Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, 10-13 July, 2011
AN EFFECTIVE NON-HT CIRCLE DETECTION FOR CENTERS AND RADII
LI-QIN JIA1, HONG-MIN LIU1, ZHI-HENG WANG1, HONG CHEN2,3
1
School of Computer Science and Technique, Henan Polytechnic University, Jiaozuo 454000, China
2
Department of physics, Anshan Normal University, Anshan 114005, China
3
College of information science and engineering, Northeastern University, Shenyang 110819, China
E-MAIL: [email protected], [email protected], [email protected], [email protected]
Abstract:
We present a non-HT circle detection algorithm applied to
search the centers and radii of circular or partially circular
components present in the image. The line coincident with the
gradient vector of each edge point and passing through the
corresponding edge point is defined first. Then, for every pixel
in the image, the number of the lines passing through this pixel
is defined as the energy of the pixel. The feature circle energy
(FCE) distribution map of the whole image is therefore obtained
and the local maxima are corresponding to the centers of
potential circles. For the detection of radius, the gradient
magnitudes in assigned region are accumulated and its variation
defined as the feature circle radius (FCR) is computed who has
maximum when the radius of the region is equal to that of the
circle. Synthetic images and natural images are used to test the
capability of the proposed method. The experimental results
indicate that the presented algorithm has excellent performance
for detection of single circle, multiple circles, concentric circles
and partial circles, and also good accuracy despite the presence
of different noises.
Keywords:
Circle detection; center detection; radius detection;
concentric circle; partial circle
1.
Introduction
In the field of pattern recognition, circle detection is
important and has been extensively studied and applied
widely, such as automatic inspection and assembly, target
locating and medical images in the past decades [1-2]. Hough
transform (HT) is the most straightforward method for the
circle-detection, which provides enough robust but requires
massive computation and memory. In order to overcome the
limitations of HT, some new approaches based on HT were
proposed. e.g., the probabilistic HT [3], the randomized HT
[4-5] and the fuzzy HT [6].The common methods of HT
based take advantage of the equation but bring about large
computation. In addition, other non-HT based methods were
also provided. Ramirez et al. [7] presented a circle detection
method based on genetic algorithms and Chung et al. [8]
presented a fast randomized algorithm to solve the
center-detection problem. These algorithms show good
accuracy or time save for the detection of circles, but the
algorithms are also complex.
A non-HT based method is proposed in this paper for
detecting the centers and radii of the circles in the images.
Firstly, Gauss template is applied to compute the gradient of
the image. Secondly, every edge point of a circle and its
gradient define a line, which passes through the center. The
feature circle energy (FCE) is defined and computed in which
the local maxima correspond to the centers. Thirdly, the
gradient magnitudes are accumulated and its variation is
computed to give the feature circle radius (FCR) which
shows the radius of the circle. This approach avoids the
voting processing in HT-based algorithm which raises the
computation efficiency, moreover results in an accurate
detection of the center and radius and easy understanding.
The application results of the proposed method to both
synthetic and natural images are presented abundantly.
The remainder of this paper is organized as follows.
Section 2 introduces the principle of the method. Section 3
gives a detailed description for this method in steps. Section 4
displays the application results of the method on both
synthetic and natural images. The conclusions are presented
in Section 5.
2.
Principle of the proposed method
For a circle, the gradient vector of an edge point points
at the center of the circle, so a line, coincident with the
gradient vector and passing through the edge point, should
also pass through the center of the circle, that is to say the
distance of the center to the line is zero. For every edge
points of the circle, there also exists a corresponding line on
which the center lying. As Fig.1 shows, the center of a circle
is indicated as O, and a line L defined by edge point A passes
through the center O. The lines defined by all the edge points
of the image are determined firstly such as line L displayed in
Fig.1. Then, for a pixel B, the distance values of pixel B to
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Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, 10-13 July, 2011
the lines are computed. The energy of pixel B is defined as
the number of the distance value equal to zero, in other words,
as the number of the lines passing through the pixel B. With
the same way, the energy of the every pixel is computed to
obtain the feature circle energy (FCE) distribution map of the
whole image. When a pixel is the center of a circle, the lines
defined by the edge points of the circle will all pass through
the pixel to get a larger energy compared with that of
non-center pixel. Therefore, detection of the local maxima in
the FCE distribution map can locate the centers of the
potential circles present in the image.
Figure 1. The energy of point B add 1 if B is on the line defined by point
A
For determining the radius of a circle, the gradient
magnitudes of the pixels in a region centering on the potential
center of the circle are accumulated in the image. If the
region is smaller than the area of the circle, the variation of
accumulation changes little. But when the radius of the
region is equal to that of the circle, the accumulation has a
large variation and keeps steady with the increase of the
radius until another concentric circle is found. So the FCR,
defined as the variation of gradient magnitudes accumulation
in the region, gives the radii on which large variations happen.
This is the basis of our method for detecting the center and
radius of the circle.
3.
Proposed Method
Based on the principle of the method, we proposed an
algorithm to detect the centers and radii of the circles.
3.1.
Compute the feature circle energy (FCE) and get the
center
Canny detector is used to get the edge of the image
and Pi (i = 1, 2, ", N ) denotes an edge point. The gradient of
each pixel in the image is computed by using Gauss template.
The line defined by the position and gradient of edge point
Pi is indicated as Li (i = 1, 2, ", N ) . For a pixel X in the
image, the distance between the X and the line Li is
computed,
which
is
presented
as
di =|| X − Pi || (i = 1,2, ", N ) and the energy of X adds 1
if di < 2 . For the whole image, the energy of every pixel is
computed and the FCE is obtained.
In order to detect the local maxima in the FCE
distribution map, a threshold T is applied, which is
determined with equation (1):
T = Mean(E ) + k ⋅ Std (E )
(1)
in which the Mean(E ) and Std (E ) respectively
represents the mean and standard deviation of the FCE
distribution map, and the factor k in the equation is a
coefficient in the range of 2~3 in a general way. A circular
region is defined whose radius is 2R1 + 1 where R1 is the
maximal searching region radius and can be 5~10 as a
general rule. The local maximum can be obtained by using
function ordfilt 2 . For a point X , it is considered as a local
maximum when its energy is equal to the maximum in the
circular region and larger than T in the FCE distribution
map. The position of the maximum corresponds to the center
of a circle.
3.2.
Compute the feature circle radius (FCR) and get the
radius
For detecting the radius of a circle, an accumulation S
is defined to accumulate the gradient magnitudes of the pixels
in the region which is centered on O (the circle center we
just get) and of a radius of R2 . The value of R2 increases
from 1 to the edge of image with 1 as interval. Since the
gradient of pixels in circle is zero (only the gradient of noise
points is not zeros when there exists noise in image), the S
changes greatly when R2 equals to the radius of circle. The
FCR is defined to describe the variation of S in which the
position of maximum indicates the radius of circle. The radii
of concentric circles consisting of two circles or more can be
obtained by searching the positions of the first several
maximum in FCR, which means the radii of concentric
circles can be determined easily using the proposed method.
4.
Experiment results
Experiment tests have been developed both on synthetic
and natural images in order to evaluate the performance of
the provided method.
4.1. Synthetic images
As shown in figure 2 (a), an image with two different
circles of size 83×158 is generated in order to evaluate the
performance of the proposed method. The FCE distribution
map is shown in figure 2 (b), in which the intensity of
potential centers of the circles are larger obviously than other
pixels. The FCR distribution map of the first detected circle
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Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, 10-13 July, 2011
(the bigger one) is shown in figure 2 (c), in which gradient
magnitudes accumulation change greatly when the searching
radius is 33, that means radius of the bigger circle is 33. As
the same way, the radius 19 of the second circle is obtained in
its corresponding FCR shown as figure 2 (d). The
corresponding detected centers and radii of circles are
displayed in figure 2 (e). The centers and radii are accurately
detected.
Figure 5 shows the noise resisting of the method. Figure
5 (a)-(c) display the original image and the edge and circle
detected results. When Gaussian noise ( σ 2 = 0.001) is added,
the edge image and the detected results is shown in Figure 5
(e) and (f) respectively. Figure 5 (h)-(j) display the results of
adding Salt & Pepper noise ( σ 2 = 0.03). The good results
show that the proposed method has good resisting for the
noise effects.
Figure 3 shows the detection result of concentric circles.
A 83×158 original image with concentric circles is shown in
Figure 3 (a). During the detection of radii, a threshold T is set
and the positions of maximal values bigger than T in
corresponding FCR are obtained. The detection result is
shown in Figure 3 (b), which indicates the centers and radii
of concentric circles accurately.
(a)
The result of proposed method on partial circles is
shown in Figure 4. The original image is shown in Figure 4 (a)
which describes a half and a quarter circle, and the detection
result is shown as (b). For the accumulating region of
gradient magnitudes starts from the center detected and the
region radius increases one pixel by one pixel, the
accumulation variation is little because only few pixel having
gradient magnitudes until the region radius equal to the radius
of circle. Test results show that the proposed method can
detect the centers effectively only using an arc of a quarter
round.
(a)The original image
Figure 3. Detection of concentric circles
(a)
(b)
Figure 4. Detection of partial circles
(a)
(b)The FCE distribution map
5
(b)
(b)
(c)
5
x 10
5
x 10
5
(d)
0
0
20
(c) The FCR of first circle
40
0
0
10
20
(e)
(f)
30
(d) The FCR of second circle
(h)
(i)
(j)
Figure 5. Detection of circles with noise
(e) Detected centers and radii of circles
Figure 2. Detection of two circles
4.2. Natural images
In this experiment the proposed algorithm is tested upon
natural images. Figure 6 shows the detection of moon. The
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Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, 10-13 July, 2011
detection result is displayed as Figure 6 (b) where the center
is marked with red “+” and the radius of the moon is obtained
to display the circle with red lines. Figure 7 shows us the
detection of golf, and result indicated as Figure 7 (b) where
the center and center are all in blue. The test demonstrates the
proposed method has excellent detection performance on
both single and multiple circles.
Figure 8 display the detection results of concentric
circles. A coin shown as Figure 8 (a) is detected. The detected
centers and radii are all shown in red. The detection results
display the proposed method can detect the center and radii
of concentric circles accurately.
The partial moon is detected and Figure 9 gives the
origin image and result which shows the detection accuracy
of the proposed method on partial circle.
Figure 9. Detection of partial moon
5.
Conclusions
In this paper, a non-HT circle detection algorithm for centers
and radii is proposed. The positions and gradient of edge points of
candidate circle defines lines and the center is the pixel passed
through by the most lines. The radius is obtained by searching the
position of gradient magnitudes accumulation greatly changes. The
experimental results indicated that our algorithm can be effectively
applied to detect the centers and radii of single circle, multiple
circles, concentric circles and partial circles, and has good accuracy
despite of the presence of different noises.
Acknowledgements
Figure 6. Detection of moon
This work is supported by National Natural Science
Foundation of China (61005033), Open Projects Program of
National Laboratory of Pattern Recognition (20090018) and
Doctor Degree Foundation of Henan Polytechnic University
(B2010-39, B2010-68).
References
Figure 7. Detection of golf
Figure 8. Detection of tire
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