Circle Detection Using Scan Lines and HistogramsMing CHEN, Feng ZHANG

�, Zhenhong DU, and Renyi LIU

Zhejiang Provincial Key Laboratory of GIS, Xixi Campus, Zhejiang University, Hangzhou 310007, China

(Received June 28, 2013; Accepted August 31, 2013)

Circle detection is significant in image processing and pattern recognition. We present a new algorithm for detectingcircles, which is based on the global geometric symmetry of circles. First, the horizontal and vertical midpointhistograms of the edge image are obtained by using scan lines. Then, we apply the peak-finding algorithm to themidpoint histograms to look for the center of the circle. The normalized radius histogram is finally used to verify theexistence of the circle and extract its radius. Synthetic images with different levels of pepper noise and real imagescontaining several circles have been taken to test the performance. Experimental results demonstrate that the proposedalgorithm has the advantage of computational efficiency as compared with the randomized Hough transform and someother algorithms. # 2013 The Japan Society of Applied Physics

Keywords: circle detection, scan line, histogram, geometric symmetry

1. Introduction

Detecting circles from digital images is a fundamentaltechnique in image processing and pattern recognition,1,2)

and plays an important role in many industrial fields suchas computer vision, automatic inspection, robotics, neuro-surgery, and target detection.3–8)

The Hough transform (HT),9) which was first presented byHough in 1962, has been widely used in image processing torecognize patterns.10–12) HT maps each edge point of objectsin an image into many accumulating cells in a parameterspace, thus converting the difficult pattern detecting in theimage space into local peak searching in the parameterspace. The method is robust to noise and easy to realizein parallel computing. However, it would require a greatdeal of computation time and memory space when appliedto detect circles since the parameter space exceeds twodimensions, which makes the method inefficient for real-time applications.

To alleviate or eliminate the drawbacks of the classicalHT, many variants have been proposed. Xu et al. presentedthe randomized Hough transform (RHT), which significantlyimproves the computational efficiency and reduces thestorage requirement via random sampling in the edgeimage and establishing a link-list data structure for theparameters.13,14) Although efficient for simple images, RHTis sensitive to noise. When it deals with multi-circle compleximages, the low sampling validity will produce greatamounts of invalid accumulations and result in a largenumber of calculations.

Chen and Chung proposed randomized circle detection(RCD), in which a possible circle is first created using threeof the four pixels randomly picked and becomes a legitimatecandidate circle if the fourth pixel falls on it.15) Then anevidence-collecting process that counts the number of edgepixels falling on the candidate circle is used to determine if itis a true circle. Experimental results show that RCD is faster

than RHT when the noise level is ranged from light tomodest. Chung and Huang presented a pruning-and-votingstrategy and plugged it into randomized circle detection(PRCD) to optimize the evidence-collecting process andreduce the required computation time.16)

Apart from random sampling, geometric properties ofcircles are also often used to reduce the search space forpossible circles. Kimme et al. and Conker identified thecenters of possible circles by using the gradient informationof edge pixels.17,18) Yip et al. explored the parallel tangentlines of edge pixels to identify the possible circles.19)

Ioannou et al. and Kim et al. first identified bisectorscorresponding to the possible chords and then utilized thepoints with large number of bisectors passing through asthe centers of the possible circles.20,21) Chiu and Liawconstructed the possible circles based on three random edgepixels only when the two side pixels are of approximatelyequal distance from the middle one.22)

However, as a result of noise and digitization error,local geometric properties such as chord and gradientinformation often suffer from poor consistency and lowlocating accuracy in digital images. To avoid theseproblems, in this paper, we present a novel algorithm basedon the global geometric symmetry of circles, which hasbeen utilized by Ho and Chen.23) Unlike Ho and Chen’salgorithm, the proposed algorithm applies the peak-findingalgorithm instead of HT to look for the center of the circle,which leads to a considerable time-saving effect. Then anormalized radius histogram is used to verify the existenceof the circle and extract its radius. Experimental resultsdemonstrate that the proposed algorithm works moreefficiently than Ho and Chen’s algorithm, RHT, RCD, andPRCD.

The remainder of this paper is organized as follows:Section 2 describes Ho and Chen’s algorithm. A novelalgorithm is presented in detail in Sect. 3. Section 4illustrates some experimental results to confirm the com-putational advantage of the proposed algorithm. Finally,some conclusions are addressed in Sect. 5.�E-mail address: zhangfeng.zju@gmail.com

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2. Ho and Chen’s Algorithm

A circle has one symmetric horizontal axis and onesymmetric vertical axis, which is the global geometricsymmetry utilized by Ho and Chen. Their algorithm steps forcircle detection can be summarized as follows:23)

Step 1. Use an edge extractor (e.g., Sobel or Canny) to findthe edge points of objects in original image f and storethe result in a binary image F .

Step 2. Initialize a blank image G and scan F from left toright and top to bottom. For each edge point Aði; jÞ in F ,if there exists another edge point A0ðk; jÞ on the samehorizontal scan line, suppose Mðu; jÞ is the midpointof AA0, obviously u ¼ ðiþ kÞ=2; Set accumulatorGðu; jÞ ¼ Gðu; jÞ þ 1.

Step 3. Apply HT to G to extract all the vertical lines andrefer to them as candidate symmetric vertical axes lv;Group all symmetric points in F relative to lv into sub-images Fh.

Step 4. Initialize a blank image L and scan each Fh fromtop to bottom and left to right. For each edge pointBði; jÞ in Fh, if there exists another edge point B0ði; kÞon the same vertical scan line, suppose Nði; vÞ isthe midpoint of BB0, obviously v ¼ ð jþ kÞ=2; Setaccumulator Lði; vÞ ¼ Lði; vÞ þ 1.

Step 5. Apply HT to L to extract all the horizontal linesand refer to them as candidate symmetric horizontalaxes lh; Group all symmetric points in Fh relative to lhinto sub-images Fhv.

Step 6. Refer to the crossed point O of lv and lh as thecenter of a circle. Initialize a blank array AR andfor each edge point in Fhv, calculate the distance rfrom it to O and set accumulator ARðrÞ ¼ ARðrÞ þ 1.The peak-finding algorithm is finally applied to ARand the peak found is considered to be the radius of acircle.

Ho and Chen’s algorithm possesses its superiority inspeed and precision as compared with algorithms that usegradient information. However, its efficiency and robustnesscan still be improved in the following four aspects:(1) It is not an efficient way to extract candidate symmetric

horizontal and vertical axes by using HT.(2) Most of the sub-images contain no circles and will cost

a large amount of unnecessary computation time.(3) If the original image contains one non-circular graphic

(e.g., a diamond) that also possess horizontal andvertical symmetry as circles do, the algorithm does notexclude it and a false circle will be detected.

(4) Edge points on concentric circles are grouped into thesame sub-image. Since the peak-finding algorithm canonly find one peak to be the radius of a circle, only oneof the concentric circles is detected at last.

3. The Proposed Algorithm

This section proposes a new algorithm and consists ofthree subsections. The first subsection demonstrates howto detect one circle or a group of concentric circles. Themethod to detect multiple circles with different radii is

presented in the second subsection. The third subsection liststhe formal steps of the proposed algorithm.

3.1 Circle detectionInitialize a blank array H and scan the edge image F from

left to right and top to bottom. For each horizontal scan lineand any two edge points on it, suppose Mði; jÞ is themidpoint of the two edge points and then set accumulatorHðiÞ ¼ HðiÞ þ 1. After all the horizontal edge point pairshave been scanned, a histogram is available from theaccumulator H. We call it the horizontal midpoint histogram(HMH) of F . Similarly, we can obtain the vertical midpointhistogram (VMH) from another accumulator V afterscanning F from top to bottom and left to right. TakeFig. 1(a), a 300� 400 edge image, as an example. Itshorizontal and vertical midpoint histograms are shown inFig. 1(b).

Note that if there is a circle C in F with center Oðx0; y0Þ,each horizontal and vertical edge point pair on C shouldcontribute one to Hðx0Þ and V ðy0Þ, respectively, whichmeans the value of Hðx0Þ and V ðy0Þ must be large.Therefore, x0 and y0 can be found via the peak-findingalgorithm, i.e., the center of C is detected.

However, there is still a need for verification of thedetected center as it may not be a circle’s center but thegeometric center of a non-circular graphic (e.g., a diamond)that also possesses horizontal and vertical symmetry. Aftera possible center is found, we can use the histogram ofthe distances of all edge points from the center to verifythe existence of the circle and extract its radius. Sincecircles with different radii have different circumferences,employing a large threshold is unfair to those circles withsmall radii. Previously, Kulpa showed that as r ! 1, thenumber of pixels on the boundary of a circle with radius r is4

ffiffiffi

2p

r.24) According to that, we divide the radius histogramby 4

ffiffiffi

2p

r to make it normalized. Then the normalizedhistogram is searched for local peaks whose values exceed acertain threshold, which does not depend on the radius of thecircle and represents the percentage of the circumference ofthe circle present in the image. If no peak is found to meetthe condition, the candidate center is verified to be false.Usually, the threshold is set to 0.7 in real applications.

This algorithm is also effective to concentric circles,which will not be all detected via Ho and Chen’s algorithm.Figure 2(a) is a 400� 300 synthetic image containing a pairof concentric circles with radii 50 and 80. After detecting apossible center ð231; 176Þ, we obtain the normalized radiushistogram as shown in Fig. 2(b). Values of two peaks atlocation 50 and 80 both exceed the given threshold 0.7, i.e.,the two concentric circles are detected both.

3.2 Multi-circle detectionUsing the method described in Sect. 3.1, we are able to

detect one circle or a group of concentric circles. In manycases, however, there are multiple circles with different radiiin an image. We will detect them one by one until thenumber of detected circles reaches the given number. Afterone circle is detected, we eliminate the edge points lying on

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it and then update the HMH and VMH in order to detect thenext circle. Instead of completely rebuilding the HMH andVMH of the edge image, we use a more efficient methoddescribed below to update them.

For each edge point E to be eliminated and each edgepoint D on the same horizontal scan line with E, supposeKði; jÞ is the midpoint of DE, and then set accumulatorHðiÞ ¼ HðiÞ � 1. After all the edge points to be eliminatedhave been processed like this, the HMH is updated. TheVMH is updated in a similar way. Utilizing the updatedHMH and VMH, we can detect the next circle as we doin Sect. 3.1. Figure 3(a) is a 300� 400 synthetic imagecontaining two circles with different radii. After detectingthe larger circle in the lower left corner and eliminating theedge points lying on it, we are able to detect the smaller onein the upper right corner.

3.3 Algorithm descriptionUnlike Ho and Chen’s algorithm, the algorithm proposed

in this paper is effective to concentric circles and able toexclude false circles. It applies the peak-finding algorithminstead of HT to look for the center of the circle, whichleads to a considerable time-saving effect. Since most ofthe sub-images contain no circles, the idea of sub-image isalso abandoned to save a large amount of unnecessarycomputation time.

In summary, the proposed algorithm consists of thefollowing six steps:Step 1. Use an edge extractor (e.g., Sobel or Canny) to find

the edge points of objects in original image f and storethe result in a binary image F .

Step 2. Initialize a blank array H and scan F from leftto right and top to bottom. For each horizontal scanline and any two edge points on it, suppose Mði; jÞ is

(a)

(b)

Fig. 2. Detecting concentric circles. (a) A synthetic imagecontaining a pair of concentric circles. (b) The normalized radiushistogram with two peaks whose values both exceed the giventhreshold 0.7.

(a)

(b)

Fig. 1. Midpoint histograms. (a) An edge image. (b) HMH and VMH of (a).

OPTICAL REVIEW Vol. 20, No. 6 (2013)486 M. CHEN et al.

the midpoint of the two edge points and then setaccumulator HðiÞ ¼ HðiÞ þ 1.

Step 3. Initialize a blank array V and scan F from top tobottom and left to right. For each vertical scan line andany two edge points on it, suppose Nði; jÞ is the midpointof the two edge points and then set accumulatorV ð jÞ ¼ V ð jÞ þ 1.

Step 4. Apply the peak-finding algorithm to H and V , andthen a candidate center is found. Utilize the methoddescribed in Sect. 3.1 to affirm whether the candidatecenter is true. If yes, the radius is extracted simulta-neously by the method, i.e., a circle is detected, and thengo to Step 6. Otherwise, go to Step 5.

Step 5. Eliminate the edge points lying on the fake circle,update H and V by using the method described inSect. 3.2, and go to Step 4.

Step 6. Check whether the number of detected circles isequal to the given number. If yes, then stop. Otherwise,

eliminate the edge points lying on the newly detectedcircle, update H and V by using the method described inSect. 3.2, and go to Step 4.

4. Experimental Results

Lots of experiments have been carried out to validate thehigh speed and accuracy of the proposed algorithm. Herethree of them are illustrated, compared with Ho and Chen’salgorithm, RHT, RCD, and PRCD. For fair competition, thedetection for each algorithm will stop only if the numberof detected circles reaches the given number. These threeexperiments are performed on a Pentium IV 2.66GHz andmemory 1GB computer using C++ language.

The first experiment is tested on a series of syntheticimages that are created by adding random pepper noise to anoriginal image at various increasing levels. The original400� 400 synthetic image with 1605 edge points is shownin Fig. 4(a). It consists of two circles where one circle is

(a) (b)

(c) (d)

Fig. 3. Detecting multiple circles. (a) A synthetic image containing two circles with different radii. (b) HMH and VMHof (a). (c) The image after eliminating the edge points lying on the larger circle. (d) HMH and VMH of (c).

OPTICAL REVIEW Vol. 20, No. 6 (2013) 487M. CHEN et al.

ðx0; y0; rÞ ¼ ð292; 262; 74Þ and the other one ð112; 82; 59Þ,and some other graphics. In order to test the robustness ofthe proposed algorithm, we randomly add different levels ofpepper noise to the original synthetic image. Here, the levelsrange from 30%, i.e., adding 482 noises, to 210%, i.e.,adding 3371 noises. The resulting image with 3371 noises isshown in Fig. 4(b).

The execution time required in each algorithm ismeasured in terms of seconds and obtained from the averageof 100 simulations. Table 1 illustrates the execution timerequired in the five algorithms. It is observed that theexecution time required in the proposed algorithm is muchless than that in the other four algorithms. As the amount ofnoise increases, the execution time required in the other fouralgorithms grows dramatically, while that in the proposedalgorithm increases at a steady rate.

All of the five algorithms can correctly detect the twocircles to each synthetic image. Figure 4(c) shows the circlesin Fig. 4(b), which are detected via the proposed algorithm.The ‘‘+’’ marks indicate the detected circle centers. Theresults are ð292; 262; 74Þ and ð112; 82; 59Þ, which coincidewith the original ones. For the clearness, the midpoint

histograms of Fig. 4(b) are shown in Fig. 4(d). The peaksrepresenting the circle centers are well separated.

The second experiment is carried out on a real image. Theoriginal 400� 400 image as Fig. 5(a) is transformed by thepretreatment into the edge image as Fig. 5(b) with 2274points. The proposed algorithm, Ho and Chen’s algorithm,RHT, RCD, and PRCD are applied to detect the circles in theedge image. The average execution time of 100 simulationsis 0.142, 0.705, 1.321, 0.819, and 0.683 s, respectively. All

Table 1. Execution time comparison for synthetic images.

Noise Average execution time (s)

ratio(%)

The proposedalgorithm

Ho and Chen’salgorithm

RHT RCD PRCD

30 0.106 0.395 0.304 0.159 0.13760 0.161 0.878 1.017 0.594 0.51490 0.228 1.401 2.858 1.891 1.612120 0.309 1.962 6.546 4.937 4.183150 0.405 2.579 13.399 14.051 11.689180 0.512 3.234 26.432 29.283 23.928210 0.634 3.943 48.735 55.652 44.756

(a) (b)

(c) (d)

Fig. 4. The first experiment. (a) The original synthetic image. (b) The image with 3371 noises. (c) The circles detectedvia the proposed algorithm. (d) HMH and VMH of (b).

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of the five algorithms are able to detect the three circlescorrectly. The detected circles with the proposed algorithmare shown in Fig. 5(c). For the clearness, the midpointhistograms of Fig. 5(b) are shown in Fig. 5(d). The peaksrepresenting the circle centers are well separated.

Another 400� 400 real image as Fig. 6(a) is tested in thethird experiment. For the edge image as Fig. 6(b) with 5701points, the proposed algorithm, Ho and Chen’s algorithm,RHT, RCD, and PRCD are applied to detect the circles in it.The average execution time of 100 simulations is 0.537,4.986, 18.179, 14.263, and 11.402 s, respectively. Thedetected four concentric circles with the proposed algorithmare shown in Fig. 6(c). For the clearness, the normalizedradius histogram is shown in Fig. 6(d). Values of four peaksat location 67, 74, 136, and 165, all exceed the giventhreshold 0.7. As a result of its fourth shortcomingmentioned in Sect. 2, Ho and Chen’s algorithm cannotcorrectly detect all the four concentric circles, while theother four algorithms can.

The proposed algorithm has the ability to detectdiscontinuous circles, e.g., the second and third concentriccircles in Fig. 6(b). However, if the defectiveness of a

circle exceeds 30%, the algorithm may not work well. Theproposed algorithm focuses on circle detection and does notdeal with arc detection. It is not able to detect ellipses eithersince they do not have the global geometric symmetry ascircles do.

5. Conclusions

In this paper, we have presented a novel algorithm fordetecting circles in digital images. Based on the globalgeometric symmetry of circles, it uses scan lines to obtainthe horizontal and vertical midpoint histograms of the edgeimage. Then the peak-finding algorithm is applied to themidpoint histograms to look for the center of the circle.Finally, a normalized radius histogram is used to verify theexistence of the circle and extract its radius. Since each ofthe circle’s parameters is accumulated in its own one-dimensional histogram, the required computation time andmemory space are reduced significantly. Experimentalresults for synthetic and real images affirm that theapplication of the global geometric symmetry makes theproposed algorithm more efficiently than RHT and someother algorithms. The proposed algorithm does not use

(a) (b)

(c) (d)

Fig. 5. The second experiment. (a) The original real image. (b) The edge image. (c) The detected circles with theproposed algorithm. (d) HMH and VMH of (b).

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gradient information, the determination of which may bevery difficult in the presence of noise and digitization error,so the accuracy will not decline. In addition, more accurateradius can be achieved by fitting the points lying on thedetected circle using the least squares method.

Acknowledgments

The authors greatly appreciate the support of all members ofZhejiang Provincial Key Laboratory of GIS. This work wasfinancially supported by the National Natural Science Foundationof China (Grant Nos. 41101356 and 41171321).

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(a) (b)

(c) (d)

Fig. 6. The third experiment. (a) The original real image. (b) The edge image. (c) The detected circles with theproposed algorithm. (d) The normalized radius histogram.

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