International audienceA new and fast method for dominant point detection and polygonal representation of a discrete curve is proposed. Starting from results of discrete geometry, the notion of maximal blurred segment of width v has been proposed, well adapted to possibly noisy and/or not connected curves. For a given width, the dominant points of a curve C are deduced from the sequence of maximal blurred segments of C in O(nlog^2n) time. Comparisons with other methods of the literature prove the efficiency of our approach

Debled-Rennesson, Isabelle

Nguyen, Thanh Phuong

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HAL Id: hal-00437293https://hal.archives-ouvertes.fr/hal-00437293Submitted on 30 Nov 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Fast and robust dominant point detection on digitalcurvesThanh Phuong Nguyen, Isabelle Debled-RennessonTo cite this version:Thanh Phuong Nguyen, Isabelle Debled-Rennesson. Fast and robust dominant point detection ondigital curves. International Conference on Image Processing, Nov 2009, Caire, Egypt. pp.953-956.￿hal-00437293￿FAST AND ROBUST DOMINANT POINTS DETECTION ON DIGITAL CURVESThanh Phuong Nguyen, Isabelle Debled-RennessonLORIA Nancy, Campus Scientifique - BP 23954506 Vandoeuvre-lès-Nancy Cedex, FranceABSTRACTA new and fast method for dominant point detection and polygonalrepresentation of a discrete curve is proposed. Starting from resultsof discrete geometry [1, 2], the notion of maximal blurred segmentof width ν has been proposed, well adapted to possibly noisy and/ornot connected curves [3]. For a given width, the dominant points ofa curveC are deduced from the sequence of maximal blurred seg-ments ofC in O(n log2 n) time. Comparisons with other methodsof the literature prove the efficacity of our approach.Index Terms— corner detection, dominant point, critical point1. INTRODUCTIONThe work on the detection of dominant points started from the re-search of Attneave [4] who said that the local maximum curvaturepoints on a curve have a rich information content and are sufficientto characterize this curve. Therefore, these points play a critical rolein curve approximation, image matching and in other domains ofmachine vision. Many works have been realised about the dominantpoint detection and an interesting survey is presented in [5]. Severalproblems have been identified in the different approaches: time com-putation, number of parameters, selection of start point, bad resultswith noisy curves, ...In this paper, we present a new fast and sequential method issuedfrom theoretical results of discrete geometry, it only requires to fixeone parameter, it is invariant to the choice of the start point and itworks naturally with general curves : possibly being noisy or dis-connected. It relies on the geometrical structure of the studied curveobtained by considering the decomposition of the curve into maxi-mal blurred segments for a given width [3].In section 2, we recall theoretical results of discrete geometryused in this paper to analyse a curve. The section 3 describes ourmethod for dominant point detection. Finally, the section 4 presentsexperimental results and comparisons with other methods.2. DECOMPOSITION OF A CURVE INTO MAXIMALBLURRED SEGMENTSThe notion of blurred segment [2] relies on the notion of arithmeticdiscrete line [6]. Anarithmetic discrete line, notedD(a, b, µ, ω),is a set of points (x, y) that verifies this double inequation:µ ≤ax − by < µ + ω with a, b,µ, ω integer parameters. Awidth νblurred segment is a set of integer points which belong to a discreteline D(a, b, µ, ω) verifying ω−1max(|a|,|b|)≤ ν. The notion ofwidthThis work is supported by the ANR in the framework of the GEODIBproject, BLAN06-2 134999.ν maximal blurred segment, used in this paper, was proposed in [3](deduced from [1,2]). We consider a discrete curveC = {Ci}i=1..nof n points, let us recall that the predicate ”Si,j , a set of points in-dexing from i to j inC, is a blurred segment of widthν” is noted byBS(i, j, ν).Definition 1Si,j is called a width ν maximal blurred segment and notedMBS(i, j, ν) iff BS(i, j, ν) and¬BS(i, j + 1, ν) and¬BS(i −1, j, ν).An algorithm is proposed in [3] to determine the sequence of max-imal blurred segments of widthν of a discrete curveC of n points.The complexity of this algorithm isO(n log2 n). For a given widthν, the sequence of the maximal blurred segments of a curveC en-tirely determines the structure ofC.Let C = {Ci}i=1..n be a discrete curve andMBSν(C) the se-quence of all maximal blurred segments of C, in which theith max-imal blurred segmentMBS(Bi, Ei, ν), is a set of point indexingfrom Bi to Ei. We recall below two important properties [3].Property 1Let MBSν(C) the sequence of widthν maximal blurred seg-ments of the curveC. Then,MBSν(C) = {MBS(B1, E1, ν),MBS(B2, E2, ν), ..., MBS(Bm, Em, ν)} and satisfiesB1 <B2 < ... < Bm. So we have:E1 < E2 < ... < Em.Property 2LetL(k), R(k) be the functions which respectively return the indicesof the left and right extremities of the maximal blurred segments onthe left and right sides of the pointCk. So:• ∀k such thatEi−1 < k ≤ Ei, thenL(k) = Bi• ∀k such thatBi ≤ k < Bi+1, thenR(k) = EiThanks to Property 2 and the sequenceMBSν(C), it is easy to ob-tain the left and right extremities of the widthν blurred segmentsstarting from each point of the studied curveC. For each point M ofC, the widthν blurred segment between M and left (resp. right) ex-tremity is calledwidth ν maximal left (resp. right) blurred segmentof this point (see Fig. 1).3. DOMINANT POINT DETECTIONWe present here a new method for dominant point detection based ontheoretical results of discrete geometry (recalled in section 2) : thesequence of maximal blurred segments of a curve permits to obtainimportant informations about the geometrical structure of the studiedMD(1,2,−2,5)OxOyD(1,−2,−3,5)ROSFig. 1. Region of support basedon maximal left (pink points) and right(blue points) blurred segments of MFig. 2. Pink zone is not a com-mon zone of successive maximalblurred segmentscurve. The width of the maximal blurred segments permits to workin different scales and permits to consider the noise present in thestudied curve. The width is the only parameter uses in our method.3.1. Dominant point and region of support (ROS)Deducing from [7], we propose in this section the notion of ROS thatis compatible with the blurred segment notion.Definition 2Widthν maximal left and right blurred segments of a point constituteits region of support (ROS) (see figure 1). The angle between them iscalled the ROS angle of this point.Remark 1The smaller the ROS angle of a point is, the higher the dominantcharacter of this point is.This remark is deduced from the work [3], wherecurvature ata point Ck is estimated as inverse of the radius of the circumcirclepassing throughCk and the extremities of its left and right widthνmaximal blurred segments. Therefore, we have a corollary of thisremark: if the ROS angle of a point is nearly180◦, this point cannotbe a dominant point.3.2. Relation between dominant points and maximal blurredsegmentsIn this paragraph, we study the relation between position of domi-nant points and maximal blurred segments.Proposition 1A dominant point of the curve must be in a common zone of succes-sive maximal blurred segments.Proof 1Let us consider the points on the pink zone (see Fig. 2) which arenot in a common zone of successive maximal blurred segments butwhich belong to one blurred segment. By applying the Property 2,the left and right end points of the blurred segments of these pointsare also in the same blurred segment. The ROS angles of these pointsare nearly180◦. Therefore these points are not candidates as domi-nant points.Let us now consider the common zone of more than 2 successivemaximal blurred segments.Proposition 2The smallest common zone of successive widthν maximal blurredsegments whose slopes are increasing or decreasing contains a can-didate as dominant point.Proof 2Let us consider k successive widthν maximal blurred segmentswhich share the smallest common zone. Without loss of general-ity, we assume that these k maximal blurred segments do not inter-sect any other smallest zone. Suppose that there are k first maximalblurred segments with the extremities below:(B1, E1), (B2, E2),...,(Bk, Ek). Their slopes satisfyslope1 < slope2 < ... < slopek(similarity to decreasing case). Due to Property 1, we must have:B1 < B2 < ... < Bk; E1 < E2 < ... < Ek. Because these max-imal blurred segments share the smallest common zone, we musthaveBk < E1. So, the smallest common zone is[Bk, E1]. By ap-plying Property 2, the left and right extremities of the points of thek partial common zones[B1, B2[, [B2, B3[, ...[Bk, E1[ respectivelyare(B1, E1), (B1, E2),...(B1, Ek). The slopes of the left blurredsegments of the points of these partial common zones are alwaysequal toslope1. On the contrary, the slopes of the right blurred seg-ments of the points of these partial common zones respectively areslope1, slope2, ... ,slopek. By a similar way, we deduce that onthe partial common zones]E1, E2],...,]Ek−1, Ek], the slopes of theright blurred segments of the points of these partial common zonesare equal toslopek and the slopes of the left blurred segments re-spectively are equal toslope2, ... ,slopek. The ROS angle of thepoints in the zone[Bk, E1) is equal to the angle(slope1, slopek)and this value is minimal for all the points indexed fromB1 to Ek,due to the hypothesis of the increasing slopes of maximal blurredsegments. Therefore, this zone contains a candidate as dominantpoint.To eliminate the weak dominant point candidates, we use the fol-lowing natural property of a maximal blurred segment, due to theshape of straight line of a maximal blurred segment and also due tothe property of corner of a dominant point.Property 3A maximal blurred segment contains at most 2 dominant points.3.3. Proposed algorithm3.3.1. AlgorithmWe propose below a heuristic strategy for localizing the position ofeach dominant point candidate.Heuristic strategy: In each smallestcommon zone of successive maximal blurred segments whose slopesare increasing or decreasing, a candidate as dominant point is de-tected as middle point of this zone. CMBS1MBSkFig. 3. A dominant point is expectedas center of smallest common zone.Let us consider a small-est zone that satisfies thiscondition. This zone con-tains a candidate as domi-nant point (cf. proposition2). By using the property 1,this zone must be the inter-section of the first and thelast maximal blurred seg-ments in the set of succes-sive maximal blurred seg-ments that share this zone (see Fig. 3). We recall that each pointin this zone has the same region of support. We then propose to lo-cate the candidate as dominant point that has geometric propertiesclose to the expected corner point. So, the candidate as dominantpoint is detected asmiddle point of the partial curve correspondingto this zone.Based on the above theoretical framework and using the heuris-tic strategy above, we present hereafter our proposed algorithm fordominant point detection. It is decomposed into two parts :• the scan of the interesting common zones of maximal blurredsegments according to the Proposition 2 and the Property 3,• the detection of dominant points in common zones of succes-sive maximal blurred segments whose slopes are increasing ordecreasing.Algorithm 1: Dominant point detectionData: C discrete curve ofn points,ν width of the segmentationResult: D set of extracted dominant pointsbeginBuild MBSν = {MBS(Bi, Ei, ν)}mi=1, {slopei}mi=1 ;m = |MBSν |; p = 1; q = 1; D = ∅ ;while p ≤ m dowhile Eq > Bp do p + +;Add (q, p − 1) to stack;q=p-1;while stack 6= ∅ doTake(q, p) from stack;Decompose{slopeq , slopeq+1, ..., slopep} into monotonesequences;Determine the last monotone sequence{sloper, ..., slopep};D = {D ∪ C⌊ r+p2⌋} ;end3.3.2. ComplexityThe complexity of our method depends on the decompositionof a curve into maximal blurred segments that can be done inO(n log2 n) time [3]. The slope estimation of maximal blurredsegments is done in linear time. On the other hand, each maximalblurred segment is considered at most twice while the curve is de-composed into common zone of maximal blurred segments whoseslopes are monotone sequence. So, in this phase, the dominant pointsare detected in linear time. Therefore the complexity of this methodis O(n log2 n).4. EXPERIMENTAL RESULTSThe figures 4 and 5 show our obtained results and compare themwith other methods (see table I) on some classical criteria: numberof dominant points (nDP), compression ratio (CR), ISE error, maxerror, and figure of merit (FOM). CR is the ratio between numberof curve points and number of detected dominant points, ISE is thesum of squared perpendicular distance of the curve points from ap-proximated polygon. As a low error of approximation leads to a lowratio of compression, Sarkar [8] propose a FOM criterion to combinethese measures:FOM = CR/ISE.Because FOM criterion is not suitable for comparation with dif-ferent dominant point number, Rosin [9] proposed other criteriato evaluate obtained result by comparing with the optimal result.Fig. 4. Detected dominant points with default parameter (width=0.9). Fromleft to right: leaf, chromosome, semicircle curves.He proposed 2 measures, fidelity for error measurement and effi-ciency for compression ratio.Fidelity = (Eopt/Eappr) ∗ 100,Efficiency = (Nopt/Nappr) ∗ 100 whereEappr andNappr arerespectively error and DP number of tested algorithm,Eopt is er-ror of optimal algorithm with the same approximated DP num-ber, Nopt is DP number of optimal algorithm with the same ap-proximated error. The merit measure is based on these measures.Merit =√Fidelity ∗ Efficiency. Our method is not as effi-cient as the top 3 of the Rosin’s list [9], that contains 30 differentmethods (see table II), but it is better than all others.Moreover, our method also gives good results on noisy curves asit can be seen on figure 6 and in the table hereafter. The noisy imageis created by using Kanungo model [10]:Without noise, nPoint=338 With noise, nPoint=370width ISE CR nDP width ISE CR nDP1 62.22 7.51 45 2 262.68 11.21 332 203.25 12.52 27 3 506.62 16.09 233 424.65 17.79 19 4 878.72 20.56 185. CONCLUSIONWe have presented a new method for dominant point detection. Thismethod utilizes recent results in discrete geometry to work naturallywith noisy curves. The width parameter permits to take into accountthe noise present in the curve. For the future work, we will compareour method with other methods that can also work with noisy curves.6. REFERENCES[1] F. Feschet and L. Tougne, “Optimal time computation of the tangent ofa discrete curve: Application to the curvature.” inDGCI, ser. LNCS,vol. 1568, 1999, pp. 31–40.[2] I. Debled-Rennesson, F. Feschet, and J. Rouyer-Degli, “Optimalblurred segments decomposition of noisy shapes in linear time,” Com-puters & Graphics, vol. 30, no. 1, 2006.[3] T. P. Nguyen and I. Debled-Rennesson, “Curvature estimation in noisycurves,” inCAIP, ser. LNCS, 2007, pp. 474–481.[4] E. Attneave, “Some informational aspects of visual perception,” Psy-chol. Rev., vol. 61, no. 3, 1954.[5] A. Masood, “Optimized polygonal approximation by dominantpointdeletion,”Pattern Recognition, vol. 41, no. 1, pp. 227–239, 2008.[6] J.-P. Reveill̀es, “Ǵeoḿetrie discr̀ete, calculs en nombre entiers et algo-rithmique,” 1991, th̀ese d’́etat. Universit́e Louis Pasteur, Strasbourg.[7] C.H.Teh and R.T.Chin, “On the detection of dominant points on thedigital curves,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 2, pp. 859–872, 1989.(a) Proposed (b) Marji [11] (c) Teh [7](d) Ansari [12] (e) Masood [13] (f) Wu [14](g) Arcelli [15]y(h) Sarkar [8]y(i) Shearer [16]Fig. 5. Dominant points of the chromosome shape[8] D. Sarkar, “A simple algorithm for detection of significant vertices forpolygonal approximation of chain-coded curves,”Pattern RecognitionLetters, vol. 14, no. 12, pp. 959–964, 1993.[9] P. L. Rosin, “Techniques for assessing polygonal approximations ofcurves,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 19, no. 6, pp.659–666, 1997.[10] T. Kanungo, R. M. Haralick, H. S. Baird, W. Stuetzle, andD. Madi-gan, “Document degradation models: Parameter estimation and modelvalidation,” inMVA, 1994, pp. 552–557.[11] M. Marji and P. Siy, “A new algorithm for dominant points detectionand polygonization of digital curves,”Pattern Recognition, vol. 36,no. 10, pp. 2239–2251, 2003.[12] N. Ansari and K. wei Huang, “Non-parametric dominant point detec-tion,” Pattern Recognition, vol. 24, no. 9, pp. 849–862, 1991.[13] A. Masood, “Dominant point detection by reverse polygonization ofdigital curves,” Image Vision Comput., vol. 26, no. 5, pp. 702–715,2008.[14] W.-Y. Wu, “An adaptive method for detecting dominant points,” PatternRecognition, vol. 36, no. 10, pp. 2231–2237, 2003.[15] C. Arcelli and G. Ramella, “Finding contour-based abstractions of pla-nar patterns,”Pattern Recognition, vol. 26, no. 10, pp. 1563–1577,1993.[16] M. Shearer and J. J. Zou, “Detection of dominant points based on noisesuppression and error minimisation,” inICITA ’05, vol. 2. USA: IEEE,2005, pp. 772–775.Shape Algo. Nd CR MaxE ISE FOMChro.Proposed 13 4.61 0.742 5.853 0.798Masood [13] 12 5 0.88 7.76 0.65Shearer [16] 10 6 6.086 0.744Wu [14] 17 3.53 0.64 5.01 0.704Marji [11] 11 5.45 0.895 9.96 0.548Teh [7] 15 4.00 0.74 7.2 0.556Ansari [12] 16 3.75 2 20.25 0.185Arcelli [15] 7 8.57Sharkar [8] 19 3.16 0.55 3.857 0.819LeafProposed 23 5.217 0.74 11.52 0.4527Masood [13] 23 5.217 0.74 10.61 0.49Shearer [16] 22 5.46 13.06 0.418Wu [14] 23 5.22 1 20.34 0.256Marji [11] 22 5.45 0.78 13.21 0.413Teh [7] 29 4.14 0.99 14.96 0.277Ansari [12] 30 4.00 2.13 25.57 0.156Arcelli [15] 16 7.50Sarkar [8] 23 5.22 0.784 13.17 0.396Semi.Proposed 21 4.857 0.84 10.41 0.466Masood [13] 19 5.37 1.00 23.9 0.23Shearer [16] 13 7.85 22.97 0.342Wu [14] 27 3.78 0.88 9.19 0.411Marji [11] 18 5.67 1 24.2 0.234Teh [7] 22 4.64 1 20.61 0.225Ansari [12] 28 3.64 1.26 17.83 0.24Arcelli [15] 10 10.20Sarkar [8] 19 5.37 1.474 17.37 0.309Table 1. Comparisons using Sarkar’s criteriaMethod nDP ISE Fidel. Effi. Merit Rosin’s rankMassod [13] 21 9.82 81.7 95.8 88.5 -Proposed 21 10.41 77.07 90.47 83.5 -Sarkar II [8] 20 13.65 66 78.9 72.2 4Sarkar I [8] 19 17.38 57.8 73.7 65.3 6Arcelli [15] 10 75.10 51.8 80.3 64.5 7Teh [7] 22 20.61 34.0 59.2 42.4 17Ansari [12] 28 17.83 18.8 49.1 28.8 26Wu [14] 27 9.01 41.09 74.07 55.17 -Table 2. Comparisons using Rosin’s criteria on semicircle curveFig. 6. From left to rigth : First line -Leaf image, segmentations forwidth=2 and 3- Second line - segmentations of noisy curve [10]for width= 2,3 and 4

Fast and robust dominant point detection on digital curves

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Fast and robust dominant point detection on digital curves

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