Octree Subdivision Using Coplanar Criterion for Hierarchical Point Simplification

Transcription

1 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton Pa-Feng Lee 1, Chen-Hsng Chang 1, Jun-Lng Tseng 2, Bn-Shyan Jong 3, and Tsong-Wuu Ln 4 1 Dept. of Electronc Engneerng, Chung Yuan Chrstan Unversty, 2 Dept. of Management Informaton System, Chn Mn Insttute of Technology, 3 Dept. of Informaton & Computer Engneerng, Chung Yuan Chrstan Unversty, 4 Dept. of Computer & Informaton Scence, Soochow Unversty {allen, hkk, arthur}@cg.ce.cycu.edu.tw, bsjong@ce.cycu.edu.tw, twln@cs.scu.edu.tw Abstract. Ths study presents a novel rapd and effectve pont smplfcaton algorthm based on pont clouds wthout usng ether normal or connectvty nformaton. Sampled ponts are clustered based on shape varatons by octree data structure, an nner pont dstrbuton of a cluster, to judge whether these ponts correlate wth the coplanar characterstcs. Accordngly, the relevant pont from each coplanar cluster s chosen. The relevant ponts are reconstructed to a trangular mesh and the error rate remans wthn a certan tolerance level, and sgnfcantly reducng number of calculatons needed for reconstructon. The herarchcal trangular mesh based on the octree data structure s presented. Ths study presents herarchcal smplfcaton and herarchcal renderng for the reconstructed model to sut user demand, and produce a unform or feature-senstve smplfed model that facltates rapd further mesh-based applcatons, especally the level of detal. 1 Introducton Due to the contnung development of computer graphcs technology, dversfed vrtual realty applcatons are beng ncreasngly adopted. Exactly how three dmensonal (3D) objects n the real world can be effcently and vvdly portrayed n vrtual scenes has recently become a crucal ssue n computer graphcs. A trangular mesh s one of the most popular data structures for representng 3D models n applcatons. Numerous methods currently exst for constructng objects usng surface reconstructon, and reconstructng a pont cloud as a trangular mesh. The data for sampled ponts are generally obtaned from a laser scanner. However, the extracted sampled ponts are frequently affected by shape varaton, causng over-samplng n the flat surface (Fg. 1). The number of trangles created rses as the number of ponts sampled from the surface of a 3D object rses, helpng to reconstruct the correct model. However, subsequent graphcs applcatons, such as morphng, anmaton, level of detal and compresson, ncrease the computaton costs. Approprate relevant ponts should be chosen to retan object features and reduce storage and calculaton costs. L.-W. Chang, W.-N. Le, and R. Chang (Eds.): PSIVT 2006, LNCS 4319, pp , Sprnger-Verlag Berln Hedelberg 2006

2 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton 55 Hence, reducng the number of trangles whle retanng surface characterstcs wthn a certan error value s worthy of research. The man research on object smplfcaton can be dvded nto two felds, mesh-based smplfcaton [1, 2, 3, 7, 9, 14] and pont-based smplfcaton [5, 6, 8, 10, 12, 13]. Mesh-based smplfcaton requres connectvty relatons to be obtaned n advance. Hence, the sampled pont data acqured by a scanner must perform trangulaton by a reconstructon algorthm before smplfcaton. Surface reconstructon extracts sampled ponts from a 3D object, and reconstructs the trangular mesh from an orgnal object wthn a certan tolerance level [4]. The mesh-based smplfcaton then attempts to reduce the number of trangles n trangular mesh whle mantanng the object qualty. Numerous good meshbased smplfcaton algorthms have been presented, ncludng vertex decmaton [14], edge contracton [1], trangle contracton [2], vertex clusterng [7], vertex par contracton [9] and feature extracton [3]. Tradtonal methods such as Quadrc Error Matrces (QEM) [9] have decmaton operatons that are generally arranged n a prorty queue accordng to an error matrx that quantfes errors caused by decmaton. Smplfcaton s performed teratvely to reduce any smoothng of pont pars caused by the decmaton operaton. Ths greedy technque can obtan the smplfed model wth the mnmum error of the orgnal model. However, these algorthms all acheve good smplfcaton effect n applcaton, but need a trangular mesh and connectvty n advance of smplfcaton. Restated, the algorthms are burdened wth a large number of computatons before smplfcaton processng. Consequently, ths process s prohbtvely expensve. Fg. 1. Over-samplng n the flat regon needlessly ncreases the number of calculatons. The smplfed model produces the same effect of sold representaton. Therefore, pont cloud smplfcaton s an attractve approach. Pont-based smplfcaton s appled before reconstructon. If sutable relevant ponts can be extracted from a pont cloud that represent surface varaton, then the number of calculatons needed for reconstructon can be sgnfcantly reduced. Dey [13] presented the frst pont cloud smplfcaton approach. Dey adopted local feature szes to detect redundancy n the nput pont cloud and ensure relevant pont denstes, thereby explotng a 3D Vorono dagram for a densely dstrbuted orgnal pont set n advance of smplfcaton. However, ths method also requres many computatons. Bossonnat and Cazals [6] presented a coarse-to-fne pont smplfcaton algorthm that randomly calculates a pont subset and bulds a 3D Delaunay trangulaton. Addtonal ponts are nserted teratvely based on ther dstance to the closest 3D facet untl the smplfed surface mesh conforms to the restrcted error value. Allegre [12] presented a convex hull for all ponts that adopts a decmaton scheme to merge adjacent ponts accordng to geometrcal and topologcal constrants. These algorthms must adopt pre-processng to retan the orgnal surface data before smplfyng the pont set, and therefore requre many computatons.

3 56 P.-F. Lee et al. Pauly [10] appled the four mesh-based smplfcaton technques to pont cloud smplfcaton. A unform ncremental clusterng method s computatonally effcent, but leads to a hgh mean error. The herarchcal clusterng method can reduce calculaton and memory, but has a margnally better mean error value than the unform ncremental clusterng method. The quadrc error-based teratve smplfcaton method obtans the best error rate, but has a major dsadvantage n that ts executon tme s senstve to the nput pont set sze. The partcle smulaton method obtans a good error rate, but requres many calculatons. Alexa [8] proposed to unformly smplfy the pont set by estmatng the dstance from a pont to the Movng Least Square (MLS) surface. Alexa also presented a re-samplng method to ensure the dstrbuton of densty. Moennng and Dodgson [5] presented an ntrnsc coarse-to-fne pont smplfcaton algorthm that guarantees unform or feature-senstve dstrbuton. They adopted the farthest pont samplng and a fast marchng algorthm to choose relevant ponts and set densty threshold to ensure pont set densty. However, ther method requres expandng the computatonal 3D Vorono dagram, and consequently requres many computatons and a large memory. Ths study presents a novel rapd and effectve pont smplfcaton algorthm based on a pont cloud wthout normal and connectvty nformaton. Ths study ntates wth a scattered sampled pont set n 3D, and the fnal output s a trangular mesh model smplfed accordng to restrctve crtera. The proposed method reduces the number of calculatons between trangulaton and establshng the connectvty relaton, and ncludes three man steps, pont smplfcaton, reconstructon guarantee and herarchcal smplfcaton. In the pont smplfcaton step, sample ponts are obtaned usng 3D acquston devces that fully represent the object surface varaton. Ths step nvestgates how to best choose the most approprate number of relevant ponts from the sampled ponts to reduce the complexty of operaton and obtan an acceptable smplfed result. Sampled ponts are clustered, based on shape varaton, by usng the octree data structure, whch s an nner pont dstrbuton of a cube, to judge whether these ponts correlate wth the coplanar propertes. The local coplanar method causes the smplfed model to have a feature-senstve property. The feature-senstve dstrbuton can acheve a small smplfcaton-based error rate, but does not permt successful reconstructon. Consequently, consderng the scattered relevant ponts n the levels of the herarchcal tree are dynamcally adjusted. Moreover, the dstrbuton densty s ncreased by dummy vertces n the regon wth excessve dfference between adjacent levels, helped by herarchcal tree nformaton. The problem of undersamplng s thus successfully solved, obtanng a good smplfed, reconstructed model. Fnally, a herarchcal trangular mesh sutable for mult-resoluton s obtaned after successfully reconstructng the smplfed pont set. Ths study presents a novel method for extractng the relevant ponts for a dense nput pont set, and adopts the reconstructon algorthm presented by Jong [4] to generate a smplfed model. Expermental results confrm that a good smplfed model, wth the advantages gven below, can be quckly obtaned. 1. Connectvty relatons do not need to be recorded. Hence, the reconstructon algorthm can be adopted, sgnfcantly reducng the computatonal cost. 2. The calculatons for extractng the relevant ponts ensure that the smplfed model has a good error rate. 3. Usng an octree data structure mantans mult-resoluton.

4 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton The herarchcal renderng and magng can be nteractvely changed by automatcally assgnng local samplng constrants. 2 Algorthm Overvew The followng steps are crucal to smplfyng the sampled pont cloud and completely reconstructng the trangular mesh. 2.1 Choose the Coplanar Varable In ths study, the pont cloud s subdvded teratvely accordng to the space coordnates untl each cluster meets ts respectve restrcted crteron. The local neghbor of a pont set for a cluster was dentfed by the formula presented by Pauly [10], based on the followng equatons: c def = 1 N q N q 3 c R (1) C f def = 0 1 N q N λ 0 = λ + λ + λ 1 t ( q c )( q c ) 2 f 0 C R λ λ λ where, gven a cluster, c denotes the gravty center; C denotes the covarance matrx; q denotes the pont n a cluster, and N denotes the number of ponts. Accordng to Eq. (2), the covarance matrx C s a symmetrcal postve semdefnte 3 3 matrx wth three egenvalues λ 0, λ 1 and λ 2. These egenvalues measure the varaton of ponts n the drecton of ther correspondng egenvectors e 0, e 1, e 2. Egenvector e 0 s a vector characterstc of the mnmum egenvalue λ 0, whch denotes the normal vector of a cluster. A cluster conforms to coplanar characterstcs f λ 0 <<λ 1 and λ 0 <<λ 2. Equaton (3) determnes whether a cluster s subdvded accordng to the coplanar varable f. The subdvson crteron for the octree s based on the coplanar varable f, whch determnes whether a node must be subdvded. Ths step ensures that dynamc subdvson s performed accordng to model surface varatons. The rough part of model s subdvded, and then addtonal relevant data can be obtaned to refne regons ncludng the object feature regons. For the flat area, a large number of sampled ponts are reduced to a sngle pont (Fg. 2) * 3 (2) ( 3 ) Fg. 2. The Bunny model (34838 ponts) as an example usng dfferent thresholds. When the coeffcent f = (a), the number of relevant ponts s 5616; when f = 0.001, the number of relevant ponts s (b).

5 58 P.-F. Lee et al. 2.2 Choose the Relevant Pont of Each Cluster The pont set s subdvded nto clusters Accordng to the coplanar varable f, and the relevant pont from each cluster s then selected. Ths pont denotes the local surface characterstc for the entre cluster. If the selected pont s the gravtatonal center of a cluster, then ether t s not the orgnal pont, or a major error has occurred, n whch case some unexpected trangles may protrude from and concave nto the surface (Fg. 3(a)). Ths study chooses the orgnal pont that s the closest to the gravtatonal center as the relevant pont (Fg. 3(b)), because selectng the closest pont can effectvely reducng the probablty of producng errors. (a) (b) Fg. 3. (a) The gravtatonal center of cluster c s chosen as the relevant pont; (b) P s chosen as the closest pont to the gravtatonal center c, and set to the relevant pont 2.3 Identfy the Near Surface and Adjust the Approprate Relevant Ponts Another potental problem caused when space subdvson occurs on the near surface, revealng that the two surfaces are extremely close to each other. The near surface n Fg. 4(a) may lead to an ncorrect judgment of flatness and cause non-manfold occurrences, because the respectve ponts belongng to two surfaces may be merged nto one relevant pont (Fg. 4(b)). Ths mstake causes nconsstent curvature and errors n topology (Fg. 4(c)). P (a) (b) (c) Fg. 4. (a) The near surface may lead to ncorrect judgment; (b) Incorrect judgment for a near surface may cause non-manfold occurrence; (c) Reducton of the near surface to p causes nconsstent curvature and topology errors The proposed near surface dentfcaton method has two parts. The frst part comprses auxlary ponts (u) that are used to detect the near surface nsde the cluster. These auxlary ponts are located at the cluster center c and the corners q (the centers of sub-clusters). For each auxlary pont u, the k closest ponts (p k ) are selected, and the cluster normal e 0 (N) s calculated to obtan a relable estmate of the sgned nner product. The near surface can be easly dstngushed accordng to the nner product sgn ( p u N ), (Fg. 5(a)). The second part assumes that the nearest neghbor r of

6 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton 59 each pont p n a cluster s found accordng to the value of ts nner product ( p r N ) n order to detect the near surface (Fg. 5(b)). If a cluster may have a near surface, then t s subdvded to ensure that the near surface does occur (Fg. 5(c)). (a) (b) (c) (d) Fg. 5. (a) Auxlary ponts at the cell center and corners are adopted to detect a near surface; (b) The nner product of the normal vector and adjacent ponts s adopted to determne whether a near surface exsts; (c) The cluster contanng a possble near surface s subdvded to avod non-manfold occurrence; (d) Produced correct surface 2.4 Adaptvely Add Dummy Vertces to Avod Under-Samplng Space subdvson can result n the obvous phenomenon of rregular densty dstrbuton between adjacent flat and feature areas. Unexpected holes durng reconstructon resultng from under-samplng, due to nsuffcent nformaton about for neghborng ponts n the local regon (Fg. 6(a)). To avod unexpected holes, the cluster based on the coplanar characterstc s agan subdvded to produce dummy vertces, and ncrease and adjust the densty of adjacent nodes. Therefore, nformaton for adjacent ponts must be consdered durng reconstructon, and the levels for neghborng nodes must be restrcted when constructng the tree structure. The nodes satsfyng the coplanar restrcton are affected by ther neghborng sub-trees and subdvson contnues. (a) (b) Fg. 6. (a) Unexpected holes durng reconstructon resultng from under-samplng. (b) Reconstructed correct base model. To avod rregular densty dstrbuton and under-samplng caused by the coplanar restrcton, subdvson contnues untl the level dfference of adjacent clusters s wthn n/2, where n denotes maxmum level of the octree even when the cluster s n accordance wth the coplanar. The refned model s called the base model, and automatcally represents dfferent densty dstrbutons accordng to model varaton. The densty can be adjusted asymptotcally at the ponts where dfferent denstes change, thus guaranteeng the accuracy of reconstructon. The experment ndcates that the number ponts n the base model are roughly 30% of that n the orgnal model, revealng that the cost of reconstructng the base model s 30% of that of reconstructng

7 60 P.-F. Lee et al. the orgnal model. Ths method ensures that the fnal trangular mesh s n accordance wth object s topology (Fg. 6(b)). The subdvded dummy vertex then reduces to ts orgnal level n the followng step. 2.5 Merge Dummy Vertex Followng correct reconstructon, the herarchcal tree-structure nformaton can recover the base model smply and effcently to a level based on the coplanar restrcton. Ths effcent mergng of a sub-tree brother reduces to ts father node s locaton at the prevous level. The expermental results ndcate that usng the coplanar varable f =0.005 can yeld a reconstructed model usng roughly 10% of the orgnal ponts. Error rates for the reconstructed scheme, the scheme by QEM and the scheme by unform subdvson pure space subdvson wthout restrctng coplanar value) were compared (Fg. 7). (a) (b) (c) Fg. 7. The expermental results obtaned by (a) QEM, (b) unform subdvson and (c) the proposed method to smplfy the same models wth smlar numbers of reconstructed model ponts. The QEM method acheves the best error rate, snce t has mesh nformaton; the proposed method adopts a coplanar restrcton to smplfy the orgnal pont cloud wthout addtonal nformaton and to perform reconstructon, therefore mantanng a good error rate. Unform subdvson has a regular dstrbuton, but causes under-samplng n the feature area. 3 Herarchcal Smplfcaton and Herarchcal Renderng for Mult-resoluton Applcatons The model can be dsplayed dynamcally and quckly based on the octree data structure after reconstructon accordng to the coplanar restrcton. Usng the octree structure for space subdvdng produces mult-resoluton, and adjustng the octree level n varous ways generates dfferent dsplay results. User-control dynamcally adjusts the resoluton level that can be dsplayed wthout further computatons. As long as the octree data structure obtaned by the prevous calculaton s adopted, suffcent smplfed data can be provded to acheve rapd and effectve smplfcaton and update the connectvty nformaton. The followng methods can be adopted to obtan dfferent renderng effects, where the user controls the number of relevant ponts.

8 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton 61 Depth Frst reducton: The relevant ponts of a unform dstrbuton are produced. The nodes on the deepest level are frst deleted and reduced to the relevant ponts of the prevous level. Returnng to the deepest level each tme results n the smplfcaton effect for a unform dstrbuton (Fg. 8(a)). Reducton by one rng neghbor coplanar measurement: Each smplfed relevant pont can denote the surface nformaton for each small regon. The varaton error denotes normal dfferences between adjacent relevant ponts, and can be adopted to estmate the coplanar degree of a relevant pont for ts adjacent regon. Smplfcaton operatons are arranged n a prorty queue accordng to the varaton error (Q* pq ) of each relevant pont par. The value of the coplanar and the relevant pont of the selected pont pars are recalculated to ensure a good error rate (Fg. 8(b)). q 6 q 1 e q 6 Q * pq6 Q * pq1 q 1 Q * pq5 q 5 p q 2 q 5 Q * pq4 p Q * pq2 q 2 Q * pq3 (a) q 4 q 3 q 4 q 3 (b) Fg. 8. (a) Depth Frst reducton reduces the deepest level each tme; (b) Reducton by one rng neghbor coplanar measurement reduces the ponts accordng to the varaton error of each relevant pont par (Q * pq=q p +Q q ) (Q p =e pq1 +e pq2 +e pq3 +e pq4 +e pq5 +e pq6 ) 4 Expermental Results The followng smplfed models were obtaned by the proposed method. Ths study confrms that the coplanar varable f = obtans a good smplfcaton result. The number of relevant ponts s slowly reduced (Fg. 9) when the coplanar varable f exceeds Restated, s an effectve value for the degree of flatness n a model. The proposed algorthm adopts as a default value, and can obtan a good error range (Table 1). The results of the proposed method are show n Fgure 10 to Fgure 12. Fg. 9. The number of relevant ponts s slowly reduced when f exceeds 0.005

9 62 P.-F. Lee et al. Table 1.The sze generated by dfferent models and smplfed error measurement by Metro tool [11] and the flatness s usng Dragon Budda Armadllo Venus Orgnal ponts base model (21%) (24%) (26%) (19%) reconstructed model (12.9%) (17.8%) (25.0%) (12.1%) Mean Error (absolute) Mean Error (relatve) Fg. 10. The reconstructed results of the Dragon model on varous levels after adoptng the Depth Frst reducton. From left to rght, Base model (90374ponts), reconstructed model (56302 ponts), ponts, ponts, ponts, and ponts (3% of orgnal). Fg. 11. The reconstructed results of the Budda model on varous levels after usng the one rng neghbor coplanar measurement. From left to rght, Orgnal model, Base model ( ponts), reconstructed model (96967 ponts), ponts, ponts, ponts, ponts; and 4963 ponts (0.9% of orgnal). (a) (b) Fg. 12. The dfferent pont dstrbuton of herarchcal renderng. (a) The depth frst reducton obtans unform dstrbuton (11154 ponts) and (b) the one rng neghbor coplanar measurement product the feature-senstve result (10014 ponts). 5 Conclusons and Future Work Ths study presents a novel method the smplfyng a pont set usng an octree structure to calculate the coplanar varable f, and spatally subdvde the sampled ponts n 3D. The nput data only contans pont coordnates. The fnal output ncludes a trangular

10 Octree Subdvson Usng Coplanar Crteron for Herarchcal Pont Smplfcaton 63 mesh and octree data structure. Reducng the level of the octree can dynamcally adjust ts result wthout needng addtonal calculatons. Ths proposed method facltates producng a unform and feature-senstve smplfed model for further meshbased applcatons. Further work wll ntegrate pont smplfcaton and reconstructon algorthms; try to permt under-samplng and produce an approprate smplfed pont set, and correctly reconstruct smplfed pont sets wthout ncreasng the dummy vertex. References 1. A. Gu ezec, Surface smplfcaton wth varable tolerance, Second Annual Intl. Symp. on Medcal Robotcs and Computer Asssted Surgery (MRCAS 95), 1995, pp B. Hamann, A Data Reducton Scheme for Trangulated Surfaces, Computer Aded Geometrc Desgn, Vol.11, 1994, pp B. S. Jong, J.L. Tseng, W. H. Yang, T. and W. Ln, Extractng Features and Smplfyng Surfaces usng Shape Operator, The 2005 IEEE Internatonal Conference on Informaton, Communcatons and Sgnal Processng (ICICS 2005), 2005, pp B. S. Jong, W. Y. Chung, P. F. Lee, and J. L. Tseng, Effcent Surface Reconstructon Usng Local Vertex Characterstcs, The 2005 Internatonal Conference on Imagng Scence, Systems, and Technology : Computer Graphcs, 2005, pp C. Moennng and N. A. Dodgson, Intrnsc pont cloud smplfcaton, In Proc. 14th GrahCon, Vol. 14, J.D. Bossonnat and F. Cazals, Coarse-to-fne surface smplfcaton wth geometrc guarantees, EUROGRAPHICS 01, Conf. Proc, 2001, pp J. Rossgnac and P. Borrel, Mult- resoluton 3D approxmatons for renderng complex scenes, Modelng n Computer Graphcs: Methods and Applcatons, 1993, pp M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D. Levn, and T. Slva, Pont Set Surfaces, In Proc. 12th IEEE Vsualzaton Conf., 2001, pp M. Garland and P. S. Heckbert, Surface smplfcaton usng quadrc error metrcs, SIGGRAPH 97 Conference Proceedngs, 1997, pp M. Pauly, M. Gross and L. P. Kobbelt, Effcent Smplfcaton of Pont-Sampled Surfaces, In Proc. 13th IEEE Vsualzaton Conf., 2002, pp P. Cgnon, C. Montan, and R. Scopgno, Metro: Measurng Error on Smplfed Surfaces, Computer Graphcs Forum, Vol.17, No.2, 1998, pp R. All`egre, R. Chane, and S. Akkouche, Convecton-drven dynamc surface reconstructon, In Proc. Shape Modelng Internatonal, 2005, pp T. K. Dey, J. Gesen and J. Hudson, Decmatng Samples for Mesh Smplfcaton, In Proc. 13th Canadan Conference on Computatonal Geometry, 2001, pp T. S. Geng, Bernd Hamann, Kenneth I. Joy, Gregory L. Schussman, and Issac J. Trotts, Constructng herarches for trangle meshes, IEEE Transactons on Vsualzaton and Computer Graphcs, Vol. 4(2), 1998, pp