ELLIPSE DETECTION USING THE HOUGH TRANSFORM

H.K. Yuen, J. Illingworth and J. KittlerDepartment of Electronics and Electrical Engineering

University of Surrey, Guildford, GU2 5XH. U.K.

We consider the problem of detecting elliptical curvesusing Hough Transform methods. Storage and efficiencyproblems are overcome by decomposing the problem intotwo stages. The first stage uses a novel constraint asthe basis for a Hough Transform to detect the ellipsecenter while the second stage finds the remaining param-eters using a simple but efficient focussing implementa-tion of the HT. The method is applicable in many situa-tions where previous HT schemes would fail. Results aredemonstrated for complex image data containing severaloverlapping and occluding ellipses.

1. Introduction.

The detection of elliptical curves or fragments ofsuch curves is an important task in computer vision asthese shapes occur commonly in many types of scene.Man made objects often have circular profiles which,when viewed obliquely, project to elliptical shapes ina 2D image. Ellipse detection is therefore a powerfulmethod of cueing into specific geometric object models.Also, the measured values of ellipse parameters providedata from which it is possible to infer the relative orien-tation of object and camera. The importance of ellipsedetection was recognised in the model based vision sys-tem, ACRONYM, developed by Brooks and BinfordW.They used the generalised cylinder as a basic modelingprimitive and had to extract lines and ellipses as ba-sic image features. Shortcomings of their system werelargely ascribed to poor segmentation and consequentunreliability of the feature extraction process.

The Hough Transform, HT, is a method of parame-ter extraction whose properties make it particularly ap-propriate for the detection of shapes within poorly seg-mented imagery'2'. It is a method which takes piecesof local evidence and votes for all parameter combina-tions which are consistent with this evidence. The votesare collected in an array of counters which is called theaccumulator array. The accumulator array is a discretepartitioning of the multidimensional space which spansthe possible parameter values. Image points on a shapevote coherently into the accumulator counter which rep-resents the parameters of the shape while votes fromisolated or random image points combine only incoher-

ently. At the end of the voting or accumulation processthose array elements containing large numbers of votesindicate strong evidence for the presence of the shapewith corresponding parameters. These properties meanthat the method is insensitive to image noise but it canstill detect shapes using the type of fragmentary evi-dence which results from both poor image segmentationand object occlusion.

The principal disadvantage of the HT is that it isdemanding both in terms of the amount of computerstorage required to represent the multidimensional pa-rameter space and the amount of computation to carryout the accumulation of votes in this space. Ellipsedetection requires the determination of 5 parameters.Each parameter has a large range of values and if uni-formly high accuracy of representation is needed overthis range then resource demands become unpracticallylarge. In this paper we consider how ellipse detectioncan be made a practical proposition by using edge di-rection information and decomposing the ellipse detec-tion problem into two sequentially executed HT stages offirst finding the center of the ellipse and then determin-ing the remaining three parameters. Although this de-composition idea has been used by Tsukune and Goto'3'and Tsuji and Matsumoto'4' we propose an alternativemethod of ellipse center finding that will work in situa-tions where the previous methods fail. In addition thesecond stage of our algorithm utilises a multiresolutionapproach to detect peaks in the HT space of the remain-ing three parameters. In Section 2 we discuss both thegeneral principles of the ellipse finding problem and spe-cific details pertaining to our method. Section 3 gives anapproximate analysis of the complexity of the methodand Section 4 includes results of experiments on com-plex image data containing several overlapping ellipses.Concluding remarks and comments are included in Sec-tion 5.

2. General principles.

The general equation of an ellipse is,

X2 + B'Y2 + 2D'XY + 2E'X + 2G'Y + C' = 0, (1)where B', D', E', G' and C" are constant coefficients nor-malized with respect to the coefficient of X2. If we use

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the standard HT method with this equation, we haveto construct a five dimensional parameter space and ifeach parameter range is divided into a intervals then theaccumulator space requires a5 storage locations. Evenfor modest values of a this becomes unpractically large.However the problem can be made much more tractableif we consider using edge direction information and de-composing it into a series of parameter estimation steps,each of which is computationally less demanding. Theexact problem decomposition can be achieved in a vari-ety of ways and this approach has been used by severalauthors'3'4'. The problem decomposition which we use

• in this paper includes a novel method to find the param-eters which describe the ellipse center. This is followedby an efficient implementation to find the three remain-ing parameters.

2.1 Stage 1: Center finding.

Center detection is probably the most importantstage in the detection of ellipses. One commonly usedmethod involves the determination of the mid-point ofthe line between two image points with parallel tangents.If these two points are on the same ellipse then the mid-point is the center of the ellipse. This principle can beused as the basis of a HT method. The first step ofthe procedure is to extract pairs of image points whosegradients are almost the same. The next step is to con-struct a two dimensional histogram of the mid-points ofsuch pairs. The histogram locations which receive highcounts become candidates for the center of an ellipse.This procedure is the one used by Tsukune and Goto'3'and Tsuji and Matsumoto'4'.

A disadvantage of the above procedure is that itfails if the image does not contain segments which aresymmetrical about the ellipse center. This type of situ-ation is common where objects are viewed so that theyare occluded by other objects or by parts of themselves.In this paper we propose another center finding proce-dure, which can be used to detect the center of an ellipsein a more general setting. The new procedure is basedon a simple property of ellipses. Suppose P(xi,yi)and Q(x2,J/2) are two points on an ellipse with non-parallel tangents intersecting at a point T(ti,tz). LetAf(mi,m2) be the mid-point of PQ. Then the centerof the ellipse must lie on the straight line TM, as illus-trated in Figure 1. The form of TM has been foundas,

with

y(ti — mi) = x(*2 — rnt) + ) (2)

yi - y 2 - s i 6 + J266-6

66(^2 - *i)

6-6

where ft and £2 are the slopes of the tangents to theellipse at the points P and Q respectively.

Lines in the form of equation 2 can be constructedfrom different pairs of image points, and they shouldintersect at the same location if they are on the sameellipse. A standard HT can be used to accumulate theselines in a two parameter accumulator array. The ac-cumulator cell which receives the highest count is thecandidate for the center of an ellipse.

2.2 Stage 2: Determination of remaining parameters.

Once the location of the center of an ellipse has beenestimated, the subset of image points which are consis-tent with it can be found by a second pass through data.The origin relative to which the points are described, canbe translated so as to coincide with the estimated ellipsecenter and then the equation describing the points of theellipse becomes

X2 + BY2 + 2DXY + C = 0, B - D2 > 0. (3)

There are several ways in which the remaining pa-rameters B, C and D can be estimated. Tsuji andMatsumoto'4' estimate all five parameters of the bestellipse formed by the subset of points using a least meansquares fitting procedure. As with all least squares pro-cedures, this is likely to be highly sensitive to spuriousoutlier points. In the work of Tsukune and Goto!3' thethree parameter estimation problem is decomposed intotwo further steps using the property that differentiationof equation 3 with respect to X yields

M

Using measured gradient values equation 4 forms the ba-sis of a two dimensional HT method to determine B andD. Once B and D are known a simple one dimensionalHT can be used to estimate C using equation 3. Thedrawback of this approach is that the range of the pa-rameters is large and even a two dimensional parameterspace can require a large amount of computer storage.Also the efficiency advantages of the decomposition intoan increasing number of stages must be weighed againstthe systematic effects of propagating errors through sev-eral stages. As the number of stages increases so doesthe reliance of the results on the accurate determinationof gray level edge directions.

In this paper we determine all three remaining pa-rameters simultaneously using equation 3 and a spaceefficient peak detection algorithm based on the AdaptiveHough Transform, AHT, method'5l. The peak detectionalgorithm locates the peak areas of the Hough Trans-form space without computing the background level infull detail. The algorithm is based on an iterative 'coarse

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to fine' accumulate and search strategy. A small fixedsize accumulator is used. In our work we use a 9x9x9accumulator. The accumulator always partitions thecurrent range of parameters into these few intervals.During the first iterative step the range of each param-eter is large and thus each parameter is only coarselyresolved. The HT is accumulated and the accumulatorcell with maximum counts is identified. The parameterrange covered by the accumulator is then centered aboutthe maximum cell and, if the parameter is not resolvedto the required accuracy, the range of that parameter isreduced to one third of its previous value. This basiccycle of accumulation and parameter range reduction isiterated and the method focusses in on, and accuratelydefines, the position of the most prominent peak in theparameter space. In our ellipse finding application thesubset of points from the first stage of the method arepredominately from a single ellipse and therefore the fo-cussing works well. Apart from its efficiency, anotheradvantage of using the focussing is that it can dynam-ically and independently vary the resolution to whicheach parameter is measured. This is particular impor-tant for estimating the parameters B, D and C, becausethe range of values of C are usually much larger thanthose of B and D and therefore it needs to be more ac-curately determined. After determination of all threeparameters, the length of the ellipse axes, a and b, andits angle of rotation, 9, can be calculated using

- 2 C

\xi - x2\ > S2, or |yi - y2\ > fa-

where Si and S2 are some pre-determined values.

Let £ be the slope of the line TM determined inequation 2. The center histogram of the new methodcan be simplified further if we notice that

x < or x> mi>

m2 if III > 1.

a = [(B

where (x, y) is the co-ordinate of the center, say. If Nis a point on the line TM such that the center of theellipse lies on MN then instead of accumulating votesfor all points along TM we only have to count the votesfor the section MN. The length, L, of MN can be pre-determined based on prior knowledge of the likely sizesof ellipses in the image.

Once the center finding HT has been accumulated,the parameters of a possible ellipse center are identifiedby determining the accumulator cell with the largestnumbers of counts. If the number of counts exceeds apredetermined threshold then the center is accepted asvalid and the subset of image points supporting this cellare identified and passed to the second stage which findsthree parameters B, D and C. Once points have beenidentified with a candidate center they are deleted andthe remaining points can again be passed through thecenter finding accumulation stage. This center findingprocedure is repeated until the maximum of the accu-mulator falls below the predetermined threshold.

6 =- 2 C

(B + 1) +

= itan-

2.3 Practical considerations.

l-B

In our proposed method of ellipse finding the HTspace for center finding may become complicated whenthere are several ellipses in an image. This is causedby background votes generated by pairing image pointson different ellipses. A complex parameter space makespeak detection a difficult task and is therefore highlyundesirable. In order to overcome or reduce these effectswe can invoke aprior knowledge of likely ellipse size orinter ellipse spacing to vote the pairing of image pointswhich are unlikely to lie on a common ellipse. It is alsonot necessary to pair two image points if they are tooclose to each other. Based on these considerations, weonly have to pair two image points, [xi, t/i) and (x2, y2),if they satisfy the conditions,

3. Complexity of the algorithm.

In this section we make an approximate calculationof the complexity of our algorithm making the simply-fying assumption that Si and 62 are suitably chosen sothat there is little pairing between points of differentellipses. If there are k ellipses and each ellipse is com-posed of Hi image points then the number of votes castby the Ph ellipse will be the product of the number ofimage point pairs in that ellipse and the number of voteswhich each pair contributes, viz

- 1 )

where L is the prespecified length of the incrementedline MN. Note that the complexity of forming pairs is0(n?) whereas the center finding methods used in [3,4]are only linearly dependent on n<. The total number ofvotes cast in determining the first centre location willbe the sum of these votes for all A; ellipses

~x2\ < Si, and (5)

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In determining the second center location the imagepoints associated with the first are removed and the HTis reaccumulated-. This process of point removal andsubsequent HT accumulation continues until all k el-lipses have been found. Thus the total number of votescast in the center finding stage is

k-lk-j

vtnt =- 1)

y=o«=i

The actual number of votes cast will be dependent onthe rii and the order in which the ellipses are found. Theminimum number of votes will occur when the largestellipses are found before smaller ones. The simplest up-per limit on the cost is to assume that no points arediscarded and therefore in finding k ellipses the accu-mulation is iterated k times. Thus the cost of finding kellipses would be O(k2) that of finding a single ellipse.A special case of particular interest, and maybe moreindicative of the average case complexity, is when all el-lipses are of the same size. In this case the cost for Vtotis O(fc(fc*^). Obviously, this method of iterative ac-cumulation is undesirably inefficient but it seems to benecessary to achieve reliable identification in a complexcenter finding space. In future work we hope to inves-tigate more efficient ways of analysing the parameterspace.

The second stage of the algorithm using the itera-tive focussing algorithm requires a 9x9x9 accumulatorarray. At each iteration, every image point has to beevaluated to determine the subset of accumulator cellsthat its parameter surface intersects. This procedurerequired 3x92 evaluations of the function in equation 3.Assuming the cost of each function evaluation is com-parable to the basic operation of determining and incre-menting a cell in stage 1 then the cost per iteration ofthe second stage is very approximately proportional to

Cx = 3 X 92 X

The maximum number of iterations required to localisethe peak is governed by the parameter whose rangeneeds to be most finely divided. However, as each it-eration produces a range reduction to | of the previousvalue the process quickly converges. In our experimentsthe focussing never required more than 10 iterations toachieve the required resolution. Therefore the cost ofthe second stage will be

In this section we demonstrate the proposed twostage algorithm using computer generated ellipse data.Figure 2 shows 67 edge points from a segment of anellipse with center at (116.8,85.72). The edge pointsalong with edge gradient directions are extracted usingSpaceks edge detection algorithm!6!. Other boundarydetection methods which accurately measure edge gra-dient or curve tangent information can be used. Forthe center finding algorithm we set the values of <5j and82 to 5 and 30 pixels respectively. The length, L, ofthe line MN was taken to be 30 pixels. The HT wasaccumulated in a 256x256 accumulator array. Usingthe new center finding algorithm, the center is foundat (116.5 ± 0.5,86.5 ± 0.5). It should be stressed thatprevious center finding methods based on parallel tan-gents axe not applicable in this situation. In the sec-ond stage of the algorithm the initial ranges for C, Band D were chosen arbitrarily large so that they en-compassed the parameter values for any realistic ellipsewhich was contained wholly within the 256x256 image.The rapid convergence inherent in the focussing HT im-plementation meant that this incurred no significant ex-tra computational cost. The parameter values eventu-ally found were 5=0.93827 ± 0.25, D=-0.17284 ± 0.25and C=-382.2588 ± 0.5. The lengths, a and b, of themajor and minor radius and the angle of rotation, 8 ofthe ellipse were calculated from these estimates and werefound to be a=21.948, 6=18.274 and 5=50.062°. Thesecompare with true values of a—22.231, 6=18.437 and0=50.675°. The image generated from the estimatedparameters is given in Figure 3.

Figure 4 shows a more complex image consisting ofeight ellipses with a total of 657 edge points. Numbershave been overlayed on the figure and these are used inTable 1 which summarises the true and estimated val-ues for the ellipse parameters. Figures 5a-5d show edgepoints that remain at some of the intermediate stagesof the method after particular edge points have been as-sociated with candidate center locations. Most of theellipses have their parameters estimated well except forthe ellipse centered at (140.24,35.78). The center of theellipse was found well but the method failed in the sec-ond stage. However it should be noted that this estima-tion is based on only 14 points and the curve fragmentis so small that it may be equally well interpreted aspart of some other ellipse. Figure 6 shows the imagegenerated from the estimated parameters.

C2 = 10 X 3 X 92 Xt = i

4. Experimental results.

5. Discussion and conclusions.

The new method of ellipse center finding which wesuggest requires less restrictive assumptions than theone commonly used. It is likely to be applicable to amuch larger range of practical problems because effects

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such as self occlusion lead to image fragments which,although elliptical, are highly asymmetric about the el-lipse center. The price paid for this increased appli-cability is an increase in the number of computations.However, the basic operation in the center finding stepis the voting for a line of points in accumulator spaceand this could become an inexpensive primitive oper-ation if implemented in dedicated hardware. The fo-cussing method used at the second stage works well andis efficient. At the moment, we overcome some of thepeak detection problems in the center finding parameterspace by deleting image points associated with candi-date centers and then reaccumulating the HT with thereduced list of image points. This is an inefficient pro-cedure and future work will attempt to develop waysof simplifying!7! the parameter space to avoid wastefulreaccumulation.

Acknowledgements.

This work was carried out as part of Alvey projectMMI-078, a collaborative project between Surrey Uni-versity, Heriot-Watt University and Computer Recogni-tion Systems.

References.

[1] Brooks R.A., Model Based Computer Vision, (Com-puter Science: Artificial Intelligence No 14), UMIResearch Press, Ann Arbor, Mich. 1981.

[2] Illingworth J. and Kittler J. » A survey of the HoughTransform.", accepted for publication in ComputerVision, Graphics and Image Processing (1988).

[3] Tsukune H. and Goto K. "Extracting elliptical fig-ures from an edge vector field.", IEEE ComputerVision and Pattern Recognition Conf, Washingtonpp 138-141 (1983).

[4] Tsuji and Matsumoto F. "Detection of ellipses bya modified Hough transformation.", IEEE Transac-tion on Computers, C-27, No 8, pp 777-781 (1978).

[5] Illingworth J. and Kittler J. "The Adaptive Houghtransform.", IEEE Trans, in Pattern Analysis andMachine Intelligence, Vol 9, No 5, pp 690-697 (1987).

[6] Spacek L.A., "Edge detection and motion detec-tion", Image and Vision Computing, Vol 4 No 1pp 43-56 (1986).

[7] Gerig G., "Linking image-space and accumulatorspace: a new approach to object recognition", 1'*IEEE Int. Conf. on Computer Vision, London, pp112-117 (1987). P P

Fig.(l) Basic geometry for center finding. The center of an ellipse must lie on the line MN.

Fig.(2) The image of a segment of an ellipse. Fig. (3) Reconstruction of the image of Fig. (2)

based on the estimates of the parameters.

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Fig.(4) The image of 8 ellipses.

(5c) (5d)

Fig.(5a-d) The remaining edge points at some of the intermediate stages after particular

edge points have been associated with candidate center locations.

270

Fig.(6) Reconstruction of the image of Fig.(4) based on the estimates.

Table 1. The true and estimated values of the parameters of ellipses as shown in Fig. (4) and Fig. (6) respectively.

Ellipses

1

2

3

4

5

6

7

8

CentersTrue

84.56110.98

120.56110.98

100.1260.24

130.4930.68

99.25140.56

130.90150.56

250.0556.90

140.2435.78

Estimated

81.5109.5

118.5110.5

100.560.5

129.531.5

98.5145.5

130.5150.5

249.557.5

141.534.5

True

25.57.25

25.57.25

12.358.35

30.6715.43

32.6721.98

22.3515.67

30.512.67

17.9511.45

RadiiEstimated

23.457.26

22.016.64

14.058.46

29.0416.38

27.5520.70

24.1415.90

32.9413.04

27.8614.13

AnglesTrue

20

20

80

145

85

33.75

128.34

32.49

of rotationEstimated

19.76

20.41

74.43

145.36

66.26

31.17

127.52

19.20

271

272