A dynamic method for dominant point detection

A dynamic method for dominant point detectionWen-Yen Wu *Department of Industrial Engineering and Management,I-Shou University,Kaohsiung 84008,TaiwanReceived 16April 2001;received in revised form 31July 2002;accepted 20November 2002AbstractDetecting dominant points is an important step for shape representation.Most of domi-nant point detection methods attend to preset or find the region of support of each point.In this paper,we demonstrate an improved method for determining the region of support in the dominant point detection.Instead of setting the regions of support for points indepen-dently,the region of support dynamically depending on the previous region of support.The experimental results show that the proposed method is effective in detecting dominant points.Ó2003Elsevier Science (USA).All rights reserved.Keywords:Curvature;Region of support;Polygonal approximation;Dominant points1.IntroductionAttneave [2]pointed out that information on a curve is concentrated at the dom-inant points.Dominant points are those points that have curvature extreme on the curve and they can suitably describe the curve for both visual perception and recog-nition.Over the years,many approaches have been proposed to detect the dominant points on digital curves.In general,the two major categories of approaches are corner detection [3,4,7,8,11,13]and polygonal approximation [1,5,6,9,10,12,14,15].The purpose of corner de-tection is to detect some potential points that can sufficiently represent the shape of the object.Corner detection approaches can detect potential significant points,but they cannot represent smooth curve appropriately.For polygonal approximation ap-proaches,sequential,iterative,and optimal algorithms are commonly used.MostofGraphical Models 64(2003)304–315*Fax:+886-7-6578536.E-mail address:wywu@.tw.1524-0703/03/$-see front matter Ó2003Elsevier Science (USA).All rights reserved.doi:10.1016/S1077-3169(02)00008-4W.-Y.Wu/Graphical Models64(2003)304–315305 the sequential techniques are simple and fast,but they have the drawbacks of missing important features.The family of iterative approaches splits and merges curves iter-atively until they meet the preset allowances.The iterative approaches suffer the sen-sitivity to the selection of the starting points for partitioning curves.For optimal approaches,they tend tofind the optimal polygonal approximation based on spec-ified criteria and error bound constraints.They have the disadvantages of large com-putation cost and memory requirement.In this paper,an improved method is proposed to determine the region of support infinding dominant points.We modify the method proposed by Teh and Chin[11] by setting the next region of support depending on the present region of support.The new choosing strategy can solve the problem of detecting too many dummy domi-nant points.In Section2,we will illustrate the proposed method for determining the region of support.Section3will present the experimental results of the new strat-egy.Some concluding remarks are then given in Section4.2.Dominant point detectionDominant points are commonly considered as points with local maximum curva-ture.In the Euclidean plane,curvature can be easily defined as the rate of change of slope as a function of arc length.Unfortunately,the above definitions do not hold for a digital curve,since there exist no exact mathematical definition for the digital curve.Therefore,the existing corner detection algorithms involve in defining an es-timation method for curvatures.Most of the algorithms use the information of the k-nearest neighbors and overcome the fundamental problem of discrete property by defining a region of support to compute curvature measure.The dominant points can then be detected by the following two common stages:(1)estimate the curvature for each point on the digital curve and(2)locate the points with local maximum cur-vatures as the dominant points.Teh and Chin[11]claimed that a precise determination of region of support was more important than the chosen curvature estimation.In order to solve the region of support setting problem,they presented a nonparametric method.Based on the local properties,the region of support of each point can be determined independently.The chord length and perpendicular distance of the point to the chord were used to de-termine the region of support.The region of support is symmetric with respect to each point.Once the region of support has been determined,various curvature mea-sures,such as k-cosine,k-curvature,and1-curvature measures,can be used to detect dominant points.However,Cornic pointed that the Teh–Chin algorithm is known to be not very ro-bust in presence of noise[3].In addition,the experimental results show that their method detects too many dummy dominant points in most cases.It is due to that their method tends tofind small region of support for each point.It has been seen that a small region of support will detect more dominant points than that of a large region of support.In addition,it is seen from the observations that the supporting regions of the points on a curve are dependent each other.That is,the determination of region ofsupport for each point can be done dynamically by using the information of the pre-vious found regions of support.The initial value of region of support is dynamically set in each stage of determination of region of support.Based on the above observa-tions,we propose a dynamic approach to determine the region of support.2.1.Break point detectionA digital closed curve C can be defined as the set consisting of n consecutive pointsC¼f pij i¼1;2;...;n g;ð1Þwhere pi is the i th point with coordinate(x i;y i),and points piÀ1and piþ1are theimmediate backward and forward neighbors of point pi (modulo n),respectively.The FreemanÕs chain code[4]of C consists of the n vectors.The vector is defined as the difference between two consecutive points,namely w i¼p i p iþually,the concept of8-connectness is used for tracing a curve.Therefore,each of the vectors is assigned to an integer varying from0to7(see Fig.1).The chain codes of C are defined as c1c2ÁÁÁc n.An example of chain codes for a curve is shown in Fig.2.Since the points on straight line cannot be considered as the dominant points,they can be excluded by tracking the chain codes[14].The survived points are candidates of dominant points and denoted as the break points.It will reduce the computation time both in determination of support region and curvature estimation,if only the break points are considered as the possible dominant points.Fig.2shows an exam-ple of linear points as well as break points on a curve.The following rule is used to identify those linear points.Rule1.If c iÀ1¼c i,then the point p i is a linear point.2.2.Determination of region of supportSuppose that l ik and d ik are the length of the chord piÀk piþkand the perpendiculardistance from point pi to the piÀkpiþk,respectively.The ratio of the deviation and thechord length is defined asr ik¼d ikl ik:ð2ÞIn Teh–ChinÕs algorithm[11],if one of the following two conditions is satisfied,the length of supporting region for point piis denoted as k.Condition A.l i;k P l i;kþ1:ð3ÞCondition B.r i;k P r i;kþ1:ð4ÞTable1Results of the proposed method and of the Teh–Chin methodImage Method n d CR MSE MAE PT Leaf Fig.2(a)Teh–Chin3724.8949.169.58 3.87ðn¼921ÞFig.2(d)Dynamic1561.40282.6935.43 1.72 Leaf Fig.2(b)Teh–Chin2824.0736.98 5.59 2.16ðn¼674ÞFig.2(e)Dynamic1544.93165.2022.40 1.12 Leaf Fig.2(c)Teh–Chin5915.5418.26 2.75 3.52ðn¼917ÞFig.2(f)Dynamic1561.13288.2836.59 1.60 Screw Fig.3(a)Teh–Chin6014.6113.54 1.87 3.57ðn¼877ÞFig.3(d)Dynamic5615.6613.73 1.43 1.13 Screw Fig.3(b)Teh–Chin5711.478.95 1.62 2.50ðn¼654ÞFig.3(e)Dynamic5511.897.730.90 1.02 Screw Fig.3(c)Teh–Chin7112.4513.46 2.37 3.40ðn¼884ÞFig.3(f)Dynamic5615.7816.31 1.80 1.42W.-Y.Wu/Graphical Models64(2003)304–315307In this paper,the determinant of support region is designed to depend on the pre-vious region of support.In fact,the starting value of k will be set to the previous re-gion of support of the break points.Suppose that there are m break points on thecurve.The length of supporting region for break point qi is denoted as k i,and is de-termined by the following algorithm.Algorithm(Determination of length of supporting region).Step1.Start with thefirst break point q1.Let i¼1and k¼0.(a)Increase k by1.(b)If Condition A or Condition B is satisfied,then k1¼k is the length of supportingregion for q1.Otherwise,go to(a).Step2.Determine the length of supporting region for qi by the following rule,fori¼2;3;...;m.Fig.3.The detected dominant points of leaf curves under different scales and orientations:(a)–(c)Teh–Chin method and(d)–(f)proposed methods.308W.-Y.Wu/Graphical Models64(2003)304–315(a)Set k¼k iÀ1.Let j¼0and s¼À1.(b)If Condition A or Condition B is satisfied,then k i¼k is the length of supportingregion for qi .Go to(a)for next i.(c)If s¼À1,then increase j by1and change s to1.Otherwise,change s to)1k¼k iÀ1þsÂj:ð5ÞGo to(b).The determination of support region for thefirst break point is similar to that of Teh–Chin method.The length of supporting region of thefirst break point is found in Step1.The strategy for adjusting k value in Eq.(5),is to vary the k value to differ from its previous length of supporting regionÆ1;Æ2;...,and so on until it meets theCondition A or Condition B.Suppose that the i th break point qi is the b i th point onthe curve.The region of support of qiis determined as the setS i¼f p j j p j is a break point for j¼b iÀk i;...;b iþk i g:ð6Þ2.3.Dynamic dominant point detection algorithmIn this paper,the k-cosine value is used as the curvature measure[8]Fig.4.The detected dominant points of screw curves under different scales and orientations:(a)–(c)Teh–Chin method and(d)–(f)proposed methods.W.-Y.Wu/Graphical Models64(2003)304–315309cos ik ¼a *ikÁb *ika *ikb *ik;ð7Þwhere a *ik ¼ðx i þk Àx i ;y i þk Ày i Þ,b *ik ¼ðx i Àk Àx i ;y i Àk Ày i Þ,Áis the inner product operator,and j Ãj is the norm of the vector.Once the curvatures of all of the break points have been calculated,the next step is to locate the points that have local maximum curvature.In addition,the points that the curvatures are less than a threshold are removed from the set of candidates for dominant points.Overall,the dynamic dominant point detection can be summarized asfollows:Fig.5.Results of the determination of support for the leaf curve ðn ¼921Þ:(a)Teh–Chin method and (b)the proposed method.310W.-Y.Wu /Graphical Models 64(2003)304–315Algorithm(Dynamic dominant point detection).Step1.Extract break points from FreemanÕs chain codes.Step2.Determine region of support for each break pointðk i;i¼1;2;...;mÞby the proposed dynamic method.pute the estimated curvatures for all of the break points(cos i;ki ,i¼1;2;...;m).Step4.Suppress those break points whose curvature less than a preset threshold . Step5.Locate the points that have local maximum curvature among their region of support as the dominant points.If two successive dominant points detected, remove the point with smaller region ofsupport.Fig.6.Results of the curvature estimation for the leaf curveðn¼921Þ:(a)Teh–Chin method and(b)the proposed method.W.-Y.Wu/Graphical Models64(2003)304–3153113.Experimental results and discussionsFive performance evaluation criteria were used in the experiment;they are the num-ber of the dominant pointsðn dÞ,compression ratio(CR),mean area error(MAE), mean square error(MSE),and processing time(PT).They are defined as follows:CR¼nn d;ð8ÞMAE¼1n dX n di¼1a i;ð9ÞMSE¼1nX ni¼1e2i;ð10Þwhere e i is the distance from pi to the approximated segment,and a i is the areaenclosed by the original curve and the approximatedsegment.Fig.7.Results of polygonal approximation(clockwise):(a)chromosome,(b)infinity,(c)leaf,and(d) semicircle.312W.-Y.Wu/Graphical Models64(2003)304–315W.-Y.Wu/Graphical Models64(2003)304–315313 In the experiment,the curvature threshold is set to)0.5.Table1lists the result of the Teh–Chin method and of the proposed method.For the leaf curves,the Teh–Chin method detects37,28,and59dominant points,respectively.However,the proposed method detects15dominant points for all of the three cases.That is, the proposed method is very insensitive to the problem of scale and orientation. The same result can also be seen for the screw curves.In addition,the proposed method have larger compression ratio than that of the Teh–Chin method.It is due to Teh–Chin method tends tofind a small region of support,and therefore de-tects many dummy dominant points.Because of the proposed method detects only the most significant dominant points,the mean square error and mean area error for the proposed method are larger than that of the Teh–Chin method for the leaf curves.However,although the numbers of dominant points of the Teh–Chin meth-theod is larger than that of the proposed method,the Teh–Chin method has both Array Fig.8.Results of polygonal approximation(counter-clockwise):(a)chromosome,(b)infinity,(c)leaf,and (d)semicircle.314W.-Y.Wu/Graphical Models64(2003)304–315larger mean square error and mean area error for the screw curves.Furthermore,the proposed method is faster than the Teh–Chin method for all cases,which can be found by comparing the processing times.The detected dominant points of leaf curves and screw curves by Teh–Chin method as well as the proposed method are shown in Figs.3and4.Fig.5shows the lengths of supporting regions for the leaf curve in Fig.3(a).From the plot of supporting lengths,it is seen that the variations of the k-values for the proposed method is larger than that of the Teh–Chin method.That is,the proposed method can dynamicallyfind a proper region of support of each point,especially for the multi-scaled curves.The estimated curvatures are shown in Fig.6.Again,it is seen that the variations of the estimated curvatures by the proposed method are lar-ger than that of the Teh–Chin method.It is due to the accurate determination of sup-porting region for each point.Since the dynamic method for determining supporting region may depend on the tracing direction of a curve,another experiment has been conducted on the curves of chromosome,infinity,leaf,and semicircle.Figs.7and8show the results of tracing curves clockwise and counter-clockwise,respectively.Although the detected domi-nant points are different in the twofigures,both of the detected dominant points are sufficient to represent the corresponding curves.4.ConclusionsIn this paper,we propose a modified approach to determine the region of support. Instead of setting the supporting regions for each point independently,the next sup-porting region dynamically depending on the present region of support.The modi-fied strategy willfind the region of support faster than the traditional method.In addition,only the break points are considered as the candidates of dominant points. It will reduce the computation time on both the determination of supporting region and curvature estimation.Further,The experimental results show that the proposed method is insensitive to noise and it is robust on scale and orientation changing. 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