Robust Curvature Estimation and Geometry Analysis of 3D point Cloud Surfaces

Transcription

1 Robust Curvature Estmaton and Geometry Analyss of 3D pont Cloud Surfaces Xaopeng ZHANG, Hongjun LI, Zhangln CHENG, Ykuan ZHANG Sno-French Laboratory LIAMA, Insttute of Automaton, CAS, Bejng , Chna Natonal Laboratory of Pattern Recognton, Insttute of Automaton, CAS, Bejng , Chna Abstract Curvatures are basc shape nformaton of a surface, whch are useful for 3D geometry analyss and 3D reconstructon. A new robust algorthm s presented to estmate prncpal curvatures n ths paper. The basc dea of ths method s local fttng of each normal secton crcle propertes wth poston nformaton and normal nformaton at a neghbor pont. In addton, a local feature curve, called as Normal Curvature Index Lnes (NCIL), s constructed to show fttng effect of all curvature estmaton methods. Wth ths accurate and robust curvature, some shape nformaton of a pont cloud surface has been obtaned, such as saddle regons, sharp rdge regons and prncple drectons. Expermental results show that ths work s more advantageous than smlar approaches. Keywords: Curvatures estmaton; Geometry analyss; Normal secton curvature; Curvature error analyss 1 Introducton As techncal development of laser scannng and mage based modelng, more and more pont cloud data are obtaned to represent 3D geometrc shapes of natural objects. Computaton of dfferental propertes of 3D dscrete geometry becomes one fundamental work, whch s helpful to reconstructon of geometrc objects, shape modelng, model smplfcaton and shape feature analyss [1]. Two of se most mportant propertes are prncpal curvatures toger wth prncpal drectons. If ponts are from a known analytc surface, curvature at every pont can be precsely computed by classc dfferental geometrc methods. However, when ponts were sampled from unknown surfaces, wth a laser-scanner for example, an estmaton of prncpal curvatures at every pont would be an nterestng and challengeable topc. Estmaton of curvatures of 3D B-Rep (Boundary Representaton) models has been presented snce 1980s. Most estmaton algorthms depend on surface fttng wth a polynomal surface at a local area. These surfaces can be a quadrc surface [, 3], a cubc surface [4], or a general polynomal surface. Some or algorthms compute curvatures at each vertex of a mesh model Correspondng author. Emal address: xpzhang@nlpr.a.ac.cn (Xaopeng ZHANG). 1

2 by measurng angle of each polygon passng through ths vertex [5, 6]. These methods work well for mesh models, and y can be appled to pont cloud model after extensons. However, t s very dffcult to mesh some scan data, for example trees. So we should try to estmate curvatures from pont cloud data drectly. Our new method depends ner on mesh reconstructon, nor on surface fttng. The nputs of our algorthm are neghbor ponts and normal vectors. The outputs nclude mnmal, maxmal, mean, Gaussan curvatures and prncple drectons. Based on se curvatures, geometrc nformaton s transferred to curvature doman. In followng, we wll brefly present our algorthm and show some applcaton of curvature n geometry analyss. Related Work.1 Curvature estmaton In past twenty years, a lot of algorthms have been proposed to estmate curvatures n lteratures. Most of m are mesh based. The movng least square method s used for curvature estmaton [8], but ths method s too complex n calculatng Gaussan curvature. The robust statstcal estmaton [9] needs more tme for computaton. The ansotropc flterng approach on normal feld and curvature tensor feld [10] focused more n pre-processng. Curvature-doman shape processng [11] emphaszes on post processng, especally geometry process operatons n curvature doman. Curvature estmaton based surface convoluton [1] s n surface fttng accordng to vertex coordnate. Accordng to a comparson [13] of several curvature estmaton methods, Taubn s approach [7] s ner best and nor worst. Therefore, we just choose Taubn approach as a representatve of se algorthms. Most of se algorthms do not employ normal vectors of neghbor ponts, so large errors may be produced. Our method wll be mproved n ths aspect. Goldfear s method [4],.e. adjacent-normal cubc approxmaton, employs normal vectors as constrants and yelds much better results. It can be safely deduced that normal vectors of neghbor ponts are mportant factors. But hs system s three tmes more complex than that of system of nput pont coordnates, whch wll ncrease computer tme and space for computaton. It sometmes produces over fttng results. Our method drectly employs normal vectors to calculate normal curvature, whch can be used to avod ths complexty whle mantanng optmal results.. Error analyss Most papers provde comparson between dfferent methods. In se comparsons, error analyss s very mportant. Tradtonally, data-error or nose-error fgure and table are employed, even n specal paper of comparson methods [13]. We wll adopt se fgures or tables too. Meanwhle, we wll adopt a new comparson fgure [15], whch s very ntutve for effect of normal curvature computaton.

3 3 Curvature Estmaton Based on Normal Secton Lnes Man steps of our method nclude two: usng estmaton of normal secton lnes for normal curvature and optmzaton of all se normal curvatures. Normal drecton at each sample pont s regarded as nput data. If normal vector nformaton s not gven, we adopt a method [14] to ft normal vectors at frst, whch s not dscussed n ths paper. 3.1 Prncples All neghbor ponts of a specfc pont on a pont cloud surface determne local shape. Because of sngular ponts or nose, bg errors may be generated f curvatures are estmated through surface fttng. So contrbuton of normal vectors should be more consdered. To estmate curvature at a pont, we wll consder contrbuton of one neghbor pont only. Ths contrbuton s converted as constructon of a normal secton curve. We wll construct a normal secton crcle and estmate normal curvature from postons and normal vectors of two ponts, objectve pont and one of ts neghbors. 3. Local fttng for normal curvatures For each pont p n pont cloud, we wll use pont coordnates and ts normal vectors, and ts neghbor ponts to estmate ts normal curvatures. Suppose re are m ponts n neghbor of pont p, and let q be -th ( = 1,,..., m) neghbor pont. The normal vector correspondng to q s M. Let { p, X, Y, N } be an orthogonal coordnates system(fg. 1), called local coordnates L at pont p. N denotes normal vector at p. X and Y are orthogonal unt vectors and y are not needed to be specfed here. In L, coordnates of p, q, and M can be shown as p (0, 0, 0), q (x, y, z ), and M (n x,, n y,, n z, ). Then we can estmate normal curvature k n of pont p wth an osculatng crcle passng through pont p and q wth normal N and M. Fg. 1 shows geometrc relaton of se varables. The normal curvature can be estmated wth radus correspondng pont q. n xy kn = sn β pq sn α n xy + n z x + y, where n xy = x n x, + y n y,, n z = n z,. (1) x + y where, α s ncluded angle between vectors N and pq, and β s between vectors N and M. Ths method employs chords, neghbor normal vectors and osculatng crcles, so we call ths method as Chord And Normal vectors (CAN) method. The dfference of ths approach from or methods s that normal vectors of neghborng ponts are drectly used to estmate prncpal curvatures of a pont. 3.3 Least square fttng wth Euler equaton The relaton of all normal curvatures wth prncpal curvatures s analyzed here. In order to estmate prncpal curvatures at a pont wth formula (1), all coordnates of neghbor ponts are transformed nto coordnates of local coordnates system. Let N = (n x,p, n y,p, n z,p ) 3

4 orthogonal unt vectors and y are not needed to be specfed gven here. In L, coordnates of p, q, and M can be that p s (0,0,0), q (x, y, z ), M (n x,,, n y,,, n z,, ). Then we can estmate normal curvature k n of pont p wth an osculatng crcle passng through pont p and q wth normal N and M. Fgure 1 shows geometrc relaton of se varables. S N N p p Y e1 e e 1 XS θ X θ α g e Y Mγ g M β O O Fg. 1: Local coordnates system L and trangle defned by osculatng crcle, neghbor pont and normal vectors. q Fgure 1 Local coordnates system L and trangle defned by Osculatng crcle, neghbor pont and normal vectors. q be normal vector of p, and φ = arccos(n z,p ), ψ = arctan(n y,p /n x,p ), n local coordnates system The normal L{p, X, curvature Y, N} can at be pont estmated p can wth be constructed radus correspondng wth X pont = q. ( sn ψ, cos ψ, 0), Y = (cos φ cos ψ, cos φ sn ψ, sn φ). Let S be plane sn β through pont p wth normal vector N. Let e kn =. (1) 1 and e be prncpal drectons at pont p, correspondng pq snα prncpal curvatures k 1 and k. e 1, e, k 1 and k are unknown. Let unknown parameter θ be ncluded angle between vectors X and e Where, α s ncluded angle between -N and pq, and β between vectors N and M. Let 1, θ be ncluded angle between vectors X and pg obtaned by projectng vector pq M pg onto nplane xy = Π S (Fg.1). 0 M pg = θ can, whch be easly s computed. projecton component Euler formula of vector smused along asdrecton pg. g s pg projecton pont kn = of kq 1 cos projectng (θ + θ) on + plane k sns, (θ γ + s θ), ncluded = 1,,..., angle m. between vectors pg and () pq. Snce The task can be wrtten as an optmzaton queston: n sn 1 cos 1 xy β = β = nz,, pq snα = pg = x + y n xy + n m z, [ ] An approxmaton mnof equaton k1 (1) cos s gven (θ + by: θ) + k sn (θ + θ) kn (3) k 1,k,θ =1 nxy x nx, + y ny, Wth method presented k n n = [15], we obtan prncpal, where nxy curvatures =, n z = n z, () and prncpal drectons. n + n x + y x + y xy z Fg. : Pont cloud torus model wth random nose. Left: (a) h=0.0 h=0.0. Rght:h=1.0. (b) h=1.0 Fgure Pont cloud torus model wth random nose. Ths method employs chord, neghbor normal vector and osculatng crcle, so we call ths method as 4 Experments Chord And Normal on vectors Analytc (CAN) method. Data The dfference of ths approach from or methods s that normal vectors of neghborng ponts are drectly used to estmate man curvatures of a pont. The precson and robustness of ths approach s shown on an analytc surface n ths secton. The procedure of 3.3. Least experment square fttng wth wth CAN Euler method equaton s performed n two steps: calculaton of normal curvature related to each neghbor pont (see expresson (1) ), and n least square fttng The relaton of all normal curvatures wth man curvatures s analyzed here. In order to estmate man (n secton 3.3) for prncpal curvatures and prncpal drectons. Gaussan curvature and mean curvatures at a pont wth formula (), all coordnates of neghbor ponts are transformed nto curvature are coordnates obtaned of from local prncpal coordnates curvatures. system. Let Experments N = ( nx, p, n are y, p, n performed z, p ) be normal wth C vector language of p, and programmng n a PC wth confguraton of Intel(R) Core(TM) CPU, G, 1.99GHz, and G memory. 4.1 Analytc surface We use a torus as a representatve of analytc surfaces. The equaton of torus s r(u, v) = ( (R + r cos u) cos v, (R + r cos u) sn v, r sn u), ( 0 u π, 0 v π ) (4) where parameters are specfed as R=, r=1. 4

5 5000 ponts are sampled from (4) accordng to a unform dstrbuton. Wth analytc method, normal vector and prncpal curvatures of each pont are calculated analytcally. Along normal drecton, nose s added to each pont whch subjects to a unform dstrbuton U ( mh, mh), where parameter m s medan of dstances of all par of ponts. Nose level was under control of postve real number h. The values of h are 0.1, 0.,..., 1.0 respectvely. Fg. shows model of 000 sample ponts wth dfferent nose levels h =0.0 and h = Comparson on absolute errors Experments are repeated 30 tmes at every nose level for each pont cloud data set n secton 4.1. The average error wll be adopted for comparson n each case. The experment results of Taubn method (legend as Taubn), Adjacent-normal cubc approxmaton method (legend as Goldfeath) and our method (legend as CAN) are explaned n Table 1. Neghbor ponts m=15. Table 1: Average absolute error comparson Gaussan curvature error mean curvature error Nose Taubn Goldfeath CAN Taubn Goldfeath CAN * * Table 1 shows that absolute error produced by Taubn method s far bgger than that by Goldfeath method and our method CAN, and CAN tends to produce better results than Goldfeath, especally when nose becomes larger. In addton, compared wth Goldfeath method, CAN algorthm s better n both tme and memory complexty, snce former has to manage a coeffcent matrx wth 3m 7, but later a m 3 one. Wth sample ponts, Goldfeath spent s, CAN s. 4.3 Error analyss wth normal curvature ndex lnes We employ normal curvature ndex lnes(ncil) method [15] to analyss error. Man curvatures and prncpal drectons are estmated respectvely wth Taubn, Goldfeath and CAN method. So we wll get three curves k n, (θ) = k 1, cos θ + k, sn θ, = 0, 1,, 3 (5) Where, k 1,0 and k,0 are true values of prncpal curvatures, and k 1,, k, ( = 1,, 3) are values estmated by th method. Usng 100th pont and ts 30 neghbor ponts, and 4 curves n expresson (5), 4 NCILs are drawn toger n Fg.3. The NCIL (blue) obtaned by CAN s more close to real curve (black). But stuaton of red one for Taubn s terrble, whch may cause large errors. The green curve for Goldfeath s moderate. 5

6 estmated respectvely wth Taubn, Goldfeath and CAN method. So we wll get three curves (7) k n, (θ ) = k1, cos θ + k, sn θ, = 0,1,,3 Where, k1,0 and k,0 are true values of man curvatures, and k1,, k, (=1,,3) are values estmated by -th method. Usng 100th pont and ts 30 neghbor ponts, and 4 curves n expresson (7), 4 NCILs are drawn toger n Fgure 3. Tao Zhang et al. /Journal of Computatonal Informaton Systems 1: (005) (a) NCILs n an orthogonal system 7 (b) NCILs n a polar system model, and se regons are same as our knowledge about hand, but Fgure 4(f) from Goldfeath Fgure 3 Normal curvature ndex lnes of three approaches (Nose level h=0.8) Fg. 3: anormal ndex4(c). lnestherefore, of threeweapproaches (Nose level h=0.8). method gves dfferent curvature regon from Fgure can safely draw a concluson that CAN Fgure 3 shows that NCIL (blue) obtaned by CAN s more close to real curve (black). But method s more robust than Goldfeath method n dscernng saddle regon. stuaton of red one for Taubn s very bad, whch may cause large errors. The green curve for Goldfeath s moderate. Normal curvature ndex lnes can be used to an analyss of effect of curvature fttng, whch helps to evaluate curvature estmaton method. Normal curvature ndex lnes are helpful to compare error of several normal curvature approxmatons, whch helps to select best one among exstng algorthms. In addton, accordng to NCIL, we can calculate determnaton coeffcent of model (3), whch s helpful to select automatcally neghbor number. 5. Applcaton on Dscrete Models nosecurvatures 0.0 (b)prncple CAN CAN Wth (a) man and drectons, we can do(d)nose some.0geometry(e)analyss of dscrete model, and 4 Saddle regon decerned by two approaches meanwhle llustrate robust offgure CAN algorthm. Fg. 4: Saddle regon decerned by two approaches. 5..Saddle Sharp rdge dscerned 5.1. regonregon dscerned 5 Wth Gaussan curvatures, saddle regon can be dscerned. Let k and k be prncple curvatures of pont If pont p s n a sharp rdge k k > δmodels, where δ > 0 and t s a subjectve parameter whch Applcaton onregon, Dscrete p. If pont p s a saddle pont, ts Gaussan curvature s, whch can be used for dscernng 1 1 H p = k1k 0 can beregon. gven accordng to pont cloud data. saddle Wth prncpal curvatures and prncple drectons, some geometry analyss can be performed on A dscrete model hand s used n experment. The heght of model s 0 unts, and t conssts of dscrete models. And meanwhle, robust of CAN algorthm can be llustrated. 658 ponts wth normal vectors. Saddle regons are labeled wth red color. Fgure 4 shows that estmated saddle regon s good wth both Goldfeath and CAN thanks to pont cloud hand model 5.1 free of nose, h=.0, Fgure 4(c) (d) andnose Fgure label saddle regon n (a) nose 0.0but when(b) CAN Fgure 4(b), (e) all(e) CAN same (f) Goldfeath Saddle regonsfgure 5 elephant model sharp rdge regons dscerned by two approaches. A dscrete model elephant was used also, where heght of ths model s 75 unts. Ths model Let k1 and k beof prncple curvatures of pont p. If pont p s a saddle pont, ts Gaussan conssted 6859 ponts wth normal vectors. Fgure 5 shows that dscerned sharp rdge regons are curvaturegood s H = k k 0, whch can be used for dscernng saddle regon. A dscrete model p both 1both wth methods thanks to model free of nose, but when nose level s hgher and hand s used n experment. The heght of model s 0 unts, and t conssts of 658 hgher, CAN method s more robust than or. ponts wth normal vectors. Saddle regons are labeled wth red color. Fg. 4 shows that estmated5.3.saddle correct through both Goldfeath and CAN thanks to Prncpleregons Drectonare Estmatng pont cloud hand model free of nose. But f h =.0, Fg. 4(b), Fg. 4(c) and Fg. 4(e) all label Anor experment s nvolved n prncple drecton estmaton. We use a rocker model. Its coordnate sameranges saddle regon, and< xse regons same about a hand, but Fg < 0.00, 0.01are < y < 0.01 are that and 0.as 000our < z <knowledge (f) from Goldfeath method gves a dfferent regon from Fg. 4(c). Therefore, a concluson can be safely drawn that CAN method s more robust than Goldfeath method n dscernng saddle regons. 6 (a) nose 0.0 (d) nose 0.05 Fgure 6 Prncple drecton estmaton of rocker model. By checkng two red crcles n Fgure 6(e) and (f), we fnd that f nose level s hgh, CAN can

7 5.. Sharp rdge regon dscerned (a) nose 0.0 (d) nose 0.05 Fgure 5 elephant model sharp rdge regons dscerned by two approaches. If pont p s n a sharp rdge regon, k1 k > δ, where δ > 0 and t s a subjectve parameter whch A dscrete model elephant was used also, where heght of ths model s 75 unts. Ths model can be gven to cloud data. Fgure 5 shows that dscerned sharp rdge regons are conssted of accordng 6859 ponts wthpont normal vectors. good wth both both methods thanks to model free of nose, but when nose level s hgher and hgher, CAN method s more robust than or Prncple Drecton Estmatng (a) nose 0.0 (d) nose 0.05 Anor experment s Fgure nvolved n prncple drecton estmaton. We a rocker model. Its coordnate 5 elephant model sharp rdge regons dscerned by twouse approaches. 0An.00 < x < 0was.model 00, wth 0also,.01 <where y <rdge 0.01 ranges are Fg. thatmodel and z < model 0by.071. s approaches. 5: elephant sharp dscerned two A dscrete elephant used regons heght of <ths 75 unts. Ths model conssted of 6859 ponts wth normal vectors. Fgure 5 shows that dscerned sharp rdge regons are good wth both both methods thanks to model free of nose, but when nose level s hgher and hgher, CAN method s more robust than or Prncple Drecton Estmatng (a) nose 0.0 (d) nose 0.05 Anor experment s nvolved n prncple drecton estmaton. We use a rocker model. Its coordnate Fgure 6 Prncple drecton estmaton of rocker model. ranges are that 0.00 < x < 0.00, 0.01 < y < 0.01 and < z < : n Prncple drecton estmaton of a f rocker model. By checkng two red Fg. crcles Fgure 6(e) and (f), we fnd that nose level s hgh, CAN can keep drectons wth that of low nose level. 5. Sharp rdge regons If pont p s n a sharp rdge regon, k1 k > δ. δ s a subjectve parameter and t can be specfed accordng to pont cloud data. An elephant model was used. Its heght s 75 unts. Ths(a)model conssts of 6859 ponts (c) wth normal vectors. Fg. 5 shows that dscerned sharp nose 0.0 Goldfeath (d) nose 0.05 rdge regons are correct wthfgure both6 Prncple methods thanks to model free of nose, but when nose drecton estmaton of rocker model. level gets hgher and hgher, CAN method s more relable than ors. By checkng two red crcles n Fgure 6(e) and (f), we fnd that f nose level s hgh, CAN can keep drectons wth that of low nose level. 5.3 Prncple drecton estmaton Anor experment s nvolved n prncple drecton estmaton. We use a rocker model. Its coordnate ranges are that 0.00 < x < 0.00, 0.01 < y < 0.01, and < z < By checkng two red crcles n Fg. 6(e) and (f), t can be be found that f nose level s hgh, CAN can keep drectons wth that of low nose level. 6 Conclusons and Furr Work A new algorthm s presented for estmatng prncpal curvatures, called as CAN, for scattered sampled pont data, and mesh vertex data as well. Experments show that ths new algorthm can be used to extract local dfferental propertes, ncludng saddle regons, sharp rdge regons and prncple drectons wth hgh relablty. The effects are better than or smlar approaches. The computaton complexty of ths new algorthm s low. The CAN algorthm s robust to data wth strong nose also. We use Normal Curvature Index Lnes (NCIL) to analyss estmaton error produced by dfferent algorthms. Wth NCIL, errors generated by curvature estmaton can be shown drectly. In future, we would lke to apply CAN method to extract features wth hgh precsons from pont cloud data, lke saddle lnes, rdge lnes and curvature lnes on a pont surface. 7

8 Acknowledgement Ths work s supported n part by Natonal Natural Scence Foundaton of Chna wth projects No , , , and ; n part by Natonal Hgh Technology Development Program (863) of Chna under Grants No. 008AA01Z301, 006AA01Z301, and n part by Bejng Muncpal Natural Scence Foundaton under Grant No References [1] Y. Mao, J. Feng, Q. Peng. Curvature Estmaton of Pont-Sampled Surfaces and Its Applcatons. ICCSA 005, LNCS 348, pp , 005. [] P. T. Sander, and S. W. Zucker. Inferrng Surface Trace and Dfferental Structure from 3-D Images. IEEE Trans. Pattern Anal. Mach. Intell. 1, 9 (Sep. 1990), [3] E. M.Stokely, and S. Y. Wu. Surface Parametrzaton and Curvature Measurement of Arbtrary 3-D Objects: Fve Practcal Methods. IEEE Trans. Pattern Anal. Mach. Intell. Vol. 14, No. 8. [4] J. Goldfear, and V. Interrante. A novel cubc-order algorthm for approxmatng prncpal drecton vectors. ACM Trans. Graph. 3, 1 (Jan. 004), [5] N. Dyn, K.Hormann, S. Km, and D. Levn. Optmzng 3D trangulatons usng dscrete curvature analyss. In Mamatcal Methods For Curves and Surfaces: Oslo 000, T. Lyche and L. L. Schumaker, Eds. 1 ed. Vanderblt Unv. Press Innovatons In Appled Mamatcs Seres. Vanderblt Unversty, Nashvlle, TN, [6] S. J. Km, C.-H. Km, and D. Levn. Surface smplfcaton usng dscrete curvature norm, n: The Thrd Israel-Korea Bnatonal Conference on Geometrc Modelng and Computer Graphcs, Seoul, Korea, October 001. [7] G. Taubn. Estmatng tensor of curvature of a surface from a polyhedral approxmaton. In Proceedngs of Ffth nternatonal Conference on Computer Vson (June 0-3, 1995). ICCV. [8] P. Yang, and X. Qan. Drect Computng of Surface Curvatures for Pont-Set Surfaces, Eurographcs Symposum on Pont-Based Graphcs (007). [9] E. Kalogeraks, P. Smar, D. Nowrouzezahra, and K. Sngh. Robust statstcal estmaton of curvature on dscretzed surfaces. In Proceedngs of Ffth Eurographcs Symposum on Geometry Processng (Barcelona, Span, July 04-06, 007). [10] M. Lu, Y. Lu, and K.Raman. Ansotropc flterng on normal feld and curvature tensor feld usng optmal estmaton ory. In Proceedngs of IEEE nternatonal Conference on Shape Modelng and Applcatons 007 (June 13-15, 007). [11] M. Egensatz, R. Sumner, and M. Pauly. Curvature-Doman Shape Processng, EUROGRAPHICS 008, Volume 7 (008), Number [1] S. Fourey, R. Malgouyres. Normals and Curvature Estmaton for Dgtal Surfaces Based on Convolutons. DGCI 008, LNCS 499, pp , 008. [13] E. Magd, O. Soldea,and E. Rvln. A comparson of Gaussan and mean curvature estmaton methods on trangular meshes of range mage data. Comput. Vs. Image Underst. 107, 3 (Sep. 007), [14] Z. Cheng, X. Zhang, B. Chen. Smple reconstructon of tree branches from a sngle range mage. Journal of Computer Scence and Technology, Vol., No. 6, pp [15] X. Zhang, H. L, Z. Cheng. Curvature Estmaton of 3D Pont Cloud Surfaces Through Fttng of Normal Secton Curvatures, Proceedngs of AsaGraph 008, pp.7-79, Tokyo, Japan, October 3-6, 008, 8