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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
1
S
INTRODUCTION
one of the basic features of shapes and objects
[1]. The traditional viewpoint of symmetry is often a
binary concept: Either an object is symmetric or it is not at
all. However, the exact mathematical definition of symmetry [2], [3] is inadequate to describe and quantify the
symmetries found in real objects and images. Most objects
have only approximate or imperfect symmetries. To
describe the imperfect symmetry quantitatively, one has
to deal with approximate symmetries.
In this paper, we propose a set of geometric symmetry
measures which can be used to quantitatively measure the
“amount” of imperfect symmetry of arbitrary shapes. This
method provides a new way to analyze and characterize the
irregularity of arbitrary shapes. We develop a new
quantitative scheme for galaxy classification using our
proposed irregular shape symmetry measures.
2
YMMETRY is
REGULARITY AND SYMMETRY
Regularity means binding, order, and harmony. For a
regular shape, all points in its contour should be arranged
following certain mathematical rules in order that the
system formed by these points achieves order and harmony
of proportions instead of disorder and disruption. Symmetry is one of the basic properties of nature. Frey [4] said,
“symmetry signifies rest and binding, asymmetry motion
and loosening, the one order and law, the other arbitrariness
. Q. Guo is with the Strangeways Research Laboratory, University of
Cambridge, Worts Causeway, Cambridge CB1 8RN, UK.
E-mail: [email protected].
. F. Guo, PO Box 081, No 216, Luoyang, Henan 471003, China.
E-mail: [email protected].
. J. Shao is with the Department of Electronics, University of Kent,
Canterbury, Kent CT2 7NT, UK. E-mail: [email protected].
Manuscript received 17 Mar. 2009; revised 16 Aug. 2009; accepted 21 Sept.
2009; published online 6 Jan. 2010.
Recommended for acceptance by K. Siddiqi.
For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number
TPAMI-2009-03-0174.
Digital Object Identifier no. 10.1109/TPAMI.2010.13.
0162-8828/10/$26.00 ß 2010 IEEE
and accident, the one formal rigidity and constraint, the
other life, play and freedom.” Mathematician Weyl [2] gave
the definition of symmetry as: “symmetry = harmony of
proportions.” “... symmetric means something like wellproportioned, well-balanced, and symmetry denotes that
sort of concordance of several parts by which they integrate
into a whole.”
We argue that one of the basic regularity features of
shapes is symmetry. No matter how complicated a
geometric shape is, the basic regularity property of the
shape in the transition from irregular shape to regular
shape is its increasing “amount” of symmetry. Fractals are
mathematical sets with a high degree of geometric complexity. However, many fractals with highly complicated
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
the properties of these shapes, which is bilateral symmetry.
Basic symmetries are bilateral symmetry (reflection), rotational symmetry, and translational symmetry. For a simple
motif, different combinations of reflection, rotation, and
translation can form very complicated symmetric shapes
and mathematical groups.
A variety of methods for dealing with the symmetry of
shape have been developed over the years. In an early
work, Blum and Nagel [6] developed medial axis transform
(MAT) for shape description. Brady and Asada [7] introduced a representation of planar shape called smoothed
local symmetries. Other medial axis-based methods have
been developed for shape representation, reconstruction,
and shape matching, such as [41], [42], [43], [44], [45], [46].
Zabrodsky et al. [1] proposed a symmetry measure for
shapes that quantifies the closeness of a given shape and its
symmetric approximation. Heijmans and Tuzikov [8]
defined symmetry measures for convex sets using Minkowski addition and the Brunn-Minkowski inequality.
Reisfeld et al. [10] proposed a symmetrical metric and the
symmetry transform was used as a context-free attention
operator. Studies of symmetry can also be found in medical
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.
Fig. 1. (a) Koch curve. (b) Sierpinski gasket. (c) Cantor set [5].
applications [22], color images [23], and 3D objects [9], [46].
However, most of the studies are generally either restricted
to certain types of symmetry, e.g., [11], [12], [14], [15], [16],
[17], [18], [19], [20], or to certain shapes, e.g., [8], [11], [13],
[14], [15]. Some studies focused on finding the symmetry
axis only, e.g., [11], [12], [21]. Furthermore, symmetry has
rarely been studied as a regularity attribute of the shapes
themselves, quantitatively and systematically. Our study
focuses on both the bilateral and rotational symmetry as
well as on finding the symmetry axis for any arbitrary
shape. Our approach to the symmetry measure is also
different from other shape representation methods, e.g., [6],
[7], [41], [42], [43], [44], [45], [46]. Following our earlier work
[47], [48], we proposed a generalized and systematic
method for characterizing the shape irregularity and
quantitatively classifying arbitrary shapes. Using our
proposed method, we are able to quantitatively classify
different galaxy shapes and define new type of galaxies.
The proposed scheme for galaxy classification utilized the
information derived from bilateral symmetry, rotational
symmetry, and the number of symmetry axes of the shape.
3
SHAPE SYMMETRY ANALYSIS AND CONTINUOUS
MEASURE
For perfect-symmetric shapes, there are two types of
rotation-based symmetries, given by Leonardo’s table (see
[2]) as follows:
C1 ; C2 ; C3 ; . . . ;
D2 ; D2 ; D3 ; . . .
ð1Þ
Fig. 2 shows a few examples of these two types of
symmetries which correspond to cyclic group (C2 and C3 )
and dihedral group (D1 and D3 ). It is clear that a group can be
used to describe the perfect symmetry of a given shape. In
this paper, we use the concept of group as a mathematic tool
to describe the series of rotation and/or reflection (geometric) motions. The main idea of our work is to derive a set
of imperfect symmetry measures for an arbitrary shape based
on these geometric transformations. Let’s take an example:
for a leaf shape
, when we rotate it 120 degrees counterclockwise three times about its bottom point, these three
rotation motions constitute a cyclic group C3 . But, when we
rotate the leaf shape 360=n degrees counterclockwise n times
about its centroid instead of the bottom end point, these
n times rotational motions constitute a cyclic group Cn .
Note that we are not interested in what kind of shape these
operations will form in the end. What we are interested in is
the measurement of the areas of the intersection of the
original shape and its transformed shapes during the
rotations. The proposed rotational central symmetry degree
(RCSD) and rotational symmetricity R can be computed
from these areas of intersection. Similarly, when we rotate the
leaf shape n times about its centroid, but, after each rotation,
we perform a reflection about a mirror line and calculate the
area of the intersection of rotated shape and reflected shape,
then the leaf shape is reflected again and continues the next
rotation. The series of rotations followed by two reflection
motions constitute a group which we call H. The proposed
bilateral central symmetry degree (BCSD) and bilateral
symmetricity B can be computed from these areas of
intersection. Groups H and Cn provide the most suitable
and rigorous mathematical tool to describe the relationships
of all of the geometric operations performed. They can also be
regarded as a kind of “coordinate system” for specifying the
position of the individual motion during the transformation
process. Furthermore, apart from parameters B and R , our
proposed measures also include the number of bilateral and
rotational symmetry axes Nb and Nr . It is relatively
straightforward to define the position and the number of
bilateral symmetry axes. But defining the rotational symmetry axis is a difficult problem in the original shape. However,
by studying the Cayley diagram of the group Cn , we are able
to define both the position and the number of the rotational
symmetry axes.
3.1
Continuous Bilateral Symmetry Measure
3.1.1 Construction of the Abelian Group H
For a given shape M, the origin of the Cartesian coordinate
system is set at its centroid. Let n be a positive number, r
be a counterclockwise rotation of M through 2/n about the
z-axis passing through the origin O and perpendicular to
the x-O-y plane. Then, I; r; r2 ; . . . ; rn1 ðrn ¼ IÞ represents
n-fold counterclockwise rotations of M about the centroid
(origin O) through i ¼ 2i=n, i ¼ 0; 1; 2; . . . ; n 1, respectively, each of which leaves the system unchanged in form;
I represents the identity operation. The symbol fII denotes a
pair of mirror reflection operations f of M about the x-axis
as mirror line. Thus, fII ¼ I ð¼ f 2 Þ. We combine every
rotational operation ri of shape M with a pair of reflection
operations fII to form a set H:
H : I; rfII ; r2 fII ; . . . ; rðn1Þ fII
ðrn fII ¼ IÞ:
ð2Þ
We have proven that the set H is a finite Abelian group of
order n under the binary operation “succession” (see the
Appendix).
A Cayley diagram is an effective graph to visualize some of
the structural properties of groups [24]. Fig. 3 illustrates a
Cayley diagram of constructed group H when n ¼ 12. We
have 12 vertices and the defining relation is r12 fII ¼ I. The
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VOL. 32,
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OCTOBER 2010
Fig. 4. Transformation of shape M. (a) Original shape M.
(b) Transformed shape (denoted M0i ) after the ith rotation.
(c) Transformed shape (denoted M00i ) after one reflection from M0i .
(d) Transformed shape (denoted M0i ) after another reflection.
Fig. 3. Cayley diagram of group H when n ¼ 12.
vertex I has been selected arbitrarily. Each word representing
an element in the group H can be interpreted as a path or a
specific sequence of directed segments of the Cayley diagram.
3.1.2 Bilateral Central Symmetry Degree and Bilateral
Symmetricity
We introduce the general definitions of bilateral central
symmetry degree, bilateral symmetricity (B ), and the
number of bilateral symmetry axes (Nb ).
Definition 1 (Bilateral central symmetry degree). Let M be a
shape in the euclidean space R2 . The group H with generators r
and fII is denoted by gpfr; fII g. Let W : wi , i ¼ 0; 1; 2; . . . ; n 1 be a set of words on all elements of group H. Let M0i be a
transformed shape from the original shape M after the ith
rotation operations, which corresponds to word w0i ¼ ri . Let M00i
be a transformed shape from the shape M0i after one reflection
operation, which corresponds
to a sequence of operations,
pffiffiffiffiffi
denoted by w00i ¼ ri fII ¼ ri f. Let A be an area. We define the
bilateral central symmetry degree, BCSDðiÞ, about the x-axis as
being the ratio of the area of the intersection of shape M0i and its
reflected shape M00i to the area of M, that is,
AðM0i \ M00i Þ H ¼ gpfr; fII g
BCSDðiÞ ¼
word wi ; i ¼ 0; 1; 2; . . . ; n 1: ð3Þ
AðMÞ
Fig. 4 shows an example of the transformations of a leaf
shape to give a pictorial representation of the set of motions
performed in Definition 1.
Definition 2 (Bilateral symmetricity). Let W : wi , i ¼
0; 1; 2; . . . ; n 1 be a set of all words on all elements of
group H. We define the bilateral symmetricity B of a shape M
relative to the group H as being the maximum value of the
bilateral central symmetry degree with the word wi ,
H ¼ gpfr; fII g
ð4Þ
B ¼ maxfBCSDðiÞg
word wi ; i ¼ 0; 1; 2; . . . ; n 1:
The value of the bilateral symmetricity B ranges from 0
to 1. When the value of B approaches the maximum value
of 1, the shape M becomes perfectly bilaterally symmetric.
When the value of B approaches the minimum value, the
shape M has the lowest bilateral symmetry. The bilateral
symmetricity B is translation, rotation, reflection, and
scaling-invariant by definition.
Definition 3 (Bilateral symmetry axis). If the word on
group H is wi ¼ ri fII , the x-axis at which the bilateral
symmetricity is obtained at word wi is defined as the bilateral
symmetry axis of the shape M. The number of bilateral
symmetry axes is denoted by Nb .
When the transformed shape M00i is in congruence with
the transformed shape M0i after ith rotation operations, the
shape M becomes perfectly bilaterally symmetric; therefore,
the bilateral symmetry axis becomes a perfectly bilateral
symmetry axis (B ¼ 1). NB is used to denote the number of
perfectly bilateral symmetry axes. The number of bilateral
symmetry axes Nb (NB ) of a given shape is counted as the
half of the number of the peaks of BCSD curves.
Remarks. For an irregular shape, bilateral symmetricity B
can be used to quantitatively measure the degree of
imperfect symmetry. If the shape is strictly symmetric,
then B ¼ 1, which corresponds to symmetry of a
dihedral group. Bilateral symmetricity B quantitatively
characterizes the “amount” of bilateral symmetry possessed by the arbitrary shape. The combination of B , Nb ,
and NB results in different levels of symmetry:
Asymmetric: The bilateral symmetricity B approaches the minimum value, B ¼ ðB Þ min . We
also have Nb > 0 ðNB ¼ 0Þ.
2. Intermediate: ðB Þmin < B < 1; Nb > 0 ðNB ¼ 0Þ,
the irregular shape is imperfectly symmetric. The
symmetry property of these shapes is intermediate
between asymmetry and perfect symmetry.
3. Symmetric: B ¼ 1; 1 NB < 1. The majority of
the regular shapes are within this range.
4. Most perfectly symmetric: The shape is both
bilaterally symmetric B ¼ 1 and rotationally
symmetric NB ! 1. Only one type of shape
satisfies these two conditions, which is a circle
in a plane or a sphere in space.
Fig. 5 illustrates the relationships between different
shapes and their corresponding values of B , Nb , and
NB . For an irregular shape, its bilateral symmetry is
mainly characterized by B . Nb plays a less important role
in describing the irregularity of the shape. With the
increasing value of B , a shape is in the transition from
completely irregular shape to the lowest possible
regularity (B ¼ 1). For a regular shape, with the increasing value of NB , the rotational symmetry of the shape is
increasing. An algorithm for computing the bilateral
symmetricity B can be summarized as follows:
1.
1.
Rotate M counterclockwise about the centroid by
i ¼ 2i=n, i ¼ 0; 1; 2; . . . , n.
GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION
Fig. 5. Relationships between different shapes and their corresponding
values of bilateral symmetricity B , the number of bilateral symmetry axis
NB , and Nb .
2.
3.
4.
Perform a reflection operation about the x-axis as
a mirror line through the centroid.
If the index of the operation i < n, calculate the
BCSD using (3). Then, perform a reflection
operation about the x-axis and go to step 1.
Otherwise, go to step 4.
Calculate B using (4).
3.2
Rotational Central Symmetry Degree and
Rotational Symmetricity
For a given shape M, a cyclic group of order n is constructed
based on the sequence of rotation operations, denoted as
Cn : I; r; r2 ; . . . ; rn1 ðrn ¼ IÞ.
Definition 4 (Rotational central symmetry degree). Let M
be a shape in the euclidean space R2 , M0i be a transformed shape
from the M : M0i ¼ Cn M. Let W : wi , i ¼ 0; 1; 2; . . . ; n 1
be a set of words on all elements of group Cn . The group Cn
with generators r is denoted by gpfrg. We define the rotational
central symmetry degree, RCSDðiÞ, as being the ratio of the
area of the intersection of shape M and its transformed
shape M0i to the area of M:
AðM \ M0i Þ Cn ¼ gpfrg
RCSDðiÞ ¼
ð5Þ
AðMÞ word wi ; i ¼ 0; 1; 2; . . . ; n 1:
Fig. 6 shows an example of the rotational transformations
of a leaf shape.
For mathematical convenience, rotation angle 2, that is
i ¼ n, is included in the following work. In order to define the
rotational symmetricity, the following discarding procedure
is required. For any arbitrary shape M, as the rotation angle i
varies from 0 to 2, its transformed shape M0i will be in
congruence with the original shape M at least once (e.g., at
the initial position when 0 ¼ 0 and n ¼ 2). That is, the
rotation operations bring the shape M into coincidence with
Fig. 6. Rotational transformation of shape M. (a) Original shape M.
(b) Transformed shape (denoted M0i ) after the ith rotation.
(c) Transformed shape (denoted M0iþ1 ) after another rotation.
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Fig. 7. A typical plot of RCSDðiÞ versus angle i (from 0 to 2) and i
(from 0 to n).
itself at these angles. For those angles near 0 (or 2) radian,
the values of computed rotational central symmetry degree
RCSD are close to 1. Particularly, the value of RCSD at angle
0 (or 2) is 1. We have RCSDð0Þ ¼ RCSDðnÞ ¼ 1. In terms of
finding the maximum value of RCSD, those RCSD data at
and near angle 0 (or 2) are pseudo-data. Therefore, we
need to discard the RCSD data both at and near 0 ¼ 0 and
n ¼ 2. The discarding procedure is illustrated as follows
(see Fig. 7):
1.
Find the value of i which gives the first minimal
RCSD value when i is increasing from 0. Noted as l1 :
l1 ¼ argfmin½RCSDðiÞji¼l1 g:
2.
Find the value of i which gives the first minimal
RCSD value when i is decreasing from n. Noted as
n l2 :
n l2 ¼ argfmin½RCSDðiÞji¼nl2 g:
3.
4.
ð6Þ
ð7Þ
Discard the data close to 0 ¼ 0, which are the
RCSDðiÞ values at i ¼ 0; 1; 2; . . . ; l1 1. Discard the
data close to n ¼ 2, which are the RCSDðiÞ values
at i ¼ n l2 þ 1; n l2 þ 2; . . . ; n.
Obtain RCSDðiÞ data at l1 i n l2 (see Fig. 7).
Definition 5 (Rotational symmetricity). Let W : wi , i ¼
0; 1; 2; . . . ; n 1 be a set of words on all elements of group Cn .
Perform the discarding procedure and find the value of l1 and
l2 . We define the rotational symmetricity R of the shape M
relative to the group Cn as being the maximum value of the
rotational central symmetry degree with word wi ,
C ¼ gpfrg
ð8Þ
R ¼ maxfRCSDðiÞg n
word wi ; l1 i n l2 :
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
bilateral symmetry axis, the rotational symmetry axis is
difficult to define in the original shape. However, we will
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Fig. 8. (a) An equilateral triangle. (b) The basic angle of rotation j and
its rotational symmetry axis in the Cayley diagram of Cn (n ¼ 12).
Fig. 9. Relationships between different shapes and their corresponding
values of rotational symmetricity R , the number of rotational symmetry
axis Nr , and NR .
show that this problem is easily solved by studying the
Cayley diagram of the group Cn .
conditions, which is a circle in a plane or a sphere
in space.
Definition 6 (Rotational symmetry axis). Let W : wi , i ¼
0; 1; 2; . . . ; n 1 be a set of words on all elements of group Cn .
Consider the Cayley diagram of group Cn which is an n-gon
whose sides are directed segments r. If the rotational
symmetricity is obtained at word wi ¼ ri , then the rotational
symmetry axis is defined as the half line extending from the
center of the n-gon and passing through vertex wi .
Fig. 9 illustrates the relationships between different
shapes and their corresponding values of R , Nr , and NR .
For an irregular shape, its rotational symmetry is characterized by R . With the increasing value of R , a shape is in the
transition from a completely irregular shape to the lowest
possible regularity (R ¼ 1). NR is an important parameter
to characterize the regular shape. With the increasing value
of NR , the rotational symmetry of the regular shape is
increasing. An algorithm for computing the rotational
symmetricity R can be summarized as follows:
The number of imperfectly rotational symmetry axes is
denoted by Nr . NR denotes the number of perfectly
rotational symmetry axes (R ¼ 1). Fig. 8 shows an
equilateral triangle, its basic angle of the rotation j , and
its rotational symmetry axes in the Cayley diagram of C12 .
For the equilateral triangle shown in Fig. 8a, its rotational
symmetricity is first obtained at word wi ¼ w4 ¼ r4 and its
basic angle of the rotation j ¼ 4 ¼ 24=12 ¼ 2=3. It can
be seen that three rotational symmetry axes are denoted by
three half lines extending from the center of the 12-gon and
passing through vertex r4 , r8 , and r12 ¼ I, respectively. The
Cayley diagram can be regarded here as a kind of
“coordinate system” for specifying the rotational symmetry
axis, the number of rotational symmetry axes, and the basic
angle of the rotation. The number of rotational symmetry
axes Nr (NR ) is counted as the number of peaks obtained in
RCSD curves at angles l1 nl2 (or l1 i n l2 )
plus 1.
Rotate M counterclockwise about the centroid by
i ¼ 2i=n, i ¼ 0; 1; 2; . . . ; n.
2. If the index of the operation i n, calculate the
RCSD using (5), then go to step 1; otherwise, go to
step 3.
3. Find the value of l1 and l2 , then select RCSD(i) data
at l1 i n l2 .
4. Compute R using (8).
Fig. 10 shows an important schematic illustration of
using the bilateral symmetricity and rotational symmetricity to characterize the regularity of shapes. For regular
shape with perfectly bilateral and rotational symmetry,
B ¼ R ¼ 1. These shapes correspond to the intersection
point of the dashed line and dashed dot line in Fig. 10.
Points on the dashed line correspond to the shapes with
B < 1 and R ¼ 1, whereas points on the dashed dot line
1.
Remarks. The combination of R , Nr , and NR results in
different levels of symmetry:
1.
2.
3.
4.
Asymmetric: The rotational symmetricity R
approaches the minimum value: R ¼ ðR Þmin .
We also have Nr > 1 (NR ¼ 1).
Intermediate: ðR Þmin < R < 1; Nr > 1 (NR ¼ 1).
The irregular shape is imperfectly rotationally
symmetric. Most of the irregular shapes have the
intermediate symmetry property.
Symmetric: R ¼ 1; 2 NR < 1. The shape is
perfectly rotationally symmetric. The majority of
regular shapes which possess the rotational
symmetry are within this range.
Most perfectly rotationally symmetric: The shape
is rotationally symmetric: R ¼ 1 and NR ! 1.
Only one type of shape satisfies these two
Fig. 10. Schematic illustration of using symmetricity to characterize the
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 . It is
denoted Rmax , which satisfies
Rmax ¼ maxðR Þ:
Fig. 11. (a) An octagon shape and its centroid (“”) and center of
symmetry (“”). (b) A breast tumor shape and its centroid and center of
symmetry.
correspond to the shapes with B ¼ 1 and R < 1. We call
these two types of shape partially regular shape as they have
either perfectly bilateral or perfectly rotational symmetry.
For irregular shapes, B < 1 and R < 1, which correspond
to points inside the square area in Fig. 10.
4
OPTIMIZATION: SEARCHING FOR THE CENTER OF
SYMMETRY
The assumption of the proposed symmetry measure is that
the best imperfect symmetry axis for computing symmetricity is near the centroid and the x-axis. This is the case for
some shapes, especially the shape with high regularity.
However, in general, calculating the symmetry measure
about a point other than centroid will give a different
value of symmetry measure. First, we give the definition of
the center of bilateral and rotational symmetry.
2
Definition 7. Let M be a shape in euclidean space R . The
center of bilateral symmetry of the shape M, denoted xB , is
defined as the point about which bilateral symmetricity B
becomes maximum.
Definition 8. Let M be a shape in euclidean space R2 . The
center of rotational symmetry of the shape M, denoted xR , is
defined as the point about which rotational symmetricity R
becomes maximum.
Fig. 11 illustrates that the center of the symmetry of the
octagon shape coincides with its centroid. For the breast
tumor shape, the positions of the center of the symmetry and
its centroid are different. In order to find the position of the
center of symmetry, an optimization procedure is required.
Due to the nonlinear and discontinuous nature of the
computing process of symmetricity in our problem, the
Nelder-Mead simplex method [25] is chosen in this work for
finding the center of symmetry. Having performed the
optimization process, the center of symmetry is obtained
and corresponding symmetricity value is optimal. The
concept of maximum symmetricity is defined as follows:
ð10Þ
For regular shapes with both strictly bilateral symmetry
and rotational symmetry, such as rectangle, square, ellipse,
circle, regular polygon, etc., Bmax ¼ B ¼ 1 and Rmax ¼
R ¼ 1. Note that, in general, the center of bilateral
symmetry xB may not be in the same position as the center
of rotational symmetry xR .
5
SYMMETRY-TYPE FACTOR AND SYMMETRY LEVEL
5.1 Definition of Symmetry-Type Factor: stf and stfc
We find that, for an arbitrary shape, a larger absolute value
of Rmax does not necessarily mean that the shape is more
rotationally symmetric. Similarly, a larger value of Bmax
does not indicate that the shape is more bilaterally
symmetric. In order to understand what type of symmetry
that a given shape possesses, we define the symmetry-type
factor as the ratio of Rmax to Bmax , that is:
Definition 11 (Symmetry-type factor). Let M be a shape in
euclidean space R2 . The symmetry-type factor of M is defined
as the ratio of Rmax to Bmax , that is,
stf ¼ Rmax =Bmax :
ð11Þ
Similarly, the symmetry-type factor of M can also be defined
using R and B , which are computed based on the centroid of
the shape, which is:
stfc ¼ R =B :
ð12Þ
The definition of the symmetry-type factor enables the
comparison of the degree of bilateral symmetry and
rotational symmetry of an arbitrary shape. It can be used
to determine the type of symmetry of an arbitrary shape:
bilateral or rotational symmetry. We have the following
observations:
ð9Þ
When stf ¼ 1, the given shape has an equal
“amount” of bilateral symmetry and rotational
symmetry (Rmax ¼ Bmax ). The shapes satisfying this
condition have the equilibrium of their bilateral and
rotational symmetries. Examples of these shapes are
equilateral triangle, square, rectangle, regular polygon, circle, etc.
When stf < 1, the given shape possesses a larger
“amount” of bilateral symmetry than rotational
symmetry. The shape is more bilaterally symmetric
than rotationally symmetric. One of the examples
satisfying this condition is the isosceles triangle.
When stf > 1, the given shape possesses a larger
“amount” of rotational symmetry than bilateral
symmetry. The shape is more rotationally symmetric
than bilaterally symmetric. One of the examples
satisfying this condition is a parallelogram.
Definition 10 (Maximum rotational symmetricity). Let M
be a shape in euclidean space R2 . The maximum rotational
symmetricity is defined as the maximum value of rotational
5.2 Definition of Symmetry Level: sl and slc
We propose the notion of symmetry level sl (slc) as the
parameter to measure the closeness of an arbitrary shape to
the ideal symmetric shape.
Definition 9 (Maximum bilateral symmetricity). Let M be a
shape in euclidean space R2 . The maximum bilateral
symmetricity is defined as the maximum value of bilateral
symmetricity about the center of bilateral symmetry xB . It is
denoted Bmax , which satisfies
Bmax ¼ maxðB Þ:
1.
2.
3.
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Fig. 12. Three celestial body shapes. (a) LkHa101, stf ¼ 0:85, sl ¼ 0:96.
(b) ngc278, stf ¼ 1:00, sl ¼ 0:98. (c) ngc234, stf ¼ 1:05, sl ¼ 0:96.
Definition 12 (Symmetry level). Let M be a shape in euclidean
space R2 . Symmetry level of M is defined as the maximal value
of Bmax and Rmax , that is:
sl ¼ maxðRmax ; Bmax Þ:
ð13Þ
Similarly, the symmetry level can also be defined as the
maximal value of R and B , which is:
slc ¼ maxðR ; B Þ:
ð14Þ
The greater the value of sl, the closer the shape is to the
ideal symmetric shape. When the value of sl approaches the
maximum value of 1, the shape becomes perfectly symmetric. The shape possesses either bilateral symmetry or
rotational symmetry, or both. For example, an isosceles
triangle has only perfectly bilateral symmetry, stf < 1, sl ¼ 1.
A parallelogram has only perfectly rotational symmetry, that
is, stf > 1, sl ¼ 1. Fig. 12 shows the comparison of the stf
values (stf < 1, stf ¼ 1, stf > 1) of three celestial body shapes
with approximately equal symmetry level sl. Fig. 13 shows a
comparison of the sl values of three celestial bodies with
approximately equal stf values (of 1). In Fig. 13, it can be seen
that, although the symmetry-type factor stf of three shapes is
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.
6
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Fig. 14. Regular shapes. (a) Triangle, NB ¼ NR ¼ 3. (b) Rectangle,
NB ¼ NR ¼ 2. (c) Square, NB ¼ NR ¼ 4. (d) Pentagon, NB ¼ NR ¼ 5.
words, those BCSD values at 2 are equal to the
BCSD values at 0 . Therefore, the number of
bilateral symmetry axes NB of a given shape is counted as
half of the number of the peak values of BCSD curves.
Where there are peaks at 0 and 2 radians, one peak is
counted. For triangle, rectangle, square, and pentagon
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
their values of NB , respectively. The maximum peak value
of BCSD gives the value of B of the given shape. Unlike the
BCSD curves, RCSD curves for all four shapes produce
peaks at angle 0 and 2. In order to obtain the rotational
symmetricity R and the number of rotational symmetry
axes NR , the discarding procedure described in Section 3.2
is necessary to remove those pseudo-RCSD data. For
example, in the RCSD curve (Fig. 15b) of the triangle shape,
the first minimal RCSD value is obtained at ¼ =3 and the
last minimal RCSD value is obtained at ¼ 5=3. Those
RCSD data whose corresponding angles are 0 < < =3
and 5=3 < < 2 are discarded. After performing the
discarding procedure, the number of rotational symmetry
axes NR is counted as the number of the peaks obtained in
RCSD curves at angles =3 5=3 plus 1. The maximum peak value of RCSD at angles =3 5=3 is R of
EVALUATION 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,
stf ¼ 0:99. (c) ngc288, sl ¼ 0:86, stf ¼ 0:99.
Fig. 15. BCSD and RCSD of triangle shape in Fig. 14. (a) BCSD versus
for triangle shape. (b) RCSD versus for triangle shape.
GUO ET AL.: IRREGULAR SHAPE SYMMETRY ANALYSIS: THEORY AND APPLICATION TO QUANTITATIVE GALAXY CLASSIFICATION
Fig. 16. A set of shapes.
triangle shape. Note that, for regular shapes, NB > 0 and/or
NR > 1. In theory, both B and R should be 1. But, due to
the computational error, these values are very close to 1.
We also study more typical shapes which are not both
perfectly bilaterally and rotationally symmetric. These
shapes are depicted in Fig. 16. We compute both bilateral
symmetricity B and rotational symmetricity R for each
shape using Definitions 2 and 5, as summarized in Table 1.
The first five (Figs. 16a, 16b, 16c, 16d, and 16e) shapes are
perfectly bilaterally symmetric but not strictly rotationally
symmetric. It can be seen from Table 1 that these shapes
have B 1 and R < 1. Shapes in Figs. 16f, 16g, 16h, 16i,
and 16j are not perfectly bilaterally symmetric but are
perfectly rotationally symmetric. These five shapes have
B < 1 and R 1. Shapes in Figs. 16k, 16l, 16m, 16n, and
16o are neither perfectly bilaterally symmetric nor perfectly
TABLE 1
Symmetricity Values for Shapes in Fig. 16
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Fig. 17. Plot of bilateral symmetricity and rotational symmetricity values
of the studied shapes in Figs. 14 and 16.
rotationally symmetric: B < 1 and R < 1. For each shape,
we also compute the number of symmetry axes NB , NR , Nb ,
and Nr (Table 1). For each shape in Figs. 14 and 16, the
computed bilateral symmetricity and rotational symmetricity values are plotted in Fig. 17.
7
APPLICATION TO QUANTITATIVE GALAXY
CLASSIFICATION IN ASTRONOMY
Galaxy classification in the universe is one of the major
challenges in astronomy. Morphological characterization of
the galaxies is the first step toward understanding the
physical properties of the galaxies and far depth of the
universe. In the 1920s, Edwin Hubble proposed a “tuning
fork” classification system [26], [27] based on the visual
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
into the above categories are regarded as irregular (Irr)
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
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Fig. 18. Comparisons of the four elliptical galaxies along with
computed parameters. (a) ngc278, sl ¼ 0:98, stf ¼ 1:00, Nb ! 1,
Nr ! 1, LSR ¼ 1:01. (b) ic418, sl ¼ 0:98, stf ¼ 1:00, Nb ¼ 2, Nr ¼ 2,
LSR ¼ 1:17. ( c ) n gc 25 4, 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.
object from the background. We use Otsu’s thresholding
method [38] to segment the galaxy images and extract the
region of interest. We find that adding a small offset (0:2
to 0.3) to Otsu’s threshold for some of the images yields a
better segmentation results. Then, we convert the gray-level
image to a black-white image using the new revised
threshold level. Next, we perform a morphological opening
operation [39], [40] on the resulting black-white image to
remove small objects. This is followed by a flood-filling
operation [39], [40] to fill all objects with holes.
7.1
Criteria for Quantitative Classification of Galaxy
Shape
7.1.1 Elliptical Galaxy (E)
In the Hubble sequence, elliptical galaxies include circular,
elliptical, and lenticular galaxies. For a galaxy with elliptical
shape, the following conditions are satisfied:
1.
sl Tm;
ð15Þ
where Tm—threshold of symmetry level sl. Empirically, Tm ¼ ½0:88-0:91.
2.
Trb1 stf Trb2;
ð16Þ
where Trb1, Trb2—threshold of symmetry-type
factor. Empirically, Trb1 ¼ ½0:95-0:97, Trb2 ¼
½1:00-1:02.
3.
N b ¼ 2 and Nr ¼ 2 ðfor elliptical and lenticularÞ;
ð17Þ
4.
Nb ! 1 and Nr ! 1 ðfor circularÞ;
ð18Þ
where Nb —the number of the bilateral symmetry axis
and Nr —the number of the rotational symmetry axis.
Once a celestial body is categorized as elliptical, we use
the ratio (LSR) of major axis to minor axis of the ellipse that
has the same normalized second central moments as the
VOL. 32,
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Fig. 19. Comparison of four spiral galaxy shapes along with computed
parameters. (a) ngc5457, stf ¼ 1:02, sl ¼ 0:79, Nr ¼ 4. (b) ngc986,
stf ¼ 1:07, sl ¼ 0:89, Nr ¼ 2. (c) ngc232, stf ¼ 1:09, sl ¼ 0:91, Nr ¼ 2.
(d) ngc210, stf ¼ 1:18, sl ¼ 0:80, Nr ¼ 2.
studied region to describe the ellipticity and differentiate
circular, elliptical, and lenticular shapes. We have the
following conditions:
Circular: 1 LSR Lcircle ;
Elliptical: Lcircle < LSR < Lellipse ;
Lenticular: LSR Lellipse ;
ð19Þ
where Lcircle —threshold of LSR, is used to differentiate circle
and ellipse, normally, Lcircle ¼ 1:1 1:2; Lellipse —threshold of
LSR, is used to differentiate ellipse and lenticular shape.
Lellipse ¼ 2:00 2:60.
Fig. 18 shows four examples of elliptical galaxies along
with computed parameters: sl, stf, Nb , Nr , LSR. It is
observed that the values of symmetry level sl of all four
shapes are greater than the threshold of symmetry level Tm
defined in (15). The values of symmetry-type factor stf of all
four shapes are within the range of threshold Trb1 and
Trb2. The most flattened elliptical galaxy in Fig. 18 is the
shape of Fig. 18d with LSR ¼ 3:00.
7.1.2 Spiral (S) and Barred Spiral (SB) Galaxy
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
interior region of the spiral arms. For all spiral galaxies, the
following condition is satisfied:
stf > Trb2;
ð20Þ
where the threshold Trb2 was introduced in (16).
Fig. 19 shows four examples of normal spiral and barred
spiral galaxies along with the increasing value of symmetrytype factor stf. We also compute and list the value of sl and
Nr for each galaxy shape in Fig. 19. Note that we only
consider the number of rotational symmetry axes since the
spiral galaxies are rotationally symmetric. Generally, with
the increasing value of stf, the galaxy tends to become a
barred spiral.
One of the common characteristics of normal spirals and
barred spirals is that their central bulge and barred core
have much higher brightness than their outstretched arms.
This can be used as an image feature to separate the normal
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
thresholding segmentation as described in the beginning of
Section 7 by increasing the amount of threshold offset in
order to remove the arms of the galaxy. We then obtain an
image of the nucleus of the galaxy. Next, various parameters 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
Fig. 20. An example of barred spiral galaxy along with segmented
results and computed parameters. (a) ngc6872 barred spiral, stf ¼ 1:05,
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.
determine whether the nucleus is a central bulge or barred
core. Fig. 20 illustrates an example of a barred spiral galaxy.
7.1.3 Hubble’s Irregular Galaxy (Irr)
In Hubble’s tuning fork classification system, the classification of the irregular galaxies is qualitative and now
considered to be too coarse. In our study, we develop
quantitative criteria for classification of irregular galaxy.
Using the parameters we proposed, we find that the
irregular galaxy can be divided into two categories: The
first one is the typical irregular galaxy (we call it “Ir1”)
which satisfies the following condition:
stf Trb1:
ð21Þ
Another class of irregular galaxy (we call it “Ir2”) is in
the transition area between the spiral galaxy (S and SB)
and the irregular galaxy Ir1. These galaxies do not satisfy
the condition (21), but, instead, they satisfy the following
conditions:
Trb1 < stf < Trb2 and sl < Tm;
ð22Þ
where the Trb1, Trb2, and Tm are described in (15) and
(16).
Many galaxies in the transition area (Ir2) possess some of
the properties of spiral and barred spiral galaxy, but their
shapes are very irregular. Fig. 21 shows examples of irregular
galaxies along with computed parameters stf and sl. Galaxies
in Figs. 21a and 21b belong to the Ir1 galaxy. It can be seen that
computed stf values are in good agreement with (21), whereas
galaxies in Figs. 21c and 21d are Ir2 galaxies and their
computed stf and sl values satisfy (22).
1739
Fig. 22. Two examples of bilateral symmetry galaxy. (a) LkHa101,
stf ¼ 0:85, sl ¼ 0:96, Nb ¼ 1. (b) cfzmcx, stf ¼ 0:63, sl ¼ 0:88, Nb ¼ 1.
symmetry axis (Nb ¼ 1). Galaxy B satisfies the following
conditions:
Nb ¼ 1;
stf < Trb1;
and
sl a stf þ b
ð23Þ
where a and b are empirical factors, a ¼ ½0:18-0:23,
b ¼ ½0:6-0:8. The conditions for our proposed irregular
galaxy Ir1 change accordingly:
stf Trb1
and sl < a stf þ b:
ð24Þ
Fig. 22 shows two example shapes of bilateral symmetry
celestial bodies. It can be seen that computed parameters
satisfy (23).
7.2.2 New Classification Scheme and Symmetry-Type
Factor (stf) versus Symmetry Level (sl) Diagram
We apply the above quantitative analysis to the classification of 55 celestial bodies, which includes 19 elliptical
galaxies, 14 spiral and barred spiral galaxies, and 22 irregular galaxies. For each celestial body, the symmetry-type
factor stf and symmetry level sl are computed and plotted
in Fig. 23 within Hubble’s scheme. It can be seen that an
elliptical galaxy only occupies a small upper rectangle
region. Irregular galaxies and spiral galaxies are separated
by the vertical solid line. On the right-hand side of the
vertical solid line are the spiral and barred spiral galaxies.
Irregular galaxies are in the left region. Clearly, our
developed criteria are effective for quantitatively classifying
the 55 galaxies based on the original Hubble’s scheme.
However, the classification is considered to be coarse due to
the qualitative and simple nature of Hubble’s method.
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. 21. Four examples of irregular galaxy. (a) ngc899, stf ¼ 0:94,
sl ¼ 0:93. (b) ngc246, stf ¼ 0:86, sl ¼ 0:89. (c) ngc346, stf ¼ 1:00,
sl ¼ 0:83. (d) ngc1952, stf ¼ 0:96, sl ¼ 0:88.
Fig. 23. Symmetry-type factor versus symmetry level diagram for
55 galaxies within Hubble’s scheme. Ellipticals (E) marked by , normal
spirals (S) and barred spirals (SB) marked by t
u, and irregulars (Irr)
marked by .
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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 t
u,
bilateral symmetry (B) marked by , and irregular (Ir) marked by .
In Fig. 24, we plot the stf versus sl diagram for the
55 galaxies using our new classification scheme described in
Section 7.1. We separate the proposed bilateral symmetry
galaxy from the original irregular galaxy using an oblique
straight line. The bilateral symmetry galaxy is situated in the
upper left region. On the left irregular galaxies, a vertical
dashed line divides the irregular galaxy into two regions: Ir1
and Ir2. The irregular galaxy Ir1 is on the left-hand side of the
dashed line, whereas the Ir2 galaxy is in the transition region
from Ir1 to rotational symmetry galaxy R. In Fig. 24, the
transition area is the rectanglular region between the vertical
dashed and solid lines and under the area of elliptical
galaxies. Using our new criteria, the galaxies can be classified
into finer categories in a quantitative fashion. Based on the
developed parameters, we propose a new classification
framework with a view to replacing or improving the
Hubble classification method. In our new scheme, all
galaxies can be classified into four categories, listed as
follows:
1.
2.
3.
4.
Bilateral symmetry and rotational symmetry galaxy
(BR). This includes all elliptical galaxies E (circular,
elliptical, and lenticular) in Hubble’s scheme as well
as those galaxies which satisfy (15)-(18). It also
includes galaxies with Bmax 1, Rmax 1, Nb ¼ 2,
Nr ¼ 2, for example, galaxy ngc6822 can be included
in this class.
Rotational symmetry galaxy (R). This includes spiral
and barred spiral galaxies in Hubble’s scheme as
well as other galaxies which appear rotationally
symmetric by visual inspection. Rotational symmetry galaxy (R) satisfies (20).
Bilateral symmetry galaxy (B). This is separated
from the irregular galaxies in Hubble’s scheme and
includes the galaxies which satisfy (23).
Irregular galaxy (Ir). This includes two types of
irregular galaxies which are typical irregular galaxies, the Ir1 galaxy satisfying (24) and the transitional irregular Ir2 galaxy satisfying (22).
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Fig. 25. The relation between B and R and n.
The proposed classification scheme is not only a quantitative method but also is able to encompass a wider range of
celestial body types than the traditional Hubble method.
8
DISCUSSION AND CONCLUSION
Our proposed imperfect symmetry measures enable characterizing arbitrary shapes in a new way. These measures are
based on geometrical operations and contain geometric
information of the shape, whereas other methods, such as
Fourier transform and moment method, lack the capability to
deal with geometric structure. The proposed methods allow
us to quantitatively classify any arbitrary shape ranging from
regular to irregular shapes, from bilaterally symmetric
shapes to rotationally symmetric shapes. Based on our study,
we give the following conjectures without proof:
Conjecture 1. For a regular shape, if the values of the bilateral
symmetricity and rotational symmetricity both equal the value
of 1, that is, B ¼ R ¼ 1, then the number of perfectly
bilateral symmetry axes equals the number of perfectly
rotational symmetry axes, that is, N B ¼ N R .
Conjecture 2. For a regular shape, if the value of the bilateral
symmetricity B is equal to the value of 1 and the number of
perfectly bilateral symmetry axes N B is greater than 1, then
the shape is rotationally symmetric, that is, R ¼ 1.
One issue regarding the error of the symmetricity
calculation is the choice of the value of n. The value of n
is used to control the amount of increment of the rotation
angle during the geometric operations. In theory, the higher
the value of n, the more precise the calculation of
symmetricity would be. Therefore, n should be sufficiently
large. We analyzed the influence of the different settings of
n on the resulting symmetricity value for the given data.
With the increasing number of n, the values of B and R
tend to approach a limit value. Fig. 25 shows one example
of the relation of B and R with respect to n. It can be seen
that the values of B and R become stable when n > 100. In
general, our study shows that the average relative error of
B due to n is less than 0.8 percent, whereas the average
relative error of R is less than 0.1 percent when n 200.
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
Fig. 26. Symmetry-type factor stfc versus symmetry level slc diagram
based on Hubble’s scheme. Ellipticals (E) marked by , normal
spirals (S) and barred spirals (SB) marked by t
u, and irregulars (Irr)
marked by .
proposal of the new way of classifying galaxy and new
categories. There are a few points worth mentioning:
1.
2.
3.
Our method is based on the binary image obtained
from the digital image segmentation. The quality of
the segmentation is important for the followed
characterization and classification of galaxy. Effective segmentation should, in theory, keep the
interested object as much as possible; at the same
time, the accuracy of the segmentation should be
maintained. It is often difficult to achieve both in
practice at the same time. Nevertheless, our study
focuses on the galaxy classification, not the image
segmentation. The segmentation method used in this
study might not be the optimal one. However, our
proposed method did produce satisfactory results
based on the current segmentation method.
The two parameters stf and sl to characterize the
shape of the galaxy are based on the maximum
symmetricity Bmax and Rmax after optimization
procedure. For comparison purposes, we also
investigate the use of defined parameter stfc and
slc which are based on the symmetricity B and R
for galaxy classification. Similarly, we compute the
stfc and slc for all 55 celestial bodies and plot these in
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
different types of galaxies that can be separated in
this diagram, the separability of using stfc and slc is
not as good as the one using stf and sl. Therefore, a
comparison study clearly shows the advantage of
using the parameter stf and sl, in other words, the
discrimination power of the parameters in the
galaxy classification is increased by performing an
optimization procedure.
The performance of our optimization problem depends on the setting of the initial value. In this study,
the initial value is set as the centroid of the shape,
rather than an arbitrary point. The centroid is the first
order moment of the given shape. The position of the
centroid is the average value of the coordinates of all
the points in the shape. Therefore, a centroid can be
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 optimization) actually coincides with its centroid. The optimization 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
barred core is a task subject to further studies.
5. It is worth emphasizing that our classification
scheme is based on shape of the segmented galaxy
image. If a galaxy is viewed close to edge-on, it is
intrinsically impossible to determine whether a
galaxy is elliptical or spiral on the basis of shape
feature alone. Some other information may be
incorporated into the scheme in order to better
understand the physical property of the galaxy.
Furthermore, the quality of the original galaxy image
also has an impact on the subsequent segmentation
and classification.
6. In future work, we intend to apply our irregular
shape symmetry analysis and quantitative criteria to
a larger data set of galaxies or celestial bodies. The
classification scheme proposed in this paper is
intended to serve as a framework and foundation
for future studies.
In summary, we have demonstrated the effectiveness of
our proposed quantitative criteria for galaxy classification
based on proposed irregular shape symmetry measures.
Our concepts have also been applied to other irregular
shape analyses, such as breast tumor classification. For
further details, see [47], [48]. The irregular shape measures
described here can be extended to 3D shapes. The method,
in principle, has the potential to be useful in many other
areas such as mathematics, artificial intelligence, image
processing, robotics, biomedicine, etc.
APPENDIX
PROOF OF FINITE ABELIAN GROUP H
Let I; r1 ; r2 ; . . . ; rn1 ðrn ¼ r0 ¼ IÞ represent n-fold counterclockwise rotations of a given shape M through i ¼ 2i=n,
i ¼ 0; 1; 2; . . . ; n 1, respectively. The symbol fII denotes a
pair of mirror reflection operation of M. We combine every
rotational motion ri of shape M with a pair of reflection
motions fII to form a set
H : I; rfII ; r2 fII ; . . . ; rðn1Þ fII
ðrn fII ¼ IÞ:
Proposition 1. Consider the set H:
H : I; rfII ; r2 fII ; . . . ; rðn1Þ fII
ðrn fII ¼ IÞ:
1742
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According to the definition of group, H is a finite group
of order n under the binary operation “succession.”
In addition, we have
where 8p; q 2 Z, and Z is the set of integers.
a
b ¼ ri1 fII ri2 fII ¼ rði1 þi2 Þ fII ¼ rði2 þi1 Þ fII ¼ ri2 fII ri1 fII
¼ b a:
Proof. Suppose 8a; b; d 2 H:
d ¼ ri4 fII :
Therefore, H is a finite Abelian group of order n under
the binary operation “succession.”
u
t
Here i1 ; i2 ; i4 2 Z. Let i3 ¼ i1 þ i2 , then i3 must be an
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;
where k is the number of rotation of 360 degrees of
shape M. i3 > 0 (k 0) indicates the counterclockwise
rotations of shape M, and i3 < 0 (k < 0) indicates the
clockwise rotations.
1.
Closure. Let c ¼ ri3 fII ¼ rði1 þi2 Þ fII , we have
a b ¼ ri1 fII ri2 fII ¼ rði1 þi2 Þ fII ¼ ri3 fII ¼ c:
a.
k ¼ 0 and 0 < i < n, we have i3 ¼ i and 0 <
i3 < n
a b ¼ rði1 þi2 Þ fII ¼ ri3 fII ¼ c; then c 2 H:
b.
k ¼ 0 and i ¼ 0, we have i3 ¼ 0
a b ¼ rði1 þi2 Þ fII ¼ ri3 fII ¼ c ¼ r0 fII ¼ I
then c 2 H:
c.
k 6¼ 0 (i3 n or i3 < 0), we have
ACKNOWLEDGMENTS
This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet
Propulsion Laboratory, California Institute of Technology,
under contract with the US National Aeronautics and Space
Administration, and the SIMBAD database, operated at
CDS, Strasbourg, France. The authors would like to thank
Professor Rangaraj Rangayyan of the University of Calgary,
Canada, for providing the mammographic tumor contours.
They also wish to thank the anonymous referees and the
Associate Editor for their comments, which have improved
the quality of this paper.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
a b ¼ rði1 þi2 Þ fII ¼ ri3 fII ¼ rknþi fII ¼ rkn ri fII ¼
Iri fII ¼ ri fII ¼ c; then c 2 H:
2.
Hence, 8a; b 2 H, we have a b 2 H.
Associativity. 8a; b; d 2 H, we have
ða bÞ d ¼ ðri1 fII ri2 fII Þ ri4 fII ¼ rði1 þi2 þi4 Þ fII ¼
ri1 fII rði2 þi4 Þ fII ¼ a ðb dÞ:
3.
Identity. 9I 2 H, here I ¼ rn fII such that 8a 2 H:
a I ¼ ri1 fII rn fII ¼ ri1 fII ¼ a;
[7]
[8]
[9]
[10]
[11]
I a ¼ rn fII ri1 fII ¼ ri1 fII ¼ a:
4.
Uniqueness: If there exists another unit element I0
in H, then I0 ¼ I0 I ¼ I.
Inverse. 8a 2 H, 9a1 2 H, here a1 ¼ ri1 fII such
that
a a1 ¼ ri1 fII ri1 fII ¼ r0 fII ¼ I;
a1 a ¼ ri1 fII ri1 fII ¼ r0 fII ¼ I:
Uniqueness: If there exists another inverse x1 of
a, then
OCTOBER 2010
¼ I a1 ¼ a1 :
rp fII rq fII ¼ rðpþqÞ fII ;
b ¼ ri2 fII ;
NO. 10,
x1 ¼ x1 I ¼ x1 ða a1 Þ ¼ ðx1 aÞ a1
H is a finite Abelian group of order n under the binary
operation defined as “succession” which can be denoted by “
”
as follows:
a ¼ ri1 fII ;
VOL. 32,
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