Efficient Non-Linear Editing of Large Point Clouds

Transcription

1 Effcent Non-Lnear Edtng of Large Pont Clouds Faban Ateanu Patrck Degener Renhard Klen Unverstät Bonn Insttut für Informatk II - Computergraphk D Bonn, Germany {ateanu, degener, rk}@cs.un-bonn.de ABSTRACT Edtng 3D models s often performed on trangular meshes. We generalze edtng operatons based on dfferental coordnates to work on pont clouds wthout explct connectvty nformaton. Ths allows a pont cloud to be nterpreted as a surface or volumetrc body upon whch physcally plausble deformatons can be appled. Our multresoluton approach allows for a real-tme edtng experence of large pont clouds wth 1M ponts wthout any offlne processng. We tested our method on a range of synthetc and real world data sets acqured by laser scanner. All of them were nteractvely edtable and produced ntutve deformaton results wthn few mnutes of edtng. Keywords: Pont clouds, non-lnear edtng, real-tme, dfferental coordnates, subsamplng. 1 INTRODUCTION Vrtual 3D objects are a common asset for moves and games. In former years those were predomnantly created manually, but n later years t has become more common to scan real objects and then edt and modfy ther vrtual counterparts. For some applcaton areas, e.g. structured lght scannng, data s avalable only as pont clouds, though edtng 3D models s predomnantly performed on trangular meshes. Despte ongong efforts n the area of surface reconstructon, trangulatng pont sets remans computatonally demandng and s error-prone n the presence of nose and outlers. Addtonally, pont clouds are usually sampled much denser than meshes and thus contan amounts of data, whch are several tmes larger than for comparable meshes. Ths n turn can negatvely nfluence the nteractvty of edtng operatons. not show these problems. The hgher ntutvty though comes at the cost of a sgnfcantly hgher computatonal effort. Space deformatons are a dfferent approach towards edtng operatons. They can conceptually be calculated quckly, but lack the ablty to respond to the structure of the edted model. Ponts whch are far apart n geodesc dstance can be close n space and thus be nfluenced by edtng operatons unntentonally. To overcome ths lmtaton, cage-based technques try to separate such geodescally dstant regons from each other, but the user s requred to create or at least adjust the cage manually. In contrast, our soluton does not requre any manual nterventon. We present a novel method to enable drect real-tme edtng of pont clouds wthout the need for pror To address ths problem specalzed edtng approaches for pont sampled models have been proposed. However, to handle the larger complexty of pont clouds, these methods resort to relatvely smple deformaton models lke transformaton nterpolaton or lnear models based on dfferental coordnates. As recently analysed by Sorkne and Botsch [SB09], a common problem of these smple deformaton models s a counterntutve deformaton behavour. For deformaton of trangle meshes ths problem s well known, and subsequently non-lnear deformaton models have been developed that do Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Fgure 1: Armadllo model after 3 edtng steps: head has been twsted by 45, left arm twsted by 90, rght arm bent by 30.

2 meshng. Our method shows ntutve, physcally plausble deformaton behavour and does not requre manual preprocessng lke cage constructon. It s based on geodesc dstances only and avods the problems of space deformaton based approaches descrbed above. Usng our method, large pont sampled geometry (Fgure 1) can be edted as a whole at nteractve frame rates. In summary our contrbuton s ths: We present an approach for edtng pont clouds whle preservng local surface detals. Due to an underlyng non-lnear deformaton model, the obtaned deformatons are physcally plausble, even for large handle transformatons. Our algorthm does not pose any specfc assumptons upon the structure of the pont cloud beng edted. The nput requres solely the vertex postons n 3D space, but no explct connectvty nformaton. The pont cloud may be rregularly sampled or contan outlers. It s even possble to handle pont sampled volumes. As pont clouds are usually sampled much denser than comparable meshes, we present a means of performng the deformaton calculatons on a coarse scale and transfer the results to the fne scale representaton of the pont cloud at nteractve frame rates. To the best of our knowledge, no other system has been presented whch could perform smlar non-lnear deformatons wthout offlne processng. 1.1 Related Work As mentoned n the ntroducton, several methods exst for edtng meshed nput data, but whch are not drectly applcable to pont clouds. In the followng we concentrate on methods targetng pont clouds. The challenge of edtng pont clouds nteractvely has been pursued n a number of papers. In an early work of Pauly et al. [PKKG03] they deform pont clouds n real-tme usng lnear nterpolatons between the handles. Mao et al. [MFXP08] use a more sophstcated method wth dfferental coordnates. Lke all lnear methods they both produce counterntutve results for large deformatons. Especally for edtng large models the requred storage can exceed the avalable man memory, or the computaton tme can become prohbtvely hgh. It s thus often necessary to use a smplfed model upon whch the edtng s performed. Wand et al. [WBB + 07] descrbe a method to vsualze data sets wth a sze of several ggabytes n a mult-resoluton out-of-core method. The drect edtng operatons are lmted to smple translatons or deletons of vertces. More complex operatons can only be performed n offlne computatons. Boubekeur et al. [BSS07] use a streamng method to perform an offlne smplfcaton of a large mesh nto a smaller pont cloud, whch can then be edted nteractvely. A second offlne streamng step apples the modfcatons to the orgnal mesh. Wcke et al. [WSG05] use the concept of thn shells to construct a network of so-called fbres on the surface of a pont cloud, whch are then used to model and calculate possble deformatons. The fbres provde a mesh representaton at a coarse scale, and the deformatons are later appled to the detaled representaton. As both of the methods of Boubekeur and Wcke requre tme-consumng steps before and after the user performs the actual edtng, the user has to wat for the post-edtng steps to complete n order to vew the fnal result. We deem ths unsutable for ad-hoc edtng, because any correctons or further edtng steps requre repeatng the whole procedure. In our method, the actual edtng of the pont cloud s performed on a dfferental representaton of ts surface. Ths has recently ganed much attenton n the context of mesh edtng. Sorkne and Botsch [SB09] compare gradent-based methods and Laplacan-based methods as the predomnant methods used for dfferental representaton-based deformatons. They dentfy two classes: lnear methods can provoke a counterntutve behavour for deformatons, snce they cannot handle both rotatons and translatons of local frames smultaneously, resultng n artfacts of dfferent knds. Non-lnear methods [BPGK06, SA07] solve these ssues, but are tme-consumng to compute. The mesh edtng framework presented by Pares et al. [PDK07] represents local frames usng quaternons. By enforcng addtonal frame constrants they produce ntutve results for translatons, rotatons and scalng. Ther non-lnear approach s computatonally demandng and thus applcable only to medum szed meshes. We generalze ther approach to be applcable to large pont clouds. Space deformatons (see e.g. [HSL + 06, XZY + 07, BPWG07, AOW + 08]) are a dfferent approach towards edtng operatons. They do not drectly manpulate the ponts postons, but warp the space and thus ndrectly manpulate the ponts. The space deformatons can conceptually be calculated quckly, but they lack the ablty to respond to the structure of the edted model. Ponts whch are far apart n geodesc dstance can be close n space and thus be nfluenced by edtng operatons unntentonally. To overcome ths lmtaton, Huang et al. [HSL + 06] used a control mesh enclosng the model to separate such geodescally ds-

3 tant regons from each other. In ther approach the user s requred to defne the control mesh. In contrast, our soluton does not requre any manual nterventon. The multscale representaton ntroduced by Pauly et al. [PKG06] and extended by Duranleau et al. [DBP08] encodes the dsplacement of ponts between the representatons at dfferent scales. It allows to edt the representaton for a specfc detal level usng a space warpng functon and then apply the deformaton to the other detal levels. Ths enhances nteractvty and controllablty of the result, but stll has the mentoned lmtatons of space deformatons. Meshless methods, of whch space deformatons are one specalzaton, provde further aspects. The phyxels proposed by Müller et al. [MKN + 04] are used to fll a volume and preserve t durng edtng operatons. The small amount of phyxels requred to fll a volume, allow for physcally plausble deformatons whch can be calculated quckly. However, as Müller et al. already menton, such volumetrc approaches are not applcable to pont clouds representng a surface. 1.2 Overvew Frst we present a bref explanaton of the mesh edtng framework of Pares et al. [PDK07] n Secton 2 as t consttutes the bass upon whch our work s bult. Our generalzaton to pont clouds follows, ncludng the concepts developed to use the mesh edtng framework wthout the need for meshng the nput data. In Secton 3 we explan our multresoluton method, whch allows edtng pont cloud data sets whch are consderably larger than comparable meshes. Although the amount of data ponts may be up to two scales of magntude hgher, our parallel GPU mplementaton stll provdes several frames per second durng edtng operatons. We conclude ths paper wth an overvew of the results (Secton 4) and our planned extensons for the future (Secton 5). 2 EDITING 2.1 Mesh Edtng As the underlyng model for a mesh Pares et al. [PDK07] employ a dfferental representaton of local surfaces. For each vertex n the regon of nterest x R they defne an orthonormal local frame F = (t 1,t2,N ) wth rght hand orentaton. F can be nterpreted as a rotaton matrx and thus be expressed n terms of a unt quaternon q. For each par of adjacent vertces (, j) n the mesh M the dfference between the quaternons s calculated as q j = q q j. Durng edtng and user nteracton the mesh surface can be reconstructed from the dfferental representaton and the equaton q q j = q j (1) whch s a lnear system. Fxng a sngle unt quaternon q 0 s suffcent for gettng a unque soluton (except for ts rotaton), by teratvely solvng the remanng equatons. Reconstructng the mesh solely based on the local frames and gnorng the geometry can lead to unntutve results, because translatons are not accounted for. To overcome ths, Pares et al. added constrants whch mpede a change of coordnates of 1-rng neghbors n the local frames F. They formulate the constrants as q (1,c j )t q = (1,x j x ) t (2) where c j := F 1 (x j x ) are the local coordnates of x j n frame F. User nput conssts of a set of handle vertces H whch specfy addtonal frame constrants q const k and postonal constrants xk const. Gven those two sets of constrants, the frame dfferences q j and the local coordnates c j, the constraned reconstructon problem s to fnd a set of local frames q l and absolute vertex coordnates x l for all vertces wthn the regon of nterest R that satsfy Equatons (1), (2) and q k = q const k x k = x const k k H. (3) As the resultng equaton system s overconstraned, Pares et al. solve t n least squares sense usng a nonlnear optmzaton procedure. They allevate the complexty ssue by partng the orgnal non-lnear problem nto several lnear equaton systems of whch the LU-matrces reman constant throughout one edtng step. Thus after specfyng a regon of nterest and edtng handles, a precomputaton step factorzes the orgnal matrces n parallel on the GPU. Durng the actual edtng the lnear equaton systems are solved on the CPU wth backsubsttuton, whch can be performed at several frames per second even for medum szed meshes wth up to 100k vertces. 2.2 Edtng Pont Clouds The ntenton to seek for a generalzaton of the edtng operatons from mesh data structures to arbtrary pont clouds, comes from the observaton that n practce raw data from range scannng devces often s avalable only n form of 3D vertex coordnates. Dependng on the devce, color nformaton may also be avalable, but t plays only a mnor role for edtng operatons.

4 Though advanced trangulaton technques lke MLS surface approxmaton exst, they are usually dependant upon a specfc structure of the pont cloud, e.g. a closed surface. Margns, outlers, or n general arbtrarly dstrbuted ponts pose severe problems. To be more robust, most methods have the undesred effect of smoothng the surface represented by the pont cloud, thus degradng the qualty of the data set. Furthermore, the computaton tme for the trangulaton becomes notably hgh for large data sets. Although the edtng framework of Pares et al. reles upon mesh data structures to ensure connectvty of the vertces and to fnd a vertex s 1-rng, the basc equaton systems are formulated wthout the need for an explct mesh. They do though requre the defnton of a local neghborhood n order to calculate the local frames. The smplest choce for a local neghborhood, whch comprses each vertex s nearest neghbors n 3D space, s suffcent for our needs and very fast to calculate. In our approach we use the k nearest neghbors of each vertex, whch are computed n a pre-processng step. To determne the nearest neghbors, we use an octree to partton the pont cloud, whch can be calculated n O(n log n) tme. The choce of k was expermentally determned, and may theoretcally be any k 2, snce we need at least two neghborng ponts to defne the local frame for x. In practce, choosng k too small, lke e.g. k = 2 leads to lne-shaped dsconnected components, whch are not treatable well as a surface by the later algorthm steps. On the other hand, choosng k too large lnearly slows down all calculatons whch terate the nearest neghbors. We found k = 5 or k = 6 to provde a good tradeoff between speed and stablty, and t s also the average number of neghbors n a regular trangulaton. Usng only the ntally found nearest neghbors for each vertex can lead to non-symmetrc relatons, especally f the nput pont cloud s rregularly sampled. Ths can prevent a complete traversaton of the graph nduced by the nearest neghbors, and splt t nto several seemngly dsconnected components. Although removng the non-symmetrc neghbors reduces the amount of data to process, t may lead to more fragmentaton. Our choce of nsertng the mssng lnks to create b-drectonal nearest neghbors relatons counteracts the possble fragmentaton. At the same tme t tes outlers to the man connected component(s). The problem wth auxlary connected components whch do not nclude user-defned constrants s that they render Equaton system (1)-(3) underdetermned, and thus provde no stable soluton. Durng the preprocessng step these unconstraned components are detected and removed from the edtable part of the model. 2.3 Area and Volume Preservaton Volume preservaton s a key feature used n mesh edtng and has also been appled to pont cloud edtng [MKN + 04]. It s however not unambguous how to defne ths property, snce a pont cloud does not possess an nherent volume, as opposed to a closed mesh. Ths s especally the case for pont clouds orgnatng from sngle-mage 3D capturng devces, whch can only capture one sde of any gven object at a tme. As our method strves to preserve the local neghborhood of each pont, the volume s only an affected property, not one whch s calculated or optmzed. In partcular, f a surface s stretched usng handles on opposte sdes, the surface s also enlarged n perpendcular drecton to preserve the local shape of each pont s 1-rng. Ths s a rather counterntutve behavour, snce most materals n nature exhbt the opposte behavour of shrnkng n perpendcular drecton when stretched n the other drecton (e.g. rubber and metal). Nevertheless, auxetc materals exst, whch do enlarge n perpendcular drecton (e.g. specal foams), and others whch do not change ther perpendcular sze at all (e.g. cork). Our method can be parameterzed to exert any one of these elastcty behavours by scalng the local frames q n each teraton before solvng the equaton systems, as detaled n [PDK07]. 3 SAMPLING The matrx factorzaton step performed durng precomputaton s necessary to reduce the calculaton tme durng the actual edtng of a model. We employed the SuperLU lbrary [DEG + 99] whch s suted well to decompose the sparse matrces. Although the matrx sze grows squarely wth the number of handle vertces, the sparsty leads to a computaton effort whch grows below quadratc. Nevertheless, from 100k vertces onwards ths requres several mnutes of precomputaton and eventually the matrx sze grows too large to be handled n man memory. Snce complex operatons on large data sets are today stll lmted n ther speed by the avalable computaton power, we devsed a multresoluton scheme smlar to Wcke et al. [WSG05]. The tme-consumng solvng of the Equaton systems (1)-(3) s performed on a coarse-scale subsample of the nput pont cloud, and the fne-scale representaton s then nterpolated. Pauly et al. [PGK02] survey dfferent pont based smplfcaton methods. For our subsamplng we chose a method from the category of herarchcal clusterng methods, snce they adapt well to pont clouds wth varyng pont samplng denstes. To generate the subsamples, we use an octree parttonng the orgnal

5 pont cloud. Takng the sample ponts from each leaf of the octree allows us to handle arbtrary pont clouds, not only those representng mplct surfaces. At the same tme, areas of the orgnal pont clouds whch have a hgh samplng densty wll receve a hgher amount of sample ponts, thus preservng local detals. Calculatng the nearest neghbors for the sample ponts can be performed wth the octree that was already calculated. For some models lke the dragon model ths can yeld undesred connectons between ponts (Fgure 2b), whch cannot be consdered adjacent n the orgnal, but become adjacent n the sampled pont cloud due to the lower pont densty. In the work presented by Xu et al. [XZY + 07] the user s requred to manually create a control mesh whch prevents such connectons between orgnally unconnected parts. Our connected sample ponts are n effect smlar to the control mesh used by Xu et al., but we sought for an alternatve workng wthout manual nterventon. The dea s to fnd the nearest neghbors n a geodesc sense nstead of the eucldean dstance. As the geodesc dstance conceptually requres a surface to be applcable, usng the nearest neghbors relaton of the orgnal pont cloud satsfes ths need, but stll makes t applcable to non-surface-lke pont clouds. Wth a breadth frst search (Algorthm 1) for each pont n the orgnal pont cloud, the geodesc nearest neghbors can be found n O(n r) tme, where n s the number of orgnal ponts and r s the orgnal-to-sample number rato. The result can be seen n Fgure 2c. After each teratve solvng step for the basc equatons, the new poston p and normal n of each orgnal pont s nterpolated from the translatons and rotatons of ts nearest sample ponts: p = s neghbors() w s (q s p s + p s ) (4) wth n = s neghbors() w s = 1 p s 2 / w s (q s n 0 ) (5) s neghbors() 1 (6) p s 2 where p s = p0 p0 s s the coordnate of pont p n the local coordnate system of p s, and q s s the quaternon descrbng the rotaton of s from ts orgnal to ts new orentaton. The weghts w s account for a smaller nfluence of neghbors whch have a greater dstance. We mplemented ths nterpolaton method on the CPU frst, but the nvolved quaternon multplcatons proved to be a lmtng factor to the achevable Fgure 2: a) Orgnal pont cloud wth sample ponts drawn n black. b) Sample ponts connected usng nearest neghbors. Wrong connectons between orgnally separated parts are created due to proxmty. c) Calculatng nearest neghbors on the orgnal pont cloud wth a breadth frst search prevents wrong connectons. Input: pont 0 ; number of sample neghbors to fnd k; Samples S; Orgnal nearest neghbors relaton N nt queue q 0 nt geodesc neghbors G( 0 ) = /0 repeat j = q.dequeue() for all n N( j) do f n not vsted then q.enqueue(n) mark n as vsted f j S then add j to G( 0 ) untl G( 0 ) k or q = /0 Algorthm 1: Geodesc nearest sample neghbors for a pont

6 Model # ponts # sample Subsamplng Precompu- Equaton Upsamplng ponts taton solvng Bunny 35k Bunny 35k Armadllo 172k Armadllo 172k Dragon 437k Dragon 437k Plan 1000k Table 1: Computaton tmes n seconds on a 2.4GHz CPU and a NVda GeForce 8800 GTX GPU. Subsamplng and precomputaton are executed once, whle equaton solvng and upsamplng occur each frame. speed, snce multplyng two quaternons ncludes 16 floatng pont multplcatons, and multplyng a quaternon wth a 3D vector ncludes 27 floatng pont multplcatons. The sheer amount of operatons requred to perform the nterpolatons for large data sets wth over 1M ponts does not allow for a true real-tme edtng experence usng ths approach, as the frame rates can drop to 1-2 frames per second. The mplementaton of the upsamplng step n CUDA for parallel computaton on the GPU s straghtforward, snce p and n can be computed ndependently for each pont on the avalable GPU processors. Our current mplementaton runnng on an off-the-shelf graphcs card (NVda GeForce 8800 GTX) performs on average 5-8 tmes faster than the CPU verson. Durng the edtng operaton, for 1M ponts we measured 8 frames per second for the GPU mplementaton, whle the CPU varant yelded only 1.4 frames per second. 4 RESULTS We tested our method on a varety of pont clouds, ncludng both artfcal and scanned models. All of the models have been used wthout manual preprocessng. Table 1 gves an overvew of the computaton tme for setup and each teraton step durng the edtng phase. Usually the model converges to ts fnal shape durng 3-5 teratons. As we have explaned before, the equaton solvng tme consumpton grows slower than O(n 2 ) where n s the number of sample ponts, whle the upsamplng step requres only lnear tme dependng on the number of ponts to be nterpolated. The vsual qualty of the result s largely ndependant of the number of sample ponts, unless t falls below a certan threshold dependng on the complexty of the model. Ths means e.g. for the Stanford Bunny (Fgure 3), whch does not have a complex structure, that the resultng pont postons have very small devatons when calculated wth 100, 1,000 or 10,000 sample ponts. Only wth fewer than about 100 sample ponts the upsamplng step produces artfacts for large deformatons (Fgure 4). For the Dragon model (Fgure 5), whch s more complex due to ts wndng body, samplng artfacts can be observed for less than 500 ponts (Fgure 4). Those artfacts could be crcumvented by a more sophstcated nterpolaton scheme whch does take nto account not only the nearest neghborng sample ponts, but a selecton of sample ponts whch are evenly dstrbuted on the surface around the pont to be nterpolated. A too low sample densty does however defeat the purpose of our nonlnear edtng method by reducng t to the lnear nterpolaton. One mnor drawback of our mplementaton usng quaternons s the nablty to perform handle rotatons of more than 360 wthn one edtng step. Ths s due to the normalzaton of quaternons whch we perform for stablty reasons when solvng the equaton systems. In practce however ths s not of concern, as t s possble to perform several edtng steps wth smaller rotatons to acheve the desred result. Fgure 3: Bunny model wth 2 edtng steps. Fgure 4: Bunny and Dragon models edted wth only 50 sample ponts each. Samplng artfacts can be observed n the head area of the Bunny and neck of the Dragon.

7 Fgure 5: Dragon model after rotaton and translaton of the head. Sesson tme ncludng subsamplng and precomputaton for each result: 1 mnute. 5 CONCLUSIONS AND FUTURE WORK We have presented a novel method to nteractvely edt large pont clouds. Usng a non-lnear deformaton model allows to produce physcally plausble deformatons even for large modfcatons. The multresoluton approach fathfully handles the coarse scale deformatons whle the model detals are preserved. Our parallel mplementaton provdes the speed necessary for real-tme edtng scenaros, shortenng the tme requred to produce a desred result. For models whch are larger than the avalable man memory, our method could be extended wth a further samplng level, for whch the upsamplng step would be performed offlne. Our method does not requre any manual precondtonng steps to create a control mesh, and s thus suted for drect edtng of arbtrary pont clouds. For the future we plan to ncorporate the deformaton features nto a model reconstructon framework whch can work on a number of scanner-acqured tme-varyng pont clouds. REFERENCES [AOW + 08] B. Adams, M. Ovsjankov, M. Wand, H. Sedel, and L. Gubas. Meshless modelng of deformable shapes and ther moton. In ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton, pages 77 86, Dubln, Ireland, ACM/Eurographcs, Eurographcs Assocaton. [BPGK06] M. Botsch, M. Pauly, M. Gross, and L. Kobbelt. Prmo: coupled prsms for ntutve surface modelng. In SGP 06: Proceedngs of the fourth Eurographcs symposum on Geometry processng, pages Eurographcs Assocaton, [BPWG07] M. Botsch, M. Pauly, M. Wcke, and M. Gross. Adaptve space deformatons based on rgd cells. Computer Graphcs Forum, 26(3): , [BSS07] T. Boubekeur, O. Sorkne, and C. Schlck. Smod: Makng freeform deformaton sze-nsenstve. In IEEE/Eurographcs Symposum on Pont-Based Graphcs 2007, September [DBP08] F. Duranleau, P. Beaudon, and P. Pouln. Multresoluton pont-set surfaces. In GI 08: Proceedngs of graphcs nterface 2008, pages , Toronto, Ont., Canada, Canada, Canadan Informaton Processng Socety. [DEG + 99] [HSL + 06] J. Demmel, S. Esenstat, J. Glbert, X. L, and J. Lu. A supernodal approach to sparse partal pvotng. SIAM J. Matrx Analyss and Applcatons, 20(3): , J. Huang, X. Sh, X. Lu, K. Zhou, L. We, S. Teng, H. Bao, B. Guo, and H. Shum. Subspace gradent doman mesh deformaton. ACM Trans. Graph., 25(3): , [MFXP08] Y. Mao, J. Feng, C. Xao, and Q. Peng. Hgh frequency geometrc detal manpulaton and edtng for pont-sampled surfaces. Vsual Computer, 24(2): , [MKN + 04] M. Müller, R. Keser, A. Nealen, M. Pauly, M. Gross, and M. Alexa. Pont based anmaton of elastc, plastc and meltng objects. In SCA 04: Proceedngs of the 2004 ACM SIGGRAPH/Eurographcs symposum on Computer anmaton, pages , Are-la-Vlle, Swtzerland, Swtzerland, Eurographcs Assocaton. [PDK07] [PGK02] [PKG06] [PKKG03] N. Pares, P. Degener, and R. Klen. Smple and effcent mesh edtng wth consstent local frames. Techncal Report CG , Unverstät Bonn, July M. Pauly, M. Gross, and L. Kobbelt. Effcent smplfcaton of pont-sampled surfaces. In VIS 02: Proceedngs of the conference on Vsualzaton 02, pages , Washngton, DC, USA, IEEE Computer Socety. M. Pauly, L. Kobbelt, and M. Gross. Pont-based multscale surface representaton. ACM Trans. Graph., 25(2): , M. Pauly, R. Keser, L. Kobbelt, and M. Gross. Shape modelng wth pont-sampled geometry. ACM Trans. Graph., 22(3): , [SA07] O. Sorkne and M. Alexa. As-rgd-as-possble surface modelng. In SGP 07: Proceedngs of the ffth Eurographcs symposum on Geometry processng, pages , Are-la-Vlle, Swtzerland, Swtzerland, Eurographcs Assocaton. [SB09] O. Sorkne and M. Botsch. Tutoral: Interactve shape modelng and deformaton. In Eurographcs, [WBB + 07] M. Wand, A. Berner, M. Bokeloh, A. Fleck, M. Hoffmann, P. Jenke, B. Maer, D. Staneker, and A. Schllng. Interactve edtng of large pont clouds. In Baoquan Chen, Matthas Zwcker, Maro Botsch, and Renato Pajarola, edtors, Symposum on Pont- Based Graphcs 2007 : Eurographcs / IEEE VGTC Symposum Proceedngs, pages 37 46, Prague, Czech Republk, Eurographcs Assocaton. [WSG05] [XZY + 07] M. Wcke, D. Stenemann, and M. Gross. Effcent anmaton of pont-sampled thn shells. Computer Graphcs Forum, 24(3): , W. Xu, K. Zhou, Y. Yu, Q. Tan, Q. Peng, and B. Guo. Gradent doman edtng of deformng mesh sequences. ACM Trans. Graph., 26(3):84, 2007.