry. The bilateral ing value of NB, the rotational symmetry of the shape is increasing. An algorithm for computing the bilateral symmetricity B is translation, rotation, reflection, and scaling-invariant by definition. symmetricity B can be summarized as follows: Definition 3 (Bilateral symmetry axis). If the word on 1. Rotate M counterclockwise about the centroid by i group H is wi ¼ r fII, the x-axis at which the bilateral i ¼ 2 i=n, i ¼ 0; 1; 2; ..., n. GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1733
Fig. 5. Relationships between different shapes and their corresponding Fig. 7. A typical plot of RCSDðiÞ versus angle i (from 0 to 2 ) and i values of bilateral symmetricity B, the number of bilateral symmetry axis (from 0 to n). NB, and Nb. itself at these angles. For those angles near 0 (or 2 ) radian, 2. Perform a reflection operation about the x-axis as the values of computed rotational central symmetry degree a mirror line through the centroid. RCSD are close to 1. Particularly, the value of RCSD at angle 3. If the index of the operation i The value of the rotational symmetricity R ranges from 0 to 1. For a given shape, the rotational symmetricity is also translation, rotation, reflection, and scaling-invariant by definition. Next, we shall introduce the definition of the rotational symmetry axis and its position. Unlike the Fig. 6. Rotational transformation of shape M. (a) Original shape M. 0 bilateral symmetry axis, the rotational symmetry axis is (b) Transformed shape (denoted Mi) after the ith rotation. 0 (c) Transformed shape (denoted Miþ1) after another rotation. difficult to define in the original shape. However, we will 1734 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 Fig. 9. Relationships between different shapes and their corresponding Fig. 8. (a) An equilateral triangle. (b) The basic angle of rotation j and values of rotational symmetricity R, the number of rotational symmetry its rotational symmetry axis in the Cayley diagram of Cn (n ¼ 12). axis Nr, and NR. show that this problem is easily solved by studying the conditions, which is a circle in a plane or a sphere Cayley diagram of the group Cn. in space. Definition 6 (Rotational symmetry axis). Let W:wi, i ¼ Fig. 9 illustrates the relationships between different 0; 1; 2; ...;n 1 be a set of words on all elements of group Cn. shapes and their corresponding values of R, Nr, and NR. Consider the Cayley diagram of group Cn which is an n-gon For an irregular shape, its rotational symmetry is character- whose sides are directed segments r. If the rotational ized by R. With the increasing value of R, a shape is in the i symmetricity is obtained at word wi ¼ r , then the rotational transition from a completely irregular shape to the lowest symmetry axis is defined as the half line extending from the possible regularity ( R ¼ 1). NR is an important parameter center of the n-gon and passing through vertex wi. to characterize the regular shape. With the increasing value of NR, the rotational symmetry of the regular shape is The number of imperfectly rotational symmetry axes is increasing. An algorithm for computing the rotational denoted by Nr. NR denotes the number of perfectly symmetricity R can be summarized as follows: rotational symmetry axes ( R ¼ 1). Fig. 8 shows an 1. Rotate M counterclockwise about the centroid by equilateral triangle, its basic angle of the rotation j, and ¼ 2 i=n, i ¼ 0; 1; 2; ...;n. its rotational symmetry axes in the Cayley diagram of C . i 12 2. If the index of the operation i n, calculate the For the equilateral triangle shown in Fig. 8a, its rotational RCSD using (5), then go to step 1; otherwise, go to symmetricity is first obtained at word w ¼ w ¼ r4 and its i 4 step 3. basic angle of the rotation j ¼ 4 ¼ 2 4=12 ¼ 2 =3. It can 3. Find the value of l1 and l2, then select RCSD(i) data be seen that three rotational symmetry axes are denoted by at l1 i n l2. three half lines extending from the center of the 12-gon and 4. Compute R using (8). passing through vertex r4, r8, and r12 ¼ I, respectively. The Fig. 10 shows an important schematic illustration of Cayley diagram can be regarded here as a kind of using the bilateral symmetricity and rotational symmetri- “coordinate system” for specifying the rotational symmetry city to characterize the regularity of shapes. For regular axis, the number of rotational symmetry axes, and the basic shape with perfectly bilateral and rotational symmetry, angle of the rotation. The number of rotational symmetry B ¼ R ¼ 1. These shapes correspond to the intersection axes Nr (NR) is counted as the number of peaks obtained in point of the dashed line and dashed dot line in Fig. 10. l i n l RCSD curves at angles l1 n l2 (or 1 2) Points on the dashed line correspond to the shapes with plus 1. B < 1 and R ¼ 1, whereas points on the dashed dot line Remarks. The combination of R, Nr, and NR results in different levels of symmetry: 1. Asymmetric: The rotational symmetricity R approaches the minimum value: R ¼ð RÞmin. We also have Nr > 1 (NR ¼ 1). 2. Intermediate: ð RÞmin < R < 1;Nr > 1 (NR ¼ 1). The irregular shape is imperfectly rotationally symmetric. Most of the irregular shapes have the intermediate symmetry property. 3. Symmetric: R ¼ 1; 2 NR < 1.Theshapeis perfectly rotationally symmetric. The majority of regular shapes which possess the rotational symmetry are within this range. 4. Most perfectly rotationally symmetric: The shape is rotationally symmetric: R ¼ 1 and NR !1. Fig. 10. Schematic illustration of using symmetricity to characterize the Only one type of shape satisfies these two regularity of shapes. GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1735 symmetricity about the center of rotational symmetry xR.Itis denoted Rmax, which satisfies Rmax¼ maxð RÞ: ð10Þ For regular shapes with both strictly bilateral symmetry Fig. 11. (a) An octagon shape and its centroid (“ ”) and center of and rotational symmetry, such as rectangle, square, ellipse, symmetry (“ ”). (b) A breast tumor shape and its centroid and center of circle, regular polygon, etc., Bmax ¼ B ¼ 1 and Rmax ¼ symmetry. R ¼ 1. Note that, in general, the center of bilateral symmetry xB may not be in the same position as the center correspond to the shapes with B ¼ 1 and R < 1. We call of rotational symmetry xR. these two types of shape partially regular shape as they have either perfectly bilateral or perfectly rotational symmetry. 5SYMMETRY-TYPE FACTOR AND SYMMETRY LEVEL For irregular shapes, B < 1 and R < 1, which correspond to points inside the square area in Fig. 10. 5.1 Definition of Symmetry-Type Factor: stf and stfc We find that, for an arbitrary shape, a larger absolute value of does not necessarily mean that the shape is more 4OPTIMIZATION:SEARCHING FOR THE CENTER OF Rmax rotationally symmetric. Similarly, a larger value of Bmax YMMETRY S does not indicate that the shape is more bilaterally The assumption of the proposed symmetry measure is that symmetric. In order to understand what type of symmetry the best imperfect symmetry axis for computing symme- that a given shape possesses, we define the symmetry-type tricity is near the centroid and the x-axis. This is the case for factor as the ratio of Rmax to Bmax, that is: some shapes, especially the shape with high regularity. Definition 11 (Symmetry-type factor). Let M be a shape in However, in general, calculating the symmetry measure euclidean space R2. The symmetry-type factor of M is defined about a point other than centroid will give a different as the ratio of Rmax to Bmax, that is, value of symmetry measure. First, we give the definition of the center of bilateral and rotational symmetry. stf ¼ Rmax= Bmax: ð11Þ Definition 7. Let M be a shape in euclidean space R2. The Similarly, the symmetry-type factor of M can also be defined center of bilateral symmetry of the shape M, denoted xB,is using R and B, which are computed based on the centroid of defined as the point about which bilateral symmetricity B the shape, which is: becomes maximum. stfc ¼ R= B: ð12Þ Definition 8. Let M be a shape in euclidean space R2. The center of rotational symmetry of the shape M, denoted xR,is The definition of the symmetry-type factor enables the defined as the point about which rotational symmetricity R comparison of the degree of bilateral symmetry and becomes maximum. rotational symmetry of an arbitrary shape. It can be used to determine the type of symmetry of an arbitrary shape: Fig. 11 illustrates that the center of the symmetry of the bilateral or rotational symmetry. We have the following octagon shape coincides with its centroid. For the breast observations: tumor shape, the positions of the center of the symmetry and 1. When stf ¼ 1, the given shape has an equal its centroid are different. In order to find the position of the “amount” of bilateral symmetry and rotational center of symmetry, an optimization procedure is required. symmetry ( Rmax ¼ Bmax). The shapes satisfying this Due to the nonlinear and discontinuous nature of the condition have the equilibrium of their bilateral and computing process of symmetricity in our problem, the rotational symmetries. Examples of these shapes are Nelder-Mead simplex method [25] is chosen in this work for equilateral triangle, square, rectangle, regular poly- finding the center of symmetry. Having performed the gon, circle, etc. optimization process, the center of symmetry is obtained 2. When stf < 1, the given shape possesses a larger and corresponding symmetricity value is optimal. The “amount” of bilateral symmetry than rotational concept of maximum symmetricity is defined as follows: symmetry. The shape is more bilaterally symmetric Definition 9 (Maximum bilateral symmetricity). Let M be a than rotationally symmetric. One of the examples shape in euclidean space R2. The maximum bilateral satisfying this condition is the isosceles triangle. symmetricity is defined as the maximum value of bilateral 3. When stf > 1, the given shape possesses a larger “amount” of rotational symmetry than bilateral symmetricity about the center of bilateral symmetry xB.Itis symmetry. The shape is more rotationally symmetric denoted Bmax, which satisfies than bilaterally symmetric. One of the examples Bmax¼ maxð BÞ: ð9Þ satisfying this condition is a parallelogram. 5.2 Definition of Symmetry Level: sl and slc Definition 10 (Maximum rotational symmetricity). Let M We propose the notion of symmetry level sl (slc) as the be a shape in euclidean space R2. The maximum rotational parameter to measure the closeness of an arbitrary shape to symmetricity is defined as the maximum value of rotational the ideal symmetric shape. 1736 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 Fig. 14. Regular shapes. (a) Triangle, NB ¼ NR ¼ 3. (b) Rectangle, N ¼ N ¼ 2. (c) Square, N ¼ N ¼ 4. (d) Pentagon, N ¼ N ¼ 5. Fig. 12. Three celestial body shapes. (a) LkHa101, stf ¼ 0:85, sl ¼ 0:96. B R B R B R (b) ngc278, stf ¼ 1:00, sl ¼ 0:98. (c) ngc234, stf ¼ 1:05, sl ¼ 0:96. words, those BCSD values at 2 are equal to the Definition 12 (Symmetry level). Let M be a shape in euclidean BCSD values at 0 . Therefore, the number of space R2. Symmetry level of M is defined as the maximal value bilateral symmetry axes NB of a given shape is counted as of Bmax and Rmax, that is: half of the number of the peak values of BCSD curves. Where there are peaks at 0 and 2 radians, one peak is sl ¼ maxð Rmax; BmaxÞ: ð13Þ counted. For triangle, rectangle, square, and pentagon Similarly, the symmetry level can also be defined as the shape, there are 6, 4, 8, and 10 peaks in their BCSD curves, respectively. Thus, the values of 3, 2, 4, and 5 are counted as maximal value of R and B, which is: their values of NB, respectively. The maximum peak value slc ¼ maxð R; BÞ: ð14Þ of BCSD gives the value of B of the given shape. Unlike the BCSD curves, RCSD curves for all four shapes produce The greater the value of sl, the closer the shape is to the peaks at angle 0 and 2 . In order to obtain the rotational ideal symmetric shape. When the value of sl approaches the symmetricity R and the number of rotational symmetry maximum value of 1, the shape becomes perfectly sym- axes NR, the discarding procedure described in Section 3.2 metric. The shape possesses either bilateral symmetry or is necessary to remove those pseudo-RCSD data. For rotational symmetry, or both. For example, an isosceles example, in the RCSD curve (Fig. 15b) of the triangle shape, triangle has only perfectly bilateral symmetry, stf < 1, sl ¼ 1. the first minimal RCSD value is obtained at ¼ =3 and the A parallelogram has only perfectly rotational symmetry, that last minimal RCSD value is obtained at ¼ 5 =3. Those is, stf > 1, sl ¼ 1. Fig. 12 shows the comparison of the stf RCSD data whose corresponding angles are 0 < < =3 values (stf < 1, stf ¼ 1, stf > 1) of three celestial body shapes and 5 =3 < <2 are discarded. After performing the with approximately equal symmetry level sl. Fig. 13 shows a discarding procedure, the number of rotational symmetry comparison of the sl values of three celestial bodies with axes NR is counted as the number of the peaks obtained in approximately equal stf values (of 1). In Fig. 13, it can be seen RCSD curves at angles =3 5 =3 plus 1. The max- that, although the symmetry-type factor stf of three shapes is imum peak value of RCSD at angles =3 5 =3 is R of approximately equal to 1, which indicates that each of the three shapes has an equal amount of bilateral and rotational symmetry, their symmetry levels sl exhibit different values. The shape in Fig. 13a has the highest sl value (0.98), indicating that it has the highest degree of symmetry among three shapes in Fig. 13. 6EVALUATION USING SYNTHETIC SHAPES In this section, the concepts developed previously are applied to some concrete examples. First, we consider four shapes, namely: a triangle, a rectangle, a square, and a pentagon. As depicted in Fig. 14, these shapes are both perfectly bilaterally symmetric and rotationally symmetric. We have B ¼ R ¼ 1 and NB ¼ NR. Both the BCSD and RCSD of the triangle shape are computed (n is set as 360) and plotted in Fig. 15 for illustration purpose. The rotation angle in this test ranges from 0 to 2 . It can be seen that BCSD curves have a period of . In other Fig. 13. Comparison of the three celestial bodies along with sl and stf values. (a) ngc278, sl ¼ 0:98, stf ¼ 1:00. (b) ngc362, sl ¼ 0:90, Fig. 15. BCSD and RCSD of triangle shape in Fig. 14. (a) BCSD versus stf ¼ 0:99. (c) ngc288, sl ¼ 0:86, stf ¼ 0:99. for triangle shape. (b) RCSD versus for triangle shape. GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1737 Fig. 17. Plot of bilateral symmetricity and rotational symmetricity values Fig. 16. A set of shapes. of the studied shapes in Figs. 14 and 16. triangle shape. Note that, for regular shapes, NB > 0 and/or rotationally symmetric: B < 1 and R < 1. For each shape, NR > 1. In theory, both B and R should be 1. But, due to we also compute the number of symmetry axes NB, NR, Nb, the computational error, these values are very close to 1. and Nr (Table 1). For each shape in Figs. 14 and 16, the We also study more typical shapes which are not both computed bilateral symmetricity and rotational symmetri- perfectly bilaterally and rotationally symmetric. These city values are plotted in Fig. 17. shapes are depicted in Fig. 16. We compute both bilateral symmetricity B and rotational symmetricity R for each 7APPLICATION TO QUANTITATIVE GALAXY shape using Definitions 2 and 5, as summarized in Table 1. CLASSIFICATION IN ASTRONOMY The first five (Figs. 16a, 16b, 16c, 16d, and 16e) shapes are perfectly bilaterally symmetric but not strictly rotationally Galaxy classification in the universe is one of the major symmetric. It can be seen from Table 1 that these shapes challenges in astronomy. Morphological characterization of have B 1 and R < 1. Shapes in Figs. 16f, 16g, 16h, 16i, the galaxies is the first step toward understanding the and 16j are not perfectly bilaterally symmetric but are physical properties of the galaxies and far depth of the perfectly rotationally symmetric. These five shapes have universe. In the 1920s, Edwin Hubble proposed a “tuning fork” classification system [26], [27] based on the visual B < 1 and R 1. Shapes in Figs. 16k, 16l, 16m, 16n, and 16o are neither perfectly bilaterally symmetric nor perfectly appearance of galaxies. In Hubble’s scheme, galaxies are divided into ellipticals (E) and spirals (unbarred spirals S and barred spirals SB) [28], [29]. Galaxies which do not fit TABLE 1 into the above categories are regarded as irregular (Irr) Symmetricity Values for Shapes in Fig. 16 galaxies. However, Hubble’s tuning fork scheme is only a subjective and qualitative method. It is highly desirable to develop a computer-based objective and quantitative morphological classification method to overcome the limitations of the Hubble scheme. Recently, there have been a number of studies on galaxy classification and morphological characterization [30], [31], [32], [33], [34], [35], [36], [37]. However, most of the previous studies on characterizing the asymmetry of the galaxy either used asymmetry as a crude measure or were restricted to rotational symmetry within the existing framework. More detailed quantitative study on asymmetry of the galaxy is not only necessary for further understanding of the galaxy morphology but also important for possible development of the classification scheme. In this section, we develop a new quantitative galaxy classification framework based on the above-defined symmetry measures. We focus on the shape characterization of the galaxy and do not consider other physical parameters such as color and luminosity. Since our developed method is applicable to the characterization of arbitrary shapes, the term galaxy in this paper is referred to a wide range of visible celestial bodies including galaxy, cluster galaxy, nebula, nebula cluster, etc. Before analyzing the galaxy shape, image segmentation is performed as a preprocessing step to separate the target 1738 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 Fig. 19. Comparison of four spiral galaxy shapes along with computed Fig. 18. Comparisons of the four elliptical galaxies along with parameters. (a) ngc5457, stf ¼ 1:02, sl ¼ 0:79, Nr ¼ 4. (b) ngc986, computed parameters. (a) ngc278, sl ¼ 0:98, stf ¼ 1:00, Nb !1, stf ¼ 1:07, sl ¼ 0:89, Nr ¼ 2. (c) ngc232, stf ¼ 1:09, sl ¼ 0:91, Nr ¼ 2. Nr !1, LSR ¼ 1:01. (b) ic418, sl ¼ 0:98, stf ¼ 1:00, Nb ¼ 2, Nr ¼ 2, (d) ngc210, stf ¼ 1:18, sl ¼ 0:80, Nr ¼ 2. LSR ¼ 1:17. (c) ngc254, sl ¼ 0:97, stf ¼ 0:98, Nb ¼ 2, Nr ¼ 2, LSR ¼ 2:03. (d) ngc3674, sl ¼ 0:97, stf ¼ 1:00, Nb ¼ 2, Nr ¼ 2, LSR ¼ 3:00. studied region to describe the ellipticity and differentiate circular, elliptical, and lenticular shapes. We have the object from the background. We use Otsu’s thresholding following conditions: method [38] to segment the galaxy images and extract the Circular: 1 LSR L ; region of interest. We find that adding a small offset ( 0:2 circle to 0.3) to Otsu’s threshold for some of the images yields a Elliptical: Lcircle< LSR < Lellipse; ð19Þ better segmentation results. Then, we convert the gray-level Lenticular: LSR Lellipse; image to a black-white image using the new revised where L —threshold of LSR, is used to differentiate circle threshold level. Next, we perform a morphological opening circle and ellipse, normally, L ¼ 1:1 1:2; L —threshold of operation [39], [40] on the resulting black-white image to circle ellipse LSR, is used to differentiate ellipse and lenticular shape. remove small objects. This is followed by a flood-filling Lellipse ¼ 2:00 2:60. operation [39], [40] to fill all objects with holes. Fig. 18 shows four examples of elliptical galaxies along 7.1 Criteria for Quantitative Classification of Galaxy with computed parameters: sl, stf, Nb, Nr,LSR.Itis Shape observed that the values of symmetry level sl of all four 7.1.1 Elliptical Galaxy (E) shapes are greater than the threshold of symmetry level Tm defined in (15). The values of symmetry-type factor stf of all In the Hubble sequence, elliptical galaxies include circular, four shapes are within the range of threshold Trb1 and elliptical, and lenticular galaxies. For a galaxy with elliptical Trb2. The most flattened elliptical galaxy in Fig. 18 is the shape, the following conditions are satisfied: shape of Fig. 18d with LSR ¼ 3:00. 1. 7.1.2 Spiral (S) and Barred Spiral (SB) Galaxy sl Tm; ð15Þ A normal spiral galaxy consists of a flattened disk and spiral arms. The central concentration of stars is known as the bulge. Barred spiral galaxy has developed a bar in the where Tm—threshold of symmetry level sl. Empiri- interior region of the spiral arms. For all spiral galaxies, the Tm 0:88-0:91 cally, ¼½ . following condition is satisfied: 2. stf > Trb2; ð20Þ Trb1 stf Trb2; ð16Þ where the threshold Trb2 was introduced in (16). Fig. 19 shows four examples of normal spiral and barred where Trb1, Trb2—threshold of symmetry-type spiral galaxies along with the increasing value of symmetry- factor. Empirically, Trb1 ¼½0:95-0:97 , Trb2 ¼ type factor stf. We also compute and list the value of sl and ½1:00-1:02 . Nr for each galaxy shape in Fig. 19. Note that we only 3. consider the number of rotational symmetry axes since the spiral galaxies are rotationally symmetric. Generally, with N ¼ 2 and N ¼ 2 ðfor elliptical and lenticularÞ; b r the increasing value of stf, the galaxy tends to become a ð17Þ barred spiral. One of the common characteristics of normal spirals and barred spirals is that their central bulge and barred core 4. have much higher brightness than their outstretched arms. This can be used as an image feature to separate the normal Nb !1and Nr !1ðfor circularÞ; ð18Þ spiral galaxies and barred spiral galaxies. The method for differentiating the normal spiral galaxies and barred spiral galaxies consists of two steps: First, we perform the same where Nb—the number of the bilateral symmetry axis thresholding segmentation as described in the beginning of and Nr—the number of the rotational symmetry axis. Section 7 by increasing the amount of threshold offset in Once a celestial body is categorized as elliptical, we use order to remove the arms of the galaxy. We then obtain an the ratio (LSR) of major axis to minor axis of the ellipse that image of the nucleus of the galaxy. Next, various para- has the same normalized second central moments as the meters such as sl, stf, LSR, Nb, and Nr are computed to GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1739 Fig. 20. An example of barred spiral galaxy along with segmented Fig. 22. Two examples of bilateral symmetry galaxy. (a) LkHa101, results and computed parameters. (a) ngc6872 barred spiral, stf ¼ 1:05, stf ¼ 0:85, sl ¼ 0:96, Nb ¼ 1. (b) cfzmcx, stf ¼ 0:63, sl ¼ 0:88, Nb ¼ 1. sl ¼ 0:71, Nr ¼ 2. (b) ngc6872 barred core, stf ¼ 0:94, sl ¼ 0:91, Nr ¼ 2, Nb ¼ 2, LSR ¼ 2:36. symmetry axis (Nb ¼ 1). Galaxy B satisfies the following determine whether the nucleus is a central bulge or barred conditions: core. Fig. 20 illustrates an example of a barred spiral galaxy. Nb ¼ 1; stf < Trb1; and sl a stf þ b ð23Þ 7.1.3 Hubble’s Irregular Galaxy (Irr) where a and b are empirical factors, a ¼½0:18-0:23 , In Hubble’s tuning fork classification system, the classifica- b ¼½0:6-0:8 . The conditions for our proposed irregular tion of the irregular galaxies is qualitative and now galaxy Ir1 change accordingly: considered to be too coarse. In our study, we develop quantitative criteria for classification of irregular galaxy. stf Trb1 and sl < a stf þ b: ð24Þ Using the parameters we proposed, we find that the Fig. 22 shows two example shapes of bilateral symmetry irregular galaxy can be divided into two categories: The celestial bodies. It can be seen that computed parameters first one is the typical irregular galaxy (we call it “Ir1”) satisfy (23). which satisfies the following condition: 7.2.2 New Classification Scheme and Symmetry-Type stf Trb1: ð21Þ Factor (stf) versus Symmetry Level (sl) Diagram Another class of irregular galaxy (we call it “Ir2”) is in We apply the above quantitative analysis to the classifica- the transition area between the spiral galaxy (S and SB) tion of 55 celestial bodies, which includes 19 elliptical and the irregular galaxy Ir1. These galaxies do not satisfy galaxies, 14 spiral and barred spiral galaxies, and 22 irre- the condition (21), but, instead, they satisfy the following gular galaxies. For each celestial body, the symmetry-type conditions: factor stf and symmetry level sl are computed and plotted Trb1 < stf < Trb2 and sl < Tm; ð22Þ in Fig. 23 within Hubble’s scheme. It can be seen that an elliptical galaxy only occupies a small upper rectangle where the Trb1, Trb2, and Tm are described in (15) and region. Irregular galaxies and spiral galaxies are separated (16). by the vertical solid line. On the right-hand side of the Many galaxies in the transition area (Ir2) possess some of vertical solid line are the spiral and barred spiral galaxies. the properties of spiral and barred spiral galaxy, but their Irregular galaxies are in the left region. Clearly, our shapes are very irregular. Fig. 21 shows examples of irregular developed criteria are effective for quantitatively classifying galaxies along with computed parameters stf and sl. Galaxies the 55 galaxies based on the original Hubble’s scheme. in Figs. 21a and 21b belong to the Ir1 galaxy. It can be seen that However, the classification is considered to be coarse due to computed stf values are in good agreement with (21), whereas the qualitative and simple nature of Hubble’s method. galaxies in Figs. 21c and 21d are Ir2 galaxies and their computed stf and sl values satisfy (22). 7.2 A New Classification Scheme of Galaxy Shapes 7.2.1 New Type of Galaxy: Bilateral Symmetry Galaxy (B) Using our proposed criteria, we can easily separate another class of galaxy from the irregular galaxy. We call it the bilateral symmetry galaxy (denoted B) with one bilateral Fig. 23. Symmetry-type factor versus symmetry level diagram for Fig. 21. Four examples of irregular galaxy. (a) ngc899, stf ¼ 0:94, 55 galaxies within Hubble’s scheme. Ellipticals (E) marked by , normal sl ¼ 0:93. (b) ngc246, stf ¼ 0:86, sl ¼ 0:89. (c) ngc346, stf ¼ 1:00, spirals (S) and barred spirals (SB) marked by tu, and irregulars (Irr) sl ¼ 0:83. (d) ngc1952, stf ¼ 0:96, sl ¼ 0:88. marked by . 1740 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 Fig. 25. The relation between B and R and n. Fig. 24. Symmetry-type factor versus symmetry level diagram based on a new classification scheme of galactic shapes. Bilateral and rotational symmetry (BR) marked by , rotational symmetry (R) marked by tu, The proposed classification scheme is not only a quanti- bilateral symmetry (B) marked by , and irregular (Ir) marked by . tative method but also is able to encompass a wider range of celestial body types than the traditional Hubble method. In Fig. 24, we plot the stf versus sl diagram for the 55 galaxies using our new classification scheme described in 8DISCUSSION AND CONCLUSION Section 7.1. We separate the proposed bilateral symmetry galaxy from the original irregular galaxy using an oblique Our proposed imperfect symmetry measures enable char- straight line. The bilateral symmetry galaxy is situated in the acterizing arbitrary shapes in a new way. These measures are based on geometrical operations and contain geometric upper left region. On the left irregular galaxies, a vertical information of the shape, whereas other methods, such as dashed line divides the irregular galaxy into two regions: Ir1 Fourier transform and moment method, lack the capability to and Ir2. The irregular galaxy Ir1 is on the left-hand side of the deal with geometric structure. The proposed methods allow dashed line, whereas the Ir2 galaxy is in the transition region us to quantitatively classify any arbitrary shape ranging from from Ir1 to rotational symmetry galaxy R. In Fig. 24, the regular to irregular shapes, from bilaterally symmetric transition area is the rectanglular region between the vertical shapes to rotationally symmetric shapes. Based on our study, dashed and solid lines and under the area of elliptical we give the following conjectures without proof: galaxies. Using our new criteria, the galaxies can be classified Conjecture 1. For a regular shape, if the values of the bilateral into finer categories in a quantitative fashion. Based on the symmetricity and rotational symmetricity both equal the value developed parameters, we propose a new classification of 1, that is, B ¼ R ¼ 1, then the number of perfectly framework with a view to replacing or improving the bilateral symmetry axes equals the number of perfectly Hubble classification method. In our new scheme, all rotational symmetry axes, that is, NB ¼ NR. galaxies can be classified into four categories, listed as Conjecture 2. For a regular shape, if the value of the bilateral follows: symmetricity B is equal to the value of 1 and the number of perfectly bilateral symmetry axes NB is greater than 1, then 1. Bilateral symmetry and rotational symmetry galaxy the shape is rotationally symmetric, that is, R ¼ 1. (BR). This includes all elliptical galaxies E (circular, elliptical, and lenticular) in Hubble’s scheme as well One issue regarding the error of the symmetricity as those galaxies which satisfy (15)-(18). It also calculation is the choice of the value of n. The value of n includes galaxies with Bmax 1, Rmax 1, Nb ¼ 2, is used to control the amount of increment of the rotation Nr ¼ 2, for example, galaxy ngc6822 can be included angle during the geometric operations. In theory, the higher in this class. the value of n, the more precise the calculation of 2. Rotational symmetry galaxy (R). This includes spiral symmetricity would be. Therefore, n should be sufficiently and barred spiral galaxies in Hubble’s scheme as large. We analyzed the influence of the different settings of well as other galaxies which appear rotationally n on the resulting symmetricity value for the given data. symmetric by visual inspection. Rotational symme- With the increasing number of n, the values of and try galaxy (R) satisfies (20). B R tend to approach a limit value. Fig. 25 shows one example 3. Bilateral symmetry galaxy (B). This is separated from the irregular galaxies in Hubble’s scheme and of the relation of B and R with respect to n. It can be seen includes the galaxies which satisfy (23). that the values of B and R become stable when n>100.In 4. Irregular galaxy (Ir). This includes two types of general, our study shows that the average relative error of irregular galaxies which are typical irregular ga- B due to n is less than 0.8 percent, whereas the average laxies, the Ir1 galaxy satisfying (24) and the transi- relative error of R is less than 0.1 percent when n 200. tional irregular Ir2 galaxy satisfying (22). We designed a set of criteria for the quantitative classification of the galaxy shapes. This leads to the GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION 1741 viewed as the initially optimized position. This can be reflected by the fact that, for most of the regular shape such as regular polygon, parallelogram, circle, ellipse, etc., the center of symmetry (after optimiza- tion) actually coincides with its centroid. The opti- mization process is essentially to find the optimal point about which the given shape becomes the most symmetric. For galaxy shape study, this optimal point is most likely either in the position of the centroid or very close to the centroid. Therefore, we regard the point obtained from optimization process as the global optimum or an approximation of the global optimum in our study. 4. In this study, we used the parameter LSR to differentiate the normal spiral and barred spiral galaxy. However, classifying the central bulge and Fig. 26. Symmetry-type factor stfc versus symmetry level slc diagram barred core is a task subject to further studies. based on Hubble’s scheme. Ellipticals (E) marked by , normal spirals (S) and barred spirals (SB) marked by tu, and irregulars (Irr) 5. It is worth emphasizing that our classification marked by . scheme is based on shape of the segmented galaxy image. If a galaxy is viewed close to edge-on, it is proposal of the new way of classifying galaxy and new intrinsically impossible to determine whether a galaxy is elliptical or spiral on the basis of shape categories. There are a few points worth mentioning: feature alone. Some other information may be 1. Our method is based on the binary image obtained incorporated into the scheme in order to better from the digital image segmentation. The quality of understand the physical property of the galaxy. the segmentation is important for the followed Furthermore, the quality of the original galaxy image characterization and classification of galaxy. Effec- also has an impact on the subsequent segmentation tive segmentation should, in theory, keep the and classification. interested object as much as possible; at the same 6. In future work, we intend to apply our irregular time, the accuracy of the segmentation should be shape symmetry analysis and quantitative criteria to maintained. It is often difficult to achieve both in a larger data set of galaxies or celestial bodies. The practice at the same time. Nevertheless, our study classification scheme proposed in this paper is focuses on the galaxy classification, not the image intended to serve as a framework and foundation segmentation. The segmentation method used in this for future studies. study might not be the optimal one. However, our In summary, we have demonstrated the effectiveness of proposed method did produce satisfactory results based on the current segmentation method. our proposed quantitative criteria for galaxy classification 2. The two parameters stf and sl to characterize the based on proposed irregular shape symmetry measures. shape of the galaxy are based on the maximum Our concepts have also been applied to other irregular symmetricity Bmax and Rmax after optimization shape analyses, such as breast tumor classification. For procedure. For comparison purposes, we also further details, see [47], [48]. The irregular shape measures investigate the use of defined parameter stfc and described here can be extended to 3D shapes. The method, slc which are based on the symmetricity B and R in principle, has the potential to be useful in many other for galaxy classification. Similarly, we compute the areas such as mathematics, artificial intelligence, image stfc and slc for all 55 celestial bodies and plot these in processing, robotics, biomedicine, etc. Fig. 26. It can be seen that the distribution of the different galaxies in the diagram is slightly different from the one in Fig. 24. Although there are three APPENDIX different types of galaxies that can be separated in this diagram, the separability of using stfc and slc is PROOF OF FINITE ABELIAN GROUP H not as good as the one using stf and sl. Therefore, a Let I;r1;r2; ...;rn 1 ðrn ¼ r0 ¼ IÞ represent n-fold counter- comparison study clearly shows the advantage of clockwise rotations of a given shape M through i ¼ 2 i=n, using the parameter stf and sl, in other words, the i ¼ 0; 1; 2; ...;n 1, respectively. The symbol fII denotes a discrimination power of the parameters in the pair of mirror reflection operation of M. We combine every galaxy classification is increased by performing an rotational motion ri of shape M with a pair of reflection optimization procedure. f 3. The performance of our optimization problem de- motions II to form a set pends on the setting of the initial value. In this study, H : I;rf ;r2f ; ...;rðn 1Þf ðrnf ¼ IÞ: the initial value is set as the centroid of the shape, II II II II rather than an arbitrary point. The centroid is the first Proposition 1. Consider the set H: order moment of the given shape. The position of the centroid is the average value of the coordinates of all 2 ðn 1Þ n H : I;rfII;r fII; ...;r fII ðr fII ¼ IÞ: the points in the shape. Therefore, a centroid can be 1742 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 32, NO. 10, OCTOBER 2010 1 1 1 1 1 1 H is a finite Abelian group of order n under the binary x ¼ x I ¼ x ða a Þ¼ðx aÞ a operation defined as “succession” which can be denoted by “ ” ¼ I a 1 ¼ a 1: as follows: According to the definition of group, H is a finite group p q ðpþqÞ r fII r fII ¼ r fII; of order n under the binary operation “succession.” In addition, we have where 8p; q 2 Z, and Z is the set of integers. i1 i2 ði1þi2Þ ði2þi1Þ i2 i1 Proof. Suppose 8a; b; d 2 H: a b ¼ r fII r fII ¼ r fII ¼ r fII ¼ r fII r fII ¼ b a: i1 i2 i4 a ¼ r fII;b¼ r fII;d¼ r fII: Therefore, H is a finite Abelian group of order n under Here i1;i2;i4 2 Z. Let i3 ¼ i1 þ i2, then i3 must be an the binary operation “succession.” tu integer. We express that i3 has a remainder i when divided by n (modulo ¼ n), i ¼ 0; 1; 2; ...;n 1. We have i3 ¼ kn þ i; k 2 Z; ACKNOWLEDGMENTS where k is the number of rotation of 360 degrees of This research has made use of the NASA/IPAC Extra- shape M. i3 > 0 (k 0) indicates the counterclockwise galactic Database (NED), which is operated by the Jet rotations of shape M, and i3 < 0 (k<0) indicates the Propulsion Laboratory, California Institute of Technology, clockwise rotations. under contract with the US National Aeronautics and Space Administration, and the SIMBAD database, operated at i ði þi Þ 1. Closure. Let c ¼ r 3 fII ¼ r 1 2 fII, we have CDS, Strasbourg, France. The authors would like to thank Professor Rangaraj Rangayyan of the University of Calgary, i1 i2 ði1þi2Þ i3 a b ¼ r fII r fII ¼ r fII ¼ r fII ¼ c: Canada, for providing the mammographic tumor contours. They also wish to thank the anonymous referees and the a. k ¼ 0 and 0 ði1þi2Þ i3 a b ¼ r fII ¼ r fII ¼ c; then c 2 H: REFERENCES [1] H. Zabrodsky, S. Peleg, and D. 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