This paper presents a method for recognition of second order surfaces (quadrics) from point clouds containing information about scanned rotational parts. The method is region growing method that exploits the scatter of data during least squares fitting of quadrics as a region growing criterion. The presented procedure is convenient for segmentation of regions with high (G1 or higher) continuity. Besides, the region seed point is automatically selected which is its comparative advantage to a number of existing methods. The applicability of the proposed method is evaluated using two case studies; the first case study refers to a synthesized signal, and the second presents the applicability of the method on a real world example.*Ovaj rad je izabran sa konferencije 12th International Scientific Conference MMA 2015 - Flexible Technologies, i publikovan u casopisu Journal of Production Engineering

Jakovljević, Živana

Marković, Veljko

Živanović, Saša

Journal of Production Engineering

This paper presents a method for recognition of second order surfaces (quadrics) from point clouds containing information about scanned rotational parts. The method is region growing method that exploits the scatter of data during least squares fitting of quadrics as a region growing criterion. The presented procedure is convenient for segmentation of regions with high (G1 or higher) continuity. Besides, the region seed point is automatically selected which is its comparative advantage to a number of existing methods. The applicability of the proposed method is evaluated using two case studies; the first case study refers to a synthesized signal, and the second presents the applicability of the method on a real world example

Machinery - Repository of the Faculty of Mechanical Engineering, University of Belgrade

12th INTERNATIONAL SCIENTIFIC CONFERENCENOVI SAD, SERBIA, SEPTEMBER 25-26, 2015Jakovljević, Ž., Marković, V., Živanović, S.RECOGNITION OF QUADRICS FROM 3D POINT CLOUDSGENERATED BY SCANNING OF ROTATIONAL PARTSAbstract: This paper presents a method for recognition of second order surfaces (quadrics) from point cloudscontaining information about scanned rotational parts. The method is region growing method that exploits thescatter of data during least squares fitting of quadrics as a region growing criterion. The presented procedure isconvenient for segmentation of regions with high (G1 or higher) continuity. Besides, the region seed point isautomatically selected which is its comparative advantage to a number of existing methods. The applicability of theproposed method is evaluated using two case studies; the first case study refers to a synthesized signal, and thesecond presents the applicability of the method on a real world example.Key words: 3D point cloud, surface recognition, quadrics segmentation, reverse engineering1. INTRODUCTIONResearch efforts and practical applications in thefield of implementation of 3D scanning devices in on-line manufacturing process control are expanding overthe years. Typical examples are assembly automationand robots navigation. The hardware of contemporary3D scanning devices, and especially optical ones, ischaracterized by high resolution, speed, and accuracy,and it is suitable for on-line applications. The sameholds for algorithms for raw data preprocessing (3Ddata registration, integration and meshing). However,there is a lack of efficient real-time applicablealgorithms for automatic recognition (segmentation andfitting) of geometric primitives from point cloud.During reverse engineering in CAD systems, geometricprimitives from point clouds are recognizedinteractively by user [1], where user recognizesgeometric primitives and then annotates a number ofpoints on them. Afterwards, software is capable offitting the best surfaces through selected points. On theother hand, implementation of 3D scanning devices incontrol applications requires fully automaticrecognition of geometric primitives from point cloud.Motivated by applications in a number of fields,such as mobile robots navigation [2] and seam posedetection in robotized welding [3], automaticrecognition of planes has attracted many researchefforts [4, 5]. Besides planar surfaces, in mechanicalengineering rotational surfaces are most frequently met.These surfaces usually consist of second order surfaces(quadrics), planes, and general tori. In the focus of thispaper is recognition of quadrics from scannedrotational surfaces.There are a number of algorithms for detection ofquadrics from point cloud, and a survey of thesemethods can be found in [6]. Generally, the methodsbelong to one of the two classes: 1) edge based, and 2)region based methods. Edge based techniques can beemployed for G0 continuous surfaces where thereexists an abrupt change between adjacent regions.Region based methods employ split and/or merge orregion growing approach. Region growing approachesstart from a seed point and grow region around it usingdifferent criteria such as local surface normal [7],average curvature [8], or principle curvature [9].Recent methods combine edge based approaches forinitial segmentation and region based approaches forsurface fitting and segmentation of regions with highercontinuity. For example, procedure proposed in [10]detects feature edges using dihedral angle; afterwardsvariational shape approximation is employed for regionpartition and general shape quadrics are fitted throughobtained regions; finally, over-segmented adjacentregions are merged using combination of Euclideandistance and normal deviation as metrics. Anotherapproach from [11] employs tensor voting basedmethod for edges detection and combination ofRANSAC, slipping motion analysis, Gauss map andprinciple curvature for recognition of shapes. Aninteresting research in the field of recognition ofplanes, spheres, cones and cylinders from sparse 3Dmeshes obtained from CAD is presented in [1]. Thepresented method recognizes feature edges usingdihedral angle, and employs region growing based onprincipal curvature for primitives’ extraction.In this paper we propose a method for automaticrecognition of quadrics from point cloud obtained byscanning of rotational parts. The proposed method is inits essence region growing method based on scatter ofdata during direct least squares fitting of second ordersurfaces. The presented method is applicable forrecognition of G1 (or higher) continuous surfaces.The remainder of the paper is structured as follows.In Section 2 we provide some theoretical backgroundand present the proposed recognition method andcorresponding algorithm. Section 3 presents theimplementation of the algorithm on a synthesized pointcloud, as well as on a real-world example. Finally, inSection 4 we give some concluding remarks.145Original scientific paper2. METHOD FOR RECOGNITION OFQUADRICS FROM SCANNED ROTATIONALPARTSThe method for recognition of quadrics from scannedrotational parts that is presented in this paper is basedon the properties of scatter matrix during least squaresfitting of second order surfaces (quadrics). Generalquadric can be represented by the following equation:010987654232221=+++++++++azayaxaxzayzaxyazayaxa(1)where ai represent surface parameters and [x y z] are thecoordinates of a point on quadric. Equation (1) can berepresented in matrix form:0=⋅ ax (2)where x=[x2 y2 z2 xy yz xz x y z 1], and a=[a1 a2… a10]Trepresents the vector of surface parameters.Coefficients a can be estimated by solving thefollowing minimization problem [12, 13]:1subject tominT2=CaaDa(3)where C is 10×10 matrix with all elements equal zeroexcept C(10,10) =1, and D is design matrix in the form:TNNNNNNNNNNNNzzzzyyyyxxxxzxzxzxzxzyzyzyzyyxyxyxyxzzzzyyyyxxxx=1...111...........................321321321332211332211332211223222122322212232221D (4)Solution of minimization problem (3) is positive eigenvalue ofCaSa = (5)whereDDS T= (6)represents scatter matrix, and λ is Lagrange multiplier.When all data in design matrix are sampled fromone exact quadric, scatter matrix will be singular [12].On the other hand, when data are sampled fromapproximate quadric, such as scanned quadric surface,the scatter matrix will be close to singular. As data thatdo not belong to the particular quadric enter designmatrix, matrix S will be farther from singular matrix.We have exploited this property of scatter matrix tocreate the method for segmentation of quadrics fromconsidered class of point clouds.The proposed method is in its essence regiongrowing method. Region growing procedure starts fromthe first point in the point cloud and adds point by pointto the region using reciprocal condition number ofscatter matrix as region growing criterion. Whenreciprocal condition number of S passes predefinedthreshold, a point that does not belong to quadric isdetected. In this situation region growing is stoppedand the coefficients a of recognized quadric areestimated. The region is considered a quadric if itcontains more than 30 points from the cloud.Otherwise, it is assumed that points do not belong toquadric, but to another type of surface.When region growing procedure stops, all pointsthat belong to recognized quadric within predefinedthreshold are excluded from point cloud; for detectionof distance between points from the cloud and surfacewe exploit estimated surface parameters. Theparameters estimation and point exclusion procedure isiteratively repeated until no more points are recognizedto belong to given quadric. The pseudo-code of thealgorithm for implementation of the procedure ispresented in Fig. 1.Since we are dealing with rotational parts, toaccelerate and facilitate computational efforts, thealgorithm starts with alignment of scanned part’srotation axis with z axis of point cloud, and withsorting of points from point cloud in descending orderalong z axis. The alignment of rotation axis along zdirection is carried out using moments of inertia for allthree axes of coordinate system, center of mass, andelementary computer graphics transforms.INPUT: x, y, z – points from the cloudthres1, thres2 – segmentation thresholdsput axis of rotation along z axiscloud=[x, y, z];sortrows(cloud,-3);m=0; trials=0;while trials<30m=m+1;surfaces(m).surf(1,:)=cloud(1,:)for i=1:length(cloud)surf_aux= [surfaces(m).surf; cloud(i,:)];calculate rcond(S) using surf_auxif rcond(S)<thres1add cloud(i,:) to surfaces(m).surf; endendif length(surfaces(m).surf)>30l1=0; l2=1000;while l1<l2l1=length(surfaces(m).surf)calculate Afor i=1:length(cloud)if distance<thres2add cloud(i,:) to surfaces(m).surfelse add cloud(i, :) to new_cloud; endendl2=length(surfaces(m).surf)endcloud=new_cloud;else trials=trials+1; endif length(cloud)<2 trials=31; endendOUTPUT: surfaces – structure containing segmented quadricsFig. 1. Pseudo-code of the algorithm forimplementation of the method for recognition ofquadrics from scanned rotational parts1463. IMPLEMENTATION OF THE PROPOSEDMETHODTo illustrate and experimentally verify the proposedprocedure, we have implemented it in recognition ofsecond order surfaces in two case studies. The first casestudy refers to a synthesized point cloud, while thesecond considers a real world example.3.1 Case study 1: synthesized point cloudThe point cloud that we use in this case studyconsists of four second order surfaces, i.e. of onesphere, two ellipsoids (spheroids in particular), and onecylinder. Sphere and cylinder are G0 continuous withadjacent surfaces, while between ellipsoids G1continuity is achieved. The CAD model of synthesizedpart is presented in Fig. 2.a, and parameters ofsynthesized surfaces in Table 1. Synthesized pointcloud is noised by white noise with signal to noise ratioof 90dB.After application of the proposed method, all foursurfaces were adequately recognized. Segmentationresults are graphically presented in Fig. 2.b, andestimated parameters in Table 1. From Table 1 it can beobserved that the algorithm was able to adequatelyrecognize surfaces. The discrepancy between estimatedsurface parameters and generated surface parameters isless than 0.3%. It should be noted that the parametersof ellipsoids are close and that accurate distinctionbetween these surfaces is a challenging task.a) b)Fig. 2. Case study 1: a) synthesized part; b)segmentation resultsSeg.no Parameters of quadric[a1×102], i=1,…10genertaed 1 [0.04 0.04 0.04 0 0 0 0 0 -0.96 -94.24]2 [0.33 0.33 0.25 0 0 0 0 0 -24 476]3 [0.32 0.32 0.11 0 0 0 0 0 -11.29 186.76]4 [1.23 1.23 0 0 0 0 0 -100]estimated1 [0.04 0.04 0.04 0.0 -0.0 -0.0 0.0 -0.0 -0.96 -94.28]2 [0.33 0.33 0.25 -0.0 -0.0 -0.0 0.0 0.0 -23.98 474.61]3 [0.32 0.32 0.11 -0.0 0.0 -0.0 0.0 -0.0 -11.29 186.76]4 [1.23 1.23 -0.0 0.0 -0.0 -0.0 0.0 0.0 0.0 -100.0]Table 1. Generated and estimated surface parameters incase study 13.1 Case study 2: real-world exampleIn second case study we consider the part presentedin Fig. 3. The part consists of ellipsoid, hyperboloid,cone and cylinder. The parameters of the designed partare presented in Table 2. The part was made on CNClathe TCN410 – echoENG. The point cloud (Fig. 3.c) isobtained by 3D digitization of the part usingZScanner® 700 [14] handheld laser scanner withresolution of 0.1mm.As presented in pseudo-code from Fig. 1, therecognition procedure starts by alignment of part’srotation axis with z axis of the coordinate system (Fig.4). After applying the whole procedure, four quadricsurfaces were adequately recognized. Segmentationresults are presented in Fig. 5, and estimatedparameters in Table 2.From estimated parameters it can be observed thatfor all four surfaces adequate surface types (ellipsoid,hyperboloid, cone and cylinder) were recognized. Thediscrepancy between designed and estimatedparameters for all coefficients except for a3 in conesegment is less than 6%. This discrepancy includes allerrors introduced during machining and scanningprocess, such as interpolation error (ellipse andhyperbola are approximated by circular and linearsegments), machining error, scanner accuracy andresolution. It should be noted that the scanner hasshown poor performance as presented in Fig. 3c. It hasvery low accuracy in vicinity of sharp edges and conesegment is located between two sharp edges.a) b)c)Fig. 3. Part from case study 2: a) designed part; b)photo of real-world part; c) scanned parta) b)Fig. 4. Point cloud from case study 2: a) before graphic147transforms; b) after alignmentSegno Parameters of quadric[a1×102], i=1,…10designed1 [0.16 0.16 0.09 0 0 0 0 0 -19.78 933.87]2 [2.04 2.04 -0.68 0 0 0 0 0  71.46 -1976.42]3 [0.33 0.33 1.0 0 0 0 0 0 66.64 1.11e+3]*1004 [0.086 0.086 0 0 0 0 0 0 0 -100]estimated1 [0.17 0.17 0.09 -0.0 0.0 -0.0 0.04 -0.02 -19.81 930.61]2 [2.04 2.04 -0.68 -0.0 0.01 -0.0 0.34 -1.11 72.68 -2033.5]3 [0.34 0.34 -0.89 -0.0 0.0 0.0 0.04 -0.29 64.4 -1.13e+3 ]*1004 [0.086 0.086 0.0 0.0 0.0 0.0 0.01 -0.04 0.0 -99.91]Table 2. Designed and estimated surface parameters incase study 2Fig. 5. Segmentation results in case study 24. CONCLUSIONIn this paper we have presented a method forrecognition of quadrics from point clouds generated byscanning of rotational parts. It is a region growingmethod based on properties of scatter matrix calculatedduring least squares fitting of second order surfaces.The method is convenient for segmentation of adjacentsegments with G1 or higher continuity, and this is itscomparative advantage to edge based procedures.Another important property of this method is that seedpoint is automatically selected as the highest point(point with maximum z coordinate) on the part.The performances of the method were presentedusing a synthesized and a real-world point cloud, andthe method has shown good recognition results.To increase the robustness of the method to poorscanner performances, the future work will address thecombination of the presented region growing methodwith edge based segmentation during preprocessing. Inaddition, recognition of tori, which are also frequentlymet in rotational parts, will be a part of future researchefforts.4. REFERENCES[1] Bénière, R., Subsol, G., Gesquière, G., Le Breton,F., Puech, W.: A comprehensive process of reverseengineering from 3D meshes to CAD models,Computer-Aided Design, Vol. 45, p.p. 1382–1393,2013.[2] Fu, G., Corradi, P., Menciassi, A., Dario, P.: AnIntegrated Triangulation Laser Scanner forObstacle Detection of Miniature Mobile Robots inIndoor Environment, IEEE/ASME Transactions onMechatronics, Vol.16, No.4, p.p.778 – 783, 2011.[3] Fang, Z., Xu, D., Tan, M.: A Vision-Based Self-Tuning Fuzzy Controller for Fillet Weld SeamTracking, IEEE/ASME Transactions onMechatronics, Vol.16, No.3, p.p.540 – 550, 2011.[4] Wang, L., Cao, J, Han, C.: Multidimensionalparticle swarm optimization-based unsupervisedplanar segmentation algorithm of unorganizedpoint clouds, Pattern Recognition, Vol.45, No.11,p.p. 4034–4043, 2012[5] Jakovljevic, Z., Puzovic, R., Pajic, M.: Recognitionof Planar Segments in Point Cloud based onWavelet Transform, IEEE Trans. on IndustrialInformatics, Vol. 11, No. 2 , p.p. 342-352, 2015.[6] Petitjean, S.: A Survey of Methods for RecoveringQuadrics in Triangle Meshes, ACM ComputingSurveys, Vol.2, No.34, p.p.1-61, 2002.[7] Rabbani, T., Van den Heuvel, F.A., Vosselman, G.:Segmentation of point clouds using smoothnessconstraint, International Archives ofPhotogrammetry, Remote Sensing and SpatialInformation Sciences, Vol.36, No.5, p.p.248-253,2006.[8] Lai, H.C., Chang, Y.H., Lai, J.Y., Development offeature segmentation algorithms for quadraticsurfaces, Advances in Engineering Software,Vol.40, No.10, p.p.1011–1022, 2009.[9] Lavoue, G., Dupont F., Baskurt A.: CurvatureTensor Based Triangle Mesh Segmentation withBoundary Rectification, Proceedings of theComputer Graphics International, p.p. 10-17,IEEE, 2004.[10] Yan, D., M., Wang, W., Liu, Y., Yang, Z.:Variational mesh segmentation via quadric surfacefitting, Computer-Aided Design Vol. 44, pp. 1072–1082, 2012.[11] Yi, B., Liu, Z., Tan, J., Cheng, F., Duan, G., Liu,L.: Shape recognition of CAD models via iterativeslippage analysis, Computer-Aided Design Vol.55, pp. 13–25, 2014.[12] Fitzgibbon, A., Pilu, M., Fisher R.B.: Direct LeastSquare Fitting of Ellipses, IEEE Transactions onPattern Analysis, Vol.21, No.5, pp.476-480, 1999.[13] Rosin, P., L.: A note on the least squares fitting ofellipses, Pattern Recognition Letters, Vol. 14, No.10, p.p. 799–808, 1993.[14] http://www.zcorp.com/documents/380_ZScanner700_PX_SpecSheet_HiRes.pdfAuthors: Dr. Živana Jakovljević, AssistantProfessor, M.Sc. Veljko Marković, Dr. SasaZivanović, Assistant Professor, University ofBelgrade, Faculty of Mechanical Engineering, KraljiceMarije 16, 11000 Belgrade, SerbiaE-mail: zjakovljevic@mas.bg.ac.rs;vmarkovic@mas.bg.ac.rs; szivanovic@mas.bg.ac.rs;Acknowledgements. This research was partially supportedby Serbian Ministry of Education, Science and TechnologicalDevelopment under research grants TR35007 and TR35020.148

Recognition of quadrics from 3d point clouds generated by scanning of rotational parts

machinery

65  JPE (2016) Vol.19 (1)                     Original Scientific  Paper Jakovljevic, Z., Markovic, V., Zivanovic, S.  RECOGNITION OF QUADRICS FROM 3D POINT CLOUDS GENERATED BY SCANNING OF ROTATIONAL PARTS  Received: 10 January 2016 / Accepted: 29 March 2016 Abstract: This paper presents a method for recognition of second order surfaces (quadrics) from point clouds containing information about scanned rotational parts. The method is region growing method that exploits the scatter of data during least squares fitting of quadrics as a region growing criterion. The presented procedure is convenient for segmentation of regions with high (G1 or higher) continuity. Besides, the region seed point is automatically selected which is its comparative advantage to a number of existing methods. The applicability of the proposed method is evaluated using two case studies; the first case study refers to a synthesized signal, and the second presents the applicability of the method on a real world example. Key words: 3D point cloud, surface recognition, quadrics segmentation, reverse engineering  Prepoznavanje kvadrika iz 3D oblaka tacaka dobijenih skeniranjem rotacionih delova. U ovom radu je predstavljen metod za prepoznavanje površi drugog reda (kvadrika) iz oblaka tacaka koji sadrže informaciju o skeniranim rotacionim delovima. Metod spada u klasu metoda zasnovanih na rastu regiona, a kao kriterijum za rast regiona koriste se svojstva informacione matrice koja se izracunava tokom regresije kvadrika. Predložen metod je pogodan za segmentaciju kontinualnih regiona (G1 ili viša kontinualnost). Pored toga, pocetna tacka za rast regiona bira se automatski što predstavlja komparativnu prednost ovog metoda u odnosu na dosada razvijene metode segmentacije kvadrika. Primenljivost razvijenog metoda je ilustrovana na dva primera; prvi primer se odnosi na sintetizovani, a drugi na oblak tacaka dobijen skeniranjem realnog objekta. Ključne reči: 3D oblak tacaka, prepoznavanje površi, segmentacija kvadrika, reverzno inženjerstvo  1. INTRODUCTION    Research efforts and practical applications in the field of implementation of 3D scanning devices in on-line manufacturing process control are expanding over the years. Typical examples are assembly automation and robots navigation. The hardware of contemporary 3D scanning devices, and especially optical ones, is characterized by high resolution, speed, and accuracy, and it is suitable for on-line applications. The same holds for algorithms for raw data preprocessing (3D data registration, integration and meshing). However, there is a lack of efficient real-time applicable algorithms for automatic recognition (segmentation and fitting) of geometric primitives from point cloud. During reverse engineering in CAD systems, geometric primitives from point clouds are recognized interactively by user [1], where user recognizes geometric primitives and then annotates a number of points on them. Afterwards, software is capable of fitting the best surfaces through selected points. On the other hand, implementation of 3D scanning devices in control applications requires fully automatic recognition of geometric primitives from point cloud.  Motivated by applications in a number of fields, such as mobile robots navigation [2] and seam pose detection in robotized welding [3], automatic recognition of planes has attracted many research efforts [4, 5]. Besides planar surfaces, in mechanical engineering rotational surfaces are most frequently met. These surfaces usually consist of second order surfaces (quadrics), planes, and general tori. In the focus of this paper is recognition of quadrics from scanned rotational surfaces.   There are a number of algorithms for detection of quadrics from point cloud, and a survey of these methods can be found in [6]. Generally, the methods belong to one of the two classes: 1) edge based, and 2) region based methods. Edge based techniques can be employed for G0 continuous surfaces where there exists an abrupt change between adjacent regions. Region based methods employ split and/or merge or region growing approach. Region growing approaches start from a seed point and grow region around it using different criteria such as local surface normal [7], average curvature [8], or principle curvature [9].  Recent methods combine edge based approaches for initial segmentation and region based approaches for surface fitting and segmentation of regions with higher continuity. For example, procedure proposed in [10] detects feature edges using dihedral angle; afterwards variational shape approximation is employed for region partition and general shape quadrics are fitted through obtained regions; finally, over-segmented adjacent regions are merged using combination of Euclidean distance and normal deviation as metrics. Another approach from [11] employs tensor voting based method for edges detection and combination of RANSAC, slipping motion analysis, Gauss map and principle curvature for recognition of shapes. An interesting research in the field of recognition of planes, spheres, cones and cylinders from sparse 3D meshes obtained from CAD is presented in [1]. The presented method recognizes feature edges using  66 dihedral angle, and employs region growing based on principal curvature for primitives’ extraction.  In this paper we propose a method for automatic recognition of quadrics from point cloud obtained by scanning of rotational parts. The proposed method is in its essence region growing method based on scatter of data during direct least squares fitting of second order surfaces. The presented method is applicable for recognition of G1 (or higher) continuous surfaces.  The remainder of the paper is structured as follows. In Section 2 we provide some theoretical background and present the proposed recognition method and corresponding algorithm. Section 3 presents the implementation of the algorithm on a synthesized point cloud, as well as on a real-world example. Finally, in Section 4 we give some concluding remarks.  2. METHOD FOR RECOGNITION OF QUADRICS FROM SCANNED ROTATIONAL PARTS  The method for recognition of quadrics from scanned rotational parts that is presented in this paper is based on the properties of scatter matrix during least squares fitting of second order surfaces (quadrics). General quadric can be represented by the following equation: 010987654232221azayaxaxzayzaxyazayaxa (1) where ai represent surface parameters and [x y z] are the coordinates of a point on quadric. Equation (1) can be represented in matrix form:     0ax                       (2) where x=[x2 y2 z2 xy yz xz x y z 1], and a=[a1 a2… a10]T represents the vector of surface parameters.  Coefficients a can be estimated by solving the following minimization problem [12, 13]:   1 subject tominT2CaaDa          (3) where C is 1010 matrix with all elements equal zero except C(10,10) =1, and D is design matrix in the form:  TNNNNNNNNNNNNzzzzyyyyxxxxzxzxzxzxzyzyzyzyyxyxyxyxzzzzyyyyxxxx1...111...........................321321321332211332211332211223222122322212232221D  (4) Solution of minimization problem (3) is positive eigen value of    CaSa    (5) where   DDS T   (6) represents scatter matrix, and  is Lagrange multiplier.  When all data in design matrix are sampled from one exact quadric, scatter matrix will be singular [12]. On the other hand, when data are sampled from approximate quadric, such as scanned quadric surface, the scatter matrix will be close to singular. As data that do not belong to the particular quadric enter design matrix, matrix S will be farther from singular matrix. We have exploited this property of scatter matrix to create the method for segmentation of quadrics from considered class of point clouds.   The proposed method is in its essence region growing method. Region growing procedure starts from the first point in the point cloud and adds point by point to the region using reciprocal condition number of scatter matrix as region growing criterion. When reciprocal condition number of S passes predefined threshold, a point that does not belong to quadric is detected. In this situation region growing is stopped and the coefficients a of recognized quadric are estimated. The region is considered a quadric if it contains more than 30 points from the cloud. Otherwise, it is assumed that points do not belong to quadric, but to another type of surface.   INPUT: x, y, z – points from the cloud         thres1, thres2 – segmentation thresholds put axis of rotation along z axis cloud=[x, y, z]; sortrows(cloud,-3); m=0; trials=0; while trials<30   m=m+1;   surfaces(m).surf(1,:)=cloud(1,:)   for i=1:length(cloud)     surf_aux= [surfaces(m).surf; cloud(i,:)];     calculate rcond(S) using surf_aux     if rcond(S)<thres1       add cloud(i,:) to surfaces(m).surf; end   end   if length(surfaces(m).surf)>30     l1=0; l2=1000;         while l1<l2      l1=length(surfaces(m).surf)       calculate A       for i=1:length(cloud)         if distance<thres2           add cloud(i,:) to surfaces(m).surf          else add cloud(i, :) to new_cloud;  end         end       l2=length(surfaces(m).surf)     end   cloud=new_cloud;   else trials=trials+1; end   if length(cloud)<2 trials=31; end end OUTPUT: surfaces – structure containing segmented quadricsFig. 1. Pseudo-code of the algorithm for implementation of the method for recognition of quadrics from scanned rotational parts  When region growing procedure stops, all points that belong to recognized quadric within predefined threshold are excluded from point cloud; for detection of distance between points from the cloud and surface we exploit estimated surface parameters. The  67 parameters estimation and point exclusion procedure is iteratively repeated until no more points are recognized to belong to given quadric. The pseudo-code of the algorithm for implementation of the procedure is presented in Fig. 1.  Since we are dealing with rotational parts, to accelerate and facilitate computational efforts, the algorithm starts with alignment of scanned part’s rotation axis with z axis of point cloud, and with sorting of points from point cloud in descending order along z axis. The alignment of rotation axis along z direction is carried out using moments of inertia for all three axes of coordinate system, center of mass, and elementary computer graphics transforms.  3.  IMPLEMENTATION OF THE PROPOSED METHOD   To illustrate and experimentally verify the proposed procedure, we have implemented it in recognition of second order surfaces in two case studies. The first case study refers to a synthesized point cloud, while the second considers a real world example.  3.1 Case study 1: synthesized point cloud  The point cloud that we use in this case study consists of four second order surfaces, i.e. of one sphere, two ellipsoids (spheroids in particular), and one cylinder. Sphere and cylinder are G0 continuous with adjacent surfaces, while between ellipsoids G1 continuity is achieved. The CAD model of synthesized part is presented in Fig. 2.a, and parameters of synthesized surfaces in Table 1. Synthesized point cloud is noised by white noise with signal to noise ratio of 90dB. After application of the proposed method, all four surfaces were adequately recognized. Segmentation results are graphically presented in Fig. 2.b, and estimated parameters in Table 1.  a) b)  Fig. 2. Case study 1: a) synthesized part; b) segmentation results  From Table 1 it can be observed that the algorithm was able to adequately recognize surfaces. The discrepancy between estimated surface parameters and generated surface parameters is less than 0.3%. It should be noted that the parameters of ellipsoids are close and that accurate distinction between these surfaces is a challenging task.  Seg. no Parameters of quadric[a1102], i=1,…10 1 [0.04 0.04 0.04 0 0 0 0 0 -0.96 -94.24] 2 [0.33 0.33 0.25 0 0 0 0 0 -24 476] 3 [0.32 0.32 0.11 0 0 0 0 0 -11.29 186.76] genertaed 4 [1.23 1.23 0 0 0 0 0 -100] 1 [0.04 0.04 0.04 0.0 -0.0 -0.0 0.0 -0.0 -0.96 -94.28] 2 [0.33 0.33 0.25 -0.0 -0.0 -0.0 0.0 0.0 -23.98 474.61]3 [0.32 0.32 0.11 -0.0 0.0 -0.0 0.0 -0.0 -11.29 186.76]estimated 4 [1.23 1.23 -0.0 0.0 -0.0 -0.0 0.0 0.0 0.0 -100.0] Table 1. Generated and estimated surface parameters in case study 1  3.1 Case study 2: real-world example In second case study we consider the part presented in Fig. 3. The part consists of ellipsoid, hyperboloid, cone and cylinder. The parameters of the designed part are presented in Table 2. The part was made on CNC lathe TCN410 – echoENG. The point cloud (Fig. 3.c) is obtained by 3D digitization of the part using ZScanner® 700 [13, 14] handheld laser scanner with resolution of 0.1mm. As presented in pseudo-code from Fig. 1, the recognition procedure starts by alignment of part’s rotation axis with z axis of the coordinate system (Fig. 4). After applying the whole procedure, four quadric surfaces were adequately recognized. Segmentation results are presented in Fig. 5, and estimated parameters in Table 2. From estimated parameters it can be observed that for all four surfaces adequate surface types (ellipsoid, hyperboloid, cone and cylinder) were recognized. The discrepancy between designed and estimated parameters for all coefficients except for a3 in cone segment is less than 6%. This discrepancy includes all errors introduced during machining and scanning process, such as interpolation error (ellipse and hyperbola are approximated by circular and linear segments), machining error, scanner accuracy and resolution. It should be noted that the scanner has shown poor performance as presented in Fig. 3c. It has very low accuracy in vicinity of sharp edges and cone segment is located between two sharp edges.  a)      b)      c)  Fig. 3. Part from case study 2: a) designed part; b) photo of real-world part; c) scanned part  68 a) b)  Fig. 4. Point cloud from case study 2: a) before graphic transforms; b) after alignment   Seg no Parameters of quadric[a1102], i=1,…10 1 [0.16 0.16 0.09 0 0 0 0 0 -19.78 933.87] 2 [2.04 2.04 -0.68 0 0 0 0 0  71.46 -1976.42] 3 [0.33 0.33 1.0 0 0 0 0 0 66.64 1.11e+3]*100 designed 4 [0.086 0.086 0 0 0 0 0 0 0 -100] 1 [0.17 0.17 0.09 -0.0 0.0 -0.0 0.04 -0.02 -19.81 930.61]2 [2.04 2.04 -0.68 -0.0 0.01 -0.0 0.34 -1.11 72.68 -2033.5]3 [0.34 0.34 -0.89 -0.0 0.0 0.0 0.04 -0.29 64.4 -1.13e+3 ]*100 estimated 4 [0.086 0.086 0.0 0.0 0.0 0.0 0.01 -0.04 0.0 -99.91] Table 2. Designed and estimated surface parameters in case study 2   Fig. 5. Segmentation results in case study 2  4. CONCLUSION  In this paper we have presented a method for recognition of quadrics from point clouds generated by scanning of rotational parts. It is a region growing method based on properties of scatter matrix calculated during least squares fitting of second order surfaces. The method is convenient for segmentation of adjacent segments with G1 or higher continuity, and this is its comparative advantage to edge based procedures. Another important property of this method is that seed point is automatically selected as the highest point (point with maximum z coordinate) on the part.   The performances of the method were presented using a synthesized and a real-world point cloud, and the method has shown good recognition results.  To increase the robustness of the method to poor scanner performances, the future work will address the combination of the presented region growing method with edge based segmentation during preprocessing. In addition, recognition of tori, which are also frequently met in rotational parts, will be a part of future research efforts.  5. REFERENCES  [1] Bénière, R., Subsol, G., Gesquière, G., Le Breton, F., Puech, W.: A comprehensive process of reverse engineering from 3D meshes to CAD models, Computer-Aided Design, Vol. 45, p.p. 1382–1393, 2013. [2] Fu, G., Corradi, P., Menciassi, A., Dario, P.: An Integrated Triangulation Laser Scanner for Obstacle Detection of Miniature Mobile Robots in Indoor Environment, IEEE/ASME Transactions on Mechatronics, Vol.16, No.4, p.p.778 – 783, 2011. [3] Fang, Z., Xu, D., Tan, M.: A Vision-Based Self-Tuning Fuzzy Controller for Fillet Weld Seam Tracking, IEEE/ASME Transactions on Mechatronics, Vol.16, No.3, p.p.540 – 550, 2011.  [4] Wang, L., Cao, J, Han, C.: Multidimensional particle swarm optimization-based unsupervised planar segmentation algorithm of unorganized point clouds, Pattern Recognition, Vol.45, No.11, p.p. 4034–4043, 2012 [5] Jakovljevic, Z., Puzovic, R., Pajic, M.: Recognition of Planar Segments in Point Cloud based on Wavelet Transform, IEEE Trans. on Industrial Informatics, Vol. 11, No. 2 , p.p. 342-352, 2015. [6] Petitjean, S.: A Survey of Methods for Recovering Quadrics in Triangle Meshes, ACM Computing Surveys, Vol.2, No.34, p.p.1-61, 2002.  [7] Rabbani, T., Van den Heuvel, F.A., Vosselman, G.: Segmentation of point clouds using smoothness constraint, International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol.36, No.5, p.p.248-253, 2006. [8] Lai, H.C., Chang, Y.H., Lai, J.Y., Development of feature segmentation algorithms for quadratic surfaces, Advances in Engineering Software, Vol.40, No.10, p.p.1011–1022, 2009.  [9] Lavoue, G., Dupont F., Baskurt A.: Curvature Tensor Based Triangle Mesh Segmentation with Boundary Rectification, Proceedings of the Computer Graphics International, p.p. 10-17, IEEE, 2004. [10] Yan, D., M., Wang, W., Liu, Y., Yang, Z.: Variational mesh segmentation via quadric surface fitting, Computer-Aided Design Vol. 44, pp. 1072–1082, 2012. [11] Yi, B., Liu, Z., Tan, J., Cheng, F., Duan, G., Liu, L.: Shape recognition of CAD models via iterative slippage analysis, Computer-Aided Design Vol. 55, pp. 13–25, 2014. [12] Fitzgibbon, A., Pilu, M., Fisher R.B.: Direct Least Square Fitting of Ellipses, IEEE Transactions on Pattern Analysis, Vol.21, No.5, pp.476-480, 1999.  [13] Rosin, P., L.: A note on the least squares fitting of ellipses, Pattern Recognition Letters, Vol. 14, No. 10, p.p. 799–808, 1993. [14] http://www.zcorp.com/documents/380_ZScanner700_PX_SpecSheet_HiRes.pdf  Authors: Dr. Zivana Jakovljevic, Assistant Professor, M.Sc. Veljko Markovic, Dr. Sasa Zivanovic, Assistant Professor, University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11000 Belgrade, Serbia E-mail: zjakovljevic@mas.bg.ac.rs; vmarkovic@mas.bg.ac.rs; szivanovic@mas.bg.ac.rs;  Acknowledgements. This research was partially supported by Serbian Ministry of Education, Science and Technological Development under research grants TR35007 and TR35020. om 

https://machinery.mas.bg.ac.rs/bitstream/123456789/5078/1/Jakovljevic_mma_2015.pdf

Recognition of quadrics from 3d point clouds generated by scanning of rotational parts

Abstract

Similar works

Full text

Available Versions

Machinery - Repository of the Faculty of Mechanical Engineering, University of Belgrade

machinery

machinery

Machinery - Repository of the Faculty of Mechanical Engineering, University of Belgrade