Pattern Recognition Letters 26 (2005) 1985–1994

www.elsevier.com/locate/patrec

Using phase information for symmetry detection

Zhitao Xiao a,b,*, Zhengxin Hou b, Changyun Miao a, Jianming Wang a

a School of Information and Communication Engineering, Tianjin Polytechnic University, No. 63 Chenglinzhuang Road,

Hedong District, Tianjin 300160, PR Chinab School of Electronic Information Engineering, Tianjin University, Tianjin 300072, PR China

Received 14 April 2004; received in revised form 3 September 2004

Available online 7 April 2005

Communicated by L. Goldfarb

Abstract

Symmetry exists widely in the real world and plays a remarkable role in perception problems. The existing symmetry

detection algorithms are mainly based on luminance or gradient information. Upon analyzing the relationship between

symmetry and phase, based on phase congruency, we propose a phase-based symmetry detection (PSD) algorithm. PSD

is calculated based on log Gabor wavelet. The symmetric points of objects are obtained by inspecting the phase infor-

mation. The feasibility analysis, phase-based symmetry detection definition, and rationality demonstrations established

the theoretic foundation for this algorithm. The experiments show that this algorithm can be applied directly to original

images without segmentation—it is invariant to rotation, luminance and contrast—and it can detect several types of

symmetries at the same time and it is compared with several other methods.

� 2005 Elsevier B.V. All rights reserved.

Keywords: Phase congruency; Symmetry detection; Phase information; Invariance

1. Introduction

Symmetry plays a remarkable role in perception

problems. For example, peaks of brain activity are

0167-8655/$ - see front matter � 2005 Elsevier B.V. All rights reserv

doi:10.1016/j.patrec.2005.02.003

* Corresponding author. Address: School of Information and

Communication Engineering, Tianjin Polytechnic University,

No. 63 Chenglinzhuang Road, Hedong District, Tianjin

300160, PR China. Tel.: +86 22 8129 5069; fax: +86 22 2452

8164.

E-mail address: zhitaoxiao@yahoo.com (Z. Xiao).

measured correspondence with visual patternshowing symmetry. Symmetry is one of the shape

features that are often used in object recognition.

Symmetry is so important to our perception that

most of the man-made objects are symmetric. Psy-

chologists think that symmetry is the fundamental

elements of perception. Since its ubiquity, the

symmetry detection is of obvious significance for

understanding and interpreting the natural images.Symmetry detection and application have become

the important part of computer vision.

ed.

1986 Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994

The more symmetric a piece of region is, the

more easily it is recognized as a figure (Zhang,

2000). Symmetry is essentially a kind of formula.

An application of this formula is, in martial dis-

guise, using various colors and shapes to changethe object�s original pattern.

Two familiar symmetry types are bilateral sym-

metry and rotational symmetry. There is another

type of symmetry, which is curve symmetry. They

are illustrated in Fig. 1.

Most of the existing symmetry detectors are

using the image brightness or gradient informa-

tion. Their major limitation is that they requireobjects to be segmented prior to any symmetry

analysis. Mara and Owens (1996) converts the

symmetry detection to eigenvalue discompose of

covariance matrix and requires that symmetric

axis of the object must be same as the axis of bin-

ary image. Thai and Healey (1998) represents sym-

metry by complex moment from power spectrum.

Sun and Si (1999) and Sun (1995) use orientationhistograms and gradient information for sym-

metry detection. The algorithm of Atallah (1985)

requires objects to be represented in terms of

points, line segments and circles. However, the

morphological algorithms can only be applied to

binary objects. A difficulty with morphological ap-

proaches is that they are very sensitive to small

variations in the outlines of objects. The groovein object outline can form several symmetric axes,

which will make the object representation more

difficult. Reisfeld et al. (1995) describes one of

the few approaches to symmetry that does not

require object recognition or segmentation. An

objection to this measure of symmetry is that it

depends on the contrast of the feature in addition

to its geometric shape. Kovesi (1996, 1997, 1999)studied the symmetry detection using phase infor-

Fig. 1. Three type of symmetry: (a) bilateral symmetry, (b)

rotational symmetry, and (c) curve symmetry.

mation and gained some helpful results. But the

test results contain the symmetry both of objects

and their background. Obviously, this is not what

we expected.

In summation, most of the existing symmetrydetectors mainly have three problems as follows.

Firstly, they are designed aim at the images con-

taining only one object. A major weakness in most

is that they require objects to be segmented prior to

any symmetry analysis.

Secondly, they primarily use gradient informa-

tion of image; however, this is not consistent with

human vision system characteristics. A bright cir-cle will be considered to be more ‘‘symmetric’’

than a low contrast one. Thus, we have no abso-

lute sense of the degree of symmetry of an object;

all one obtains are locations in the image where

symmetry is locally maximal. Most symmetry

detectors can ensure the invariance of rotation,

but they are not invariant to contrast and bright-

ness. The basic reason is that they compute usingthe gradient information.

Finally, they are usually effective only for one

or two type symmetries, such as bilateral or/and

rotational symmetry, that is, they are not general.

A new symmetry detection algorithm—phase-

based symmetry detection (PSD) algorithm—is

presented in this paper. PSD can be directly ap-

plied to the original images without segmenta-tion—it is invariant to rotation, brightness and

contrast—it can detect several kinds of symmetries

simultaneously, such as bilateral symmetry, rota-

tional symmetry and curve symmetry.

2. Feasibility of symmetry detection by phase

information

Kovesi (1996) argued that an important aspect

of symmetry is the periodicity that it implies in

the structure of the object. Accordingly it is per-

haps natural that one should use a frequency-

based approach in attempting to recognize and

analyze symmetry in images. Indeed, an inspection

of the Fourier series of some simple functionsmakes this very apparent. At points of symmetry

and asymmetry, we find readily identifiable pat-

terns of phase. Fig. 2 shows the Fourier series

Fig. 2. Construction of (a) square waveform and (b) triangular

waveform from their Fourier series. In both diagrams the first

few terms of the respective Fourier series are plotted.

Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994 1987

representation of both a square wave and a trian-

gular wave. Phase congruency and local energy

model (Morrone and Owens, 1987) use the fact

that all sinusoidal waves are in phase at the steppoints of square wave (or peaks/troughs of trian-

gular wave) to detect edges. One can see that the

axis of symmetry corresponds to the point where

all the frequency components are at either the min-

imum or maximum points in their cycles, that is,

where all the frequency components are at the

most symmetric points in their cycles (the mid-

point of the square wave and the peaks/troughsof the triangular wave). So, it is feasibility to detect

symmetry using phase information.

3. Phase-based symmetry detection

It should be noted that this paper only consid-

ers local symmetry of intensity values in images,that is, a low-level view of symmetry. Overall geo-

metric symmetries that might exist in the image are

not considered.

PC2ðxÞ ¼ max/ðxÞ2½0;2p�

Pn AnðxÞ cosð/nðxÞ � /ðxÞÞ � j sinð/nðxÞ � /ðxÞÞj� �

PnAnðxÞ þ e

3.1. Phase congruency and Kovesi’s symmetry

detector

Given a complex function f = g + ih, and let G

and H be the Fourier transforms of g and h.

Then, g and h are in quadrature if they are

Hilbert transform of each other, namely, H(x) =

�iG(x) sign (x) (where sign (x) is 1 for xP 0

and �1 otherwise). The quadrature filter used in

this work is described in Section 3.2.

The convolution of a complex kernel (whose

real and imaginary part are in quadrature) with areal image results in a complex image. Its norm

is called energy and its argument is called phase.

Consider 1-dimensional signal. The Fourier ser-

ies expansion of a signal, I(x), is

IðxÞ ¼Xn

An cosðnxxþ /n0Þ ¼Xn

An cosð/nðxÞÞ

where An represents the amplitude of the nth co-

sine component, x is a constant (usually 2p), and/no(x) is the phase offset of the nth component.

The function /n(x) represents the local phase of

the Fourier component at position x. Then, phase

congruency and local energy (Kovesi, 1996) are de-fined as

PCðxÞ ¼ max/ðxÞ2½0;2p�

PnAnðxÞ cosð/nðxÞ � /ðxÞÞP

nAnðxÞ þ e

EðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2ðxÞ þ H 2ðxÞ

q

respectively, where H(x) is the Hilbert transform

of I(x), e is a small constant to avoid division by

zero.

Using the new measure of phase deviation,

D/ðxÞ ¼ cosð/nðxÞ � /ðxÞÞ � j sinð/nðxÞ � /ðxÞÞj

a new measure of phase congruency is defined by

Kovesi (1996) as

where, as before, e is a small constant to avoid

division by zero, and b c denotes that the enclosedquantity is itself if it is positive, and zero for all

other values.

Calculating PSD involves convolution at multi-

scales with a complex kernel whose real and

imaginary parts are in quadrature. We choose

1988 Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994

log Gabor wavelet (Xiao et al., 2002) to cal-

culate PSD. On the linear frequency scale, the

transfer function of log Gabor function has the

form

gðxÞ ¼ e�ðlogðx=x0ÞÞ2

2ðlogðb=x0ÞÞ2

where x0 is the filter�s center frequency. To obtain

constant shape ratio filter, the term b/x0 must also

be held constant for varying x0.Log Gabor wavelet has some important attri-

butes. Log Gabor wavelet, by definition, has no

DC component. The transfer function of the log

Gabor has extended tails at the high frequency

end. Thus it should be able to encode natural

images more efficiently than ordinary Gabor func-

tions, which would over-represent the low fre-

quency components and under-represent the highfrequency components in any encoding process.

And it is consistent with measurements on human

visual system that indicates we have cells response

that are symmetric on the log frequency scale

(Kovesi, 1996).

The convolution output from each quadrature

pair of filters at a location in the signal can be con-

sidered to represent a response vector in the com-plex plane: the real component coming from the

even-symmetric filter output and imaginary com-

ponent from the odd-symmetric output. This can

be illustrated in Fig. 3. The amplitude specifies

the length of each vector and the phase specifies

its angle. Note that the wavelet filters are scaled

geometrically, so their center frequencies vary

accordingly. Due to the singularity in the loga-

Fig. 3. The convolution output from each quadrature pair of

filters at a location in the signal can be considered to represent a

response vector.

rithm function at the origin, one cannot construct

an analytic expression for log Gabor function in

the spatial domain. One can design the filters in

the frequency domain and multiply with the signal

of the frequency domain, and then perform anumerical inverse Fourier transform to get the

result in spatial domain.

The above discussion about phase congruency

has been limited to 1-D signals. Applying the

1-D analysis in multiple orientations and forming

a weighted sum can extend the 1-D analysis to

2-D. In general, selecting six orientations can get

good compromise between computational loadand accuracy. Kovesi (1996, 1999) described the

extension to 2-D in detail, so this will not be

repeated here.

It is readily to adapt the existing measures of

phase congruency to construct equivalent mea-

sures of symmetry. As shown in Fig. 4, at the point

of symmetry, the absolute value of the even-sym-

metric filter outputs will be large and the absolutevalue of the odd-symmetric filter outputs will be

small. From these, Kovesi (1996, 1997) present a

measure of symmetry as

SymðxÞ ¼P

n AnðxÞ½j cosð/nðxÞÞj � j sinð/nðxÞÞj�b cPnAnðxÞ þ e

An important contribution of Kovesi�s work is

that the measure of symmetry is normalized,

dimensionless measure. It is independent of the

brightness or contrast of image features. That is,

unlike symmetry measures developed by others,

an absolute sense of the level of symmetry is

obtained. This measure has been narrated in 1-

D signals. Its extension to 2-D can be done byusing the same techniques as phase congruency

use.

3.2. Definition of PSD (phase-based symmetry

detector)

From Fig. 2, we can see that, for the bright ob-

jects in the dark background, phase at the pointsof symmetry is zero; for the dark objects in the

bright background, phase at the points of symme-

try is p. For convenience, we narrate the symmetry

detection only for the bright objects in dark back-

Fig. 4. At a point of symmetry the local phase pattern will be such that only even symmetric filters will be responding (a), and at a

point of asymmetry only odd symmetric filters be responding (b).

Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994 1989

ground. The case for dark objects in bright back-

ground is similar to this.Phase-based symmetry detection (PSD) is de-

fined based on one particular phase (zero phase)

and phase congruency, that is, phase congruency

is computed simply on this special phase.

Assuming that the support of the kernel is not

too small, the phase varies continuously. Thus,

zero crossing of the real part, for example, occurs

between adjacent pixels with opposite real partsigns. Therefore, the location of zero phase can

be found just by considering the sign of responses

in adjacent pixels. Let Snjoi(x) be the response when

the scale (n) being j and the orientation (o) being i,

at a particular location (x), its real part be Rnjoi (x),

and its imaginary part be Mnjoi(x), i.e. SnjoiðxÞ ¼RnjoiðxÞ þ iMnjoiðxÞ. If for two adjacent pixels p

and q, Rnjoi (p) > 0, Mnjoi(p) > 0, Rnjoi(q) > 0,Mnjoi(q) < 0, then there is a zero crossing of

imaginary part between p and q, and the real part

is positive in both p and q. This corresponds to

zero phase.

In the general case, let d be the distance betweentwo adjacent pixels in certain orientation. Accord-

ing to signs of the real parts and imaginary parts

of the response, Rnjoi(p � (d/2)), Mnjoi(p � (d/2)),Rnjoi (p + (d/2)), Mnjoi(p + (d/2)), the zero phase

points can be identified. Here d must be even.

Alternatively, we can judge the signs of Rnjoi (p),

Mnjoi(p), Rnjoi(p + d), Mnjoi(p + d). Now there is no

limit on d.Let the phase of the results be /njoi(p), and the

phase congruency value be PCnjoi(p), over all log

Gabor filters when the scale (n) being j and the ori-entation (o) being i, at a particular location (p).

Here, PCnjoi(p) is calculated according to the PC

formula of Kovesi (1996).

At the points of symmetry, the absolute value of

even-symmetric filter outputs will be large and theabsolute value of odd-symmetric filter outputs will

be small. Therefore, here the energy is absolute

value of the difference of even-symmetric filter

and odd-symmetric filter. Considering the relation-

ships between symmetry and phase synchronously,

we can get the following definition.

Definition: Phase-based symmetry detector

(PSD) maps a pixel, p, an orientation, o, and a

scale, n, to a phase congruency value, PCnjoi(p),

and a special phase, /njoi (p),

PSDðp; n; oÞ ¼ ðPCnjoiðpÞ;/njoiðpÞÞ:

Here,

PCnjoiðpÞ ¼sumEðpÞ

sumAðpÞ þ e¼

Pk;qEnkoqðpÞP

k;qAnkoqðpÞ þ e;

where both of the following conditions should be

met:

(1) sumEðpÞ ¼ fP

k;qEnkoqðpÞj/nkoqðpÞ ¼ 0;8k; qg,that is, sumE(p) is the total energy when

phases are congruent under all scales and ori-

entations and phases are zero,

(2) sumAðpÞ ¼ fP

k;qAnkoqðpÞj/nkoqðpÞ ¼ 0; 8k;qg,that is, sumA(p) is the total amplitude when

phases are congruent under all scales and ori-

entations and phases are zero. e is a small

positive constant to avoid division by zero.Here, e is taken as 0.0001.

Here, ‘‘phases are congruent’’ means phases are

the same value over all orientations and scales at

a certain position. PSD is calculated from sumE(p)

divided by sumA(p). Therefore, it is a normalized

1990 Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994

quantity, which can ensure the value of PSD is

invariant to brightness and contrast.

Note that the energy sumE(p) is calculated

according to Kovesi (1996).

3.3. Computational complexity of PSD

From the definition of PSD, it can be seen that

the computational load of PSD is fairly high. It

needs FFT, IFFT, and matrix multiplication dur-

ing its implementation. If the image size involved

is N · N, it will need about N2 real multiplications,

2N2 log2N complex multiplications, and 4N2 log2Ncomplex additions. Here the radix-2 FFT and

IFFT are assumed.

4. Rationality of PSD

Let the signal be I(x), the quadrature filter ker-

nel is kno, where n represents scale, and o repre-sents orientation. Thus, when the scale is n and

orientation is o, the local energy is

EnoðxÞ ¼ IðxÞ � knok k ¼ InoðxÞk kBecause log Gabor wavelet (Xiao et al., 2002) does

not exist analytic expression in spatial domain, we

show the rationality of PSD using Mexico waveletby two examples. In spatial domain, this wavelet

has the form

gnðxÞ ¼1

n1� x2

n2

� �e� x2

2n2

Firstly, consider a delta function located at 0,

which is the ideal symmetric object with scale zero.The result of the convolution is just the filter ker-

nel itself. Given a scale n, at the origin, the convo-

lution result is In(0) = 1/n + 0i. Therefore, the local

phase is /n(0) = 0, and local energy is En(0) = 1/n2.

According to the definition of PSD, the symmetric

phase is 0.

Next example is a rectangle function,

hðxÞ ¼1; jxj 6 t

0; jxj > t

�

This is a symmetric object located at 0 with scale t.

Given a scale n, the convolution result at 0 is

Inð0Þ ¼Z t

�tgnðxÞdx

� �þ 0i

Therefore, the local phase at 0 is /n(0) = 0. This is

again consistent with the definition of PSD.

5. Experimental results

To demonstrate the performance of PSD, PSD

is applied to several kinds of images and comparedwith other methods. In this paper, three different

resolutions (scales) and six different orientations

for each resolution are chosen. The minimum scale

used is 20 pixels. Increasing the number of resolu-

tions and/or orientations can improve the accu-

racy of the results, but the improved effect is not

clearly, and the computation load will increase

rapidly. The experiments show that the parame-ters used here can get good compromise between

computational load and accuracy. It should be

emphasized that the same parameters of PSD are

used in all of the experiment results given in this

paper except in Section 5.3, in which the scale

parameter�s influence on detection results are

tested.

5.1. Performance demonstration

(1) PSD can be applied directly to the original

images without segmentation. This is shown

in Figs. 5(a), (b) and 6. However, the tradi-

tional detectors would require segmentation

in advance.

(2) It is invariant to rotation, brightness andcontrast. Fig. 6 illustrates it.

(3) It can detect bilateral symmetry, rotational

symmetry and curve symmetry at the same

time, that is, it is a fairly general detector.

See Fig. 5(a) and (b).

5.2. Comparisons with other methods

(1) Comparisons between PSD and KSD: KSD

represents the symmetry detector of Kovesi

(1996, 1997). The comparison between PSD

and KSD is given in Fig. 5(b) and (c). Obvi-

ously, the result of PSD is what we wanted.

Fig. 5. Comparison between PSD and other methods. (a) Original image, (b) detection result of PSD, (c) detection result of KSD, and

(d) thinning result.

Fig. 6. PSD is invariant to rotating, brightness and contrast. (a) Original images�s contrast increased 60, (b) original image�s brightnessincreased 60, and (c) original images rotated by 90�. (a 0), (b 0), and (c 0) are the symmetry detection results of (a), (b), and (c),

respectively.

Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994 1991

(2) Comparisons between PSD and thinning

method: This comparison is shown in Fig.

5(b) and (d). The morphology thinning

method can only be applied to binary

images. From the thinning result, we can

see that its difficulty is that it is very sensitive

to small variations in the outlines of objects.

And small spots inside the object will affectthe results of thinning method seriously but

do not influence the result of PSD on the

whole. These sensitive will make the object

representation more difficult.

(3) Comparisons between PSD and PCA: PCA

means the principal component analysis

based symmetry detection method (Lu et al.,

1999). For image containing many objects,

PCA will need segmentation, so the exampleused here contains only one object. From

Fig. 7. Comparison between PSD and PCA. (a) Original plane,

(b) symmetry detection by PCA, and (c) symmetry detection by

PSD.

Fig. 8. Comparison between PSD and PCA for noised images.

(a) Plane with 10% Gauss distribution noise, (b) PCA of (a), (c)

PSD of (a); (a 0) plane with 20% uniform distribution noise, (b 0)

PCA of (a 0), and (c 0) PSD of (a 0).

1992 Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994

Fig. 7, we can see that the symmetry detec-

tion result of plane by PCA is only one main

axis, but the detection result by PSD not

only contains the main axis, but also the

detailed axes. When noise is added, the

symmetry detection results are shown inFig. 8. It can be seen that the detection

results of PCA are affected seriously, but the

results of PSD are not affected on the whole.

These can illustrate the anti-noise ability of

PSD.

5.3. The scale parameter’s influence on results

The number of scale and the minimum scale are

tested in different values. In Fig. 9(a) is the original

image; Fig. 9(b)–(d) are the detected results when

the minimum scale is 16, 20 and 24 respectively

(the unit is pixel). If the minimum scale is too

small, when number of scale is fixed, then the

Fig. 9. PSD results when the

much larger objects� symmetry cannot be detected(see Fig. 10(b)). If the minimum scale is too large,

then very small and/or thin objects in image will

not be detected (shown in Fig. 10(c)). According

to the experiences, we select 20 for the minimum

scale. About the number of scale, selecting 3–5 is

well. Here we use 3. We find that the results change

little even when it increases to 20. Fig. 11(b) and

(c) are the PSD results when the number of scaleis 3 and 20 respectively. From tests it is shown that

slightly changing these parameters has no distinct

effect on the detection results.

minimum scale varies.

Fig. 11. PSD results when the number of scale is changed.

Fig. 10. PSD results when the minimum scale is too small and too large.

Z. Xiao et al. / Pattern Recognition Letters 26 (2005) 1985–1994 1993

6. Conclusions

The main limitations of most existing symmetry

detection algorithms are discussed. The feasibility

of using phase information for symmetry detection

is analyzed. Then, based on phase congruency and

Kovesi�s symmetry detector, a novel phase-basedsymmetry detection algorithm (PSD) is given and

its rationality is demonstrated. And its computa-

tional complexity is analyzed simply. Finally, the

PSD performance is illustrated by experiments

and is compared with other techniques, such as

Kovesi�s symmetry detector, principal component

analysis based symmetry detector and thinning

method. The test results show that this algorithmcan be applied directly to original images without

segmentation—it is invariant to rotation, bright-

ness and contrast—it can detect bilateral symme-

try, rotational symmetry and curve symmetry at

the same time.

Acknowledgement

Sponsored by Tianjin Higher Education Science

and Technology Development Foundation under

grant 20041304, and China Postdoctoral Science

Foundation under grant 2003034328.

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