Consolidation of Unorganized Point Clouds for Surface Reconstruction

Transcription

1 Consoldaton of Unorganzed Pont Clouds for Surface Reconstructon Hu Huang Dan L Unversty of Brtsh Columba Hao Zhang Ur Ascher Smon Fraser Unversty Danel Cohen-Or3 3 Tel-Avv Unversty Abstract We consoldate an unorganzed pont cloud wth nose, outlers, non-unformtes, and n partcular nterference between close-by surface sheets as a preprocess to surface generaton, focusng on relable normal estmaton. Our algorthm ncludes two new developments. Frst, a weghted locally optmal projecton operator produces a set of denosed, outler-free and evenly dstrbuted partcles over the orgnal dense pont cloud, so as to mprove the relablty of local PCA for ntal estmate of normals. Next, an teratve framework for robust normal estmaton s ntroduced, where a prorty-drven normal propagaton scheme based on a new prorty measure and an orentaton-aware PCA work complementarly and teratvely to consoldate partcle normals. The prorty settng s renforced wth front stoppng at thn surface features and normal flppng to enable robust handlng of the close-by surface sheet problem. We demonstrate how a pont cloud that s wellconsoldated by our method steers conventonal surface generaton schemes towards a proper nterpretaton of the nput data. Introducton Surface reconstructon from pont cloud data has been an extensvely studed problem n computer graphcs [Turk and Levoy 994; Carr et al. 00; Cazals and Gesen 006; Ohtake et al. 003; Kazhdan et al. 006]. Typcally acqured by a laser scanner, the raw nput ponts are often unorganzed, lackng nherent structure or orentaton nformaton. Orented normals at the ponts play a crtcal role n surface reconstructon, as they locally defne the reconstructed surface to frst order and dentfy the nsde/outsde and hence topology of the underlyng shape. Although photometrc stereo may be appled to estmate normals from captured mages, such estmates are not always relable due to less than deal acquston condtons such as specular reflectons, materal artfacts, and shadowng [Sun et al. 007]. Indeed, surface normal acquston s a delcate process [Ma et al. 007] requrng a well-controlled envronment and careful calbraton wth the process of pont acquston. We take as nput an unorganzed pont cloud whch may contan outlers, nose, and non-unformtes n thckness and spacng, due to acquston errors or msalgnment of multple scans. Based on pont postons alone, we consoldate [Alexa et al. 003] the pont cloud. Ths preprocessng phase for surface reconstructon ncludes denosng, outler removal, thnnng, orentaton, and redstrbuton of the nput ponts. Durng the process, we defer and avod any surface generaton, a phase that s hghly susceptble to varous data artfacts. Decouplng the two phases can effectvely avod premature and erroneous decsons n surface reconstructon. Photo. Raw scan. RBF reconstructons. Fgure : Data consoldaton, especally accurate normal estmaton, from a nosy, unorganzed, raw pont cloud s crucal to obtanng a correct surface reconstructon. The rght-most result s produced after applyng our pont cloud consoldaton scheme. A central task to pont consoldaton s normal estmaton. The classcal scheme for estmatng unsgned normal drectons s prncpal component analyss (PCA), whch can be unrelable due to thck pont cloud, non-unform dstrbuton, or close-by surfaces, as shown n Fgure. The most wdely appled approach to consstent normal orentaton [Hoppe et al. 99] s va normal propagaton, where propagaton between close-by ponts whose unsgned normal drectons make a small angle s gven prorty. However, under dffcult scenaros such as the presence of close-by surfaces, propagaton errors do occur, as shown n Fgure 3(a-b). Close-by surface sheets also challenge sharp feature detecton, a problem relevant to normal propagaton. As shown n Fgure 3(c), a thn surface feature,.e., a sharp feature delmtng close-by surfaces, does not admt a b-modal dstrbuton of unsgned normal drectons or a good ft usng multple surfaces. Thus, prevous approaches to sharp feature detecton, e.g., [Page et al. 00; Fleshman et al. 005], whle generally robust, are not desgned to handle such cases. We address the above ssues by combnng two technques. Frst, to make local PCA more robust, we denose, remove outlers, and down-sample the otherwse dense nput pont cloud to obtan a thnned and evenly dstrbuted set of ponts, called partcles [Pauly et al. 00] to dstngush from the nput ponts. In ths frst step, we modfy and extend the locally optmal projecton (LOP) operator of Lpman et al. [007] to deal wth non-unform dstrbutons common n raw data. We then estmate partcle normals va an teratve predctor-corrector scheme (see [Ascher and Petzold 998] for orgns and analoges for ths term). The predctor uses PCA to predct unsgned normal drectons. Ths s followed by partcle orentaton va a prorty-drven normal propagaton scheme. The obtaned partcle orentatons are then utlzed to correct or consoldate estmates of normal drectons va an orentaton-aware PCA, and the normal propagaton scheme s re-appled. A novel contrbuton n the orentaton scheme s a dstance measure whch prortzes normal propagaton and trggers proper normal flppng. The new measure combnes Eucldean and angular dstances wth propagaton drectons to robustly handle the close-by surface sheet problem. The teratve approach to normal estmaton by way of normal correcton s also new and shown to be effectve and necessary. We demonstrate that the result of our algorthm, a clean and unformly dstrbuted pont set endowed wth relable normals, leads

2 (a) (b) (c) (d) Fgure : Classcal PCA leads to naccurate estmates of unsgned normal drectons (black lnes) near a thck pont cloud (a), nonunform dstrbuton (b), or close-by surfaces (c-d). (a) (b) (c) Fgure 3: Close-by surfaces cause erroneous normal propagaton when prortzed only by Eucldean and angular dstances: (a) between partcles on opposte surfaces; (b) through a thn surface feature. (c): A thn surface feature does not admt a b-modal dstrbuton of normal drectons or a good ft usng multple surfaces. to qualty up-samplng and Delaunay-based surface reconstructon [Amenta et al. 00; Dey and Gesen 00]. Above all, t enables conventonal surface generaton schemes whch rely on pont normals, such as radal bass functon (RBF) [Carr et al. 00] and Posson [Kazhdan et al. 006] technques, to obtan a fne nterpretaton of the nput data n varous challengng stuatons. In Fgure, the two erroneous RBF reconstructons are obtaned from a raw scan after preprocessng by LOP (30 and 00 teratons, respectvely) and normal estmaton va classcal PCA and normal orentaton [Hoppe et al. 99]. The fnal mage shows RBF result from the same nput after applyng 00 teratons of our mproved LOP operator and then our normal estmaton scheme. Background and related work The lterature on surface reconstructon s vast. Delaunay technques [Cazals and Gesen 006] typcally produce a mesh whch nterpolates the nput ponts but contans rough geometry when the ponts are nosy. These methods often provde provable guarantees under prescrbed samplng crtera [Amenta and Bern 998] or nose models [Dey and Goswam 006] that are generally not realzable by real-world data. Addtonal assumptons, such as even pont dstrbuton, may also be requred n other pont set processng whch clam guarantees [Mtra et al. 004]. Approxmate reconstructon works mostly wth mplct surface representatons followed by so-surfacng. Most notable are methods whch carry out mplct modelng va tangent plane estmates [Hoppe et al. 99], RBF [Carr et al. 00], or Posson felds [Kazhdan et al. 006], all of whch requre orented normals. Accurate normal estmates, even pont dstrbuton, and suffcent samplng densty (va proper up-samplng) can all be acheved by pont consoldaton. Dervng a new pont set from a gven pont cloud has been consdered n the context of defnng pont set surfaces. Well-known defntons nclude movng least squares (MLS) [Alexa et al. 003] and extremal surfaces [Amenta and Kl 004]. In practce, these defntons can be appled to smooth or down-sample a raw pont cloud. There are also algorthms for pont cloud smoothng [Lange and Polther 005] and smplfcaton [Pauly et al. 00] guded by local geometry analyss, such as curvature estmaton. To better deal wth outlers and delcate surface structures, Lpman et al. [007] develop a hghly effectve, parameterzaton-free projecton operator (LOP). However, we have observed that LOP can fal to converge, oscllatng near a soluton nstead, and t may not work well when the dstrbuton of the nput ponts s hghly non-unform. Pont normals are essental for surface reconstructon as they provde local frst-order surface approxmatons and nsde/outsde drectons. Most normal estmaton schemes rely on PCA n some form [Hoppe et al. 99; Pauly et al. 00; Alexa et al. 003; Mtra et al. 004; Lange and Polther 005]. Classcal PCA reles on Eucldean dstances between ponts. More recently, drectonal nformaton has been taken nto account when computng an mproved centrod of a set of ponts n a local neghborhood [Amenta and Kl 004; Lehtnen et al. 008], replacng Eucldean dstances by Mahalanobs dstances. Nehab et al. [005] combne postonal and normal measurements va photometrc stereo to produce a surface whch conforms to both. As PCA normals are un-orented, computng a consstent normal orentaton requres addtonal work. Ths problem turns out to be surprsngly dffcult [Hoppe et al. 99; Mello et al. 003], and challenges due to close-by surfaces (Fgure 3) have not been specfcally addressed n prevous works. Asde from estmatng pont normals va purely geometrc means, acquston mechansms such as photometrc stereo [Woodham 980; Nehab et al. 005] are also possble but they are often subject to error caused by surface or llumnaton artfacts. If a scanner can return the outward drecton at each pont, a vector from the pont to the scanner head, such a drecton may be used to orent the normal towards the outsde by nsstng that the two drectons make an acute angle. However, naccurately estmated normal drectons va classcal PCA or near orthogonalty between the normal and outsde drectons can be sources for error, where the latter s lkely to occur near close-by surface sheets. We beleve that the normal orentaton problem s one whch requres a global consstency evaluaton and s not entrely solvable only through purely local consderatons. In practce, the outward drectons are stll not wdely avalable from current acquston devces and they can be a source of sgnfcant nose, especally for hand-held or other scanners whch contnually change head postons. 3 Improved weghted LOP (WLOP) The LOP operator [Lpman et al. 007] takes as nput a nosy pont cloud, possbly wth outlers, and outputs a new pont set whch more fathfully adheres to the underlyng shape. LOP operates well on raw data wthout relyng on a local parameterzaton of the ponts or on ther local orentaton. Gven an unorganzed set of ponts P = {p j} j J R 3, LOP defnes a set of projected ponts X = {} I R 3 by a fxed pont teraton where, gven the current terate X k, k = 0,,..., the next terate X k+ s to mnmze X X X p j θ( ξj ) k + λ η( x k )θ( δk ), I j J I\{} wth ξj k = x k p j and δ k = xk x k. In practce, n = I s often sgnfcantly smaller than m = J. Intutvely, LOP dstrbutes the ponts by approxmatng ther l medans to acheve robustness to outlers and data nose. Here s the -norm, θ(r) = e r /(h/4) s a rapdly decreasng smooth weght functon wth support radus h defnng the sze of the nfluence neghborhood, and η(r), the repulson term, s another decreasng functon penalzng ponts that get too close to other ponts n X. The balancng terms {λ } I vary at each pont but depend on just one parameter, 0 µ <.5, controllng the repulson force. Throughout our experments, we set µ = 0.45 and h may be adjusted, but the default value of h = 4 p d bb/m, where d bb s the dagonal length of the boundng box of the nput model, generally works very well. New repulson term LOP often works well, but we have found that the orgnal repulson functon η(r) = /(3r 3 ) may drop too quckly to guarantee suffcent penalty when r s large. Ths could

3 (a) η(r) = 3r 3 : σ = (b) η(r) = r: σ = Fgure 4: Partcle dstrbutons after LOP wth dfferent repulsons. For ths llustraton, all partcles are properly orented wth backface cullng. Vsually and from the σ measure, we see that the new repulson term η(r) = r produces a more regular dstrbuton. (a) Orgnal. (b) LOP: σ = 0.. (c) WLOP: σ = Fgure 6: WLOP vs. LOP: (a) The Lena mage s mapped onto a curved surface wth three holes to produce a pont set wth pont denstes proportonal to mage ntenstes. Then, randomly takng /0 of the ponts n (a) as ntal set, the results of LOP and WLOP projectons are shown n (b) and (c), respectvely. Non-unformty of the LOP result s manfested by traces of Lena n (b)..4 x 0 4 x η(r) = /(3r 3 ) η(r) = r (a) For the hand n Fgure LOP WLOP (b) For Lena n Fgure 6. Fgure 5: Plots of dstances X k+ X k /n between consecutve terates to llustrate convergence behavor. (a) Wthout densty weghts, the old repulson term leads to oscllaton near a soluton and the new term results n smooth convergence. (b) Wth densty weghts, WLOP apparently retans such convergence property. lead to a lack of clear-cut convergence and an undesrably rregular pont dstrbuton, especally when n m. To ths end, we propose to use the new repulson η(r) = r, whch decreases more gently and penalzes more at larger r, yeldng both better convergence and a more locally regular pont dstrbuton, as shown n Fgures 4(b) and 5(a). As a rough quanttatve regularty measure for pont dstrbutons, we use the varance of dstances to nearest neghbors at the ponts, whch we denote by σ throughout. Densty weghts The frst term n the optmzaton crtera above for LOP s closely related to the multvarate medan, also referred to as the l medan, whch leads to projecton ponts movng toward the local dstrbuton center. If the gven pont cloud s hghly non-unform, as n the example gven by Fgure 6(a), projecton by LOP tends to follow the trend of such non-unformty, no matter what ntal set X 0 we choose. Ths may be desrable n certan cases, e.g., to allow hgher pont denstes near shape features. In other cases, e.g., normal estmaton, one may prefer unform pont dstrbuton everywhere. To acheve ths, we propose to ncorporate locally adaptve densty weghts nto LOP, resultng n WLOP. Let us defne the weghted local denstes for each pont p j n P and n X durng the kth teraton by v j = + P j J\{j} θ( pj p j ) and w k = + P I\{} θ( δk ), k = 0,,.... Then the projecton for pont x k+ fnally becomes x k+ = X α k j p /vj j j J Pj J (αk j /vj) + µ X I\{} δ k w k β k P I\{} (wk β k ), where αj k = θ( ξk j ) and β k ξ j k = θ( δ k ) η ( δ k ). Thus, the δ k attracton of pont clusters n the gven set P s relaxed by the (a) Raw scan. (b) LOP (old η). (c) LOP (new η). (d) WLOP. Fgure 7: WLOP vs. LOP on the raw scan of a Japanese lady (a). The quanttatve measure of pont regularty takes on values: (b) σ = 0.4; (c) σ = 0.8; (d) σ = 0.09, ndcatve of mprovement. weghted local densty v n the frst term, and the repulson force from ponts n dense areas s strengthened by the weghted local densty w n the second term. LOP vs. WLOP Note that LOP wth the new repulson term s a specal case of WLOP by settng all densty weghts to. In ths case, t s possble to show contracton of the fxed pont teraton near an assumed soluton. For the more general WLOP wth adaptve weghts, emprcally, we have consstently obtaned error plots that are ndcatve of convergence; see Fgure 5(b) for an example. In addton to the synthetc Lena example, we also show mproved pont regularty provded by the new repulson and densty weghts on a raw scan example n Fgure 7. 4 Normal estmaton and consoldaton After WLOP, we obtan a thnned, outler-free, and unformly dstrbuted set of partcles, denoted by x,..., x n. For the next step, normal estmaton, we start wth the predctor step based on ntal unsgned normal drectons estmated va classcal weghted PCA [Pauly et al. 00]. The neghborhood sze h and weght functon θ for PCA are the same as those for WLOP. For subsequent corrector teratons, we employ an orentaton-aware PCA (Secton 4.) to consoldate the normals, where partcle orentatons are obtaned va normal propagaton (Secton 4.).

4 4. Normal propagaton We wsh to fnd an optmal assgnment of partcle orentatons to maxmze a certan consstency crteron. Hoppe et al. [99] use the sum of v, v j over all pars of partcles that are suffcently close to model consstency, where v and v j are the partcle normal drectons. We refer to t as the tradtonal propagaton scheme and adopt the same prorty-drven propagaton strategy whle ntroducng a new prorty measure for relable propagaton under problems such as those arsng from close-by surfaces. Overall scheme Frst, a source partcle s selected, where a relable normal drecton can be obtaned. Then we perform a conservatve check to dentfy certan partcles at thn surface features, where the advancng front of normal propagaton s forced to stop. Then the orentaton, startng from the source, s propagated, as permtted, va a prorty-drven traversal of the partcles. Specfcally, once a partcle s orented, ts k-nearest neghbors (knns), where k = 6 by default, are added nto a prorty queue. Potental orentaton errors may happen possbly due to the greedy approach or erroneous propagaton near undetected thn surface features. Thus an addtonal error check s performed whch may trgger one or more normal flps. Ths s followed by another propagaton pass, and these may be terated untl no more orentaton changes. Source selecton Typcally, an extremal partcle, e.g., one wth the maxmum x coordnate, s chosen [Hoppe et al. 99]. However, t s not unusual for such an extremal partcle to be at a sharp feature and cause erroneous results, e.g., see Fgure 3(b). We propose to pck a source over a flat regon a partcle whose unsgned drecton has the least angular varaton from those of ts knns. As the partcles have been denosed and evenly dstrbuted by WLOP, a desrable source can be relably found. The orentaton chosen at the source s less mportant, as a flppng of all the consstent normals, f deemed necessary, s smple to carry out at the end. Dstance measure Prevous consderatons for the dstance or prorty measure whch drves normal propagaton nclude both Eucldean and angular dstances but could stll lead to error, e.g., see Fgure 3(a). What has been mssng s the drecton of propagaton. The ntuton here s that a correct normal propagaton should less lkely be along the local normal drecton t should be along the local tangental drecton, whch, for two close-by partcles, we approxmate smply by the vector connectng the partcles. Let and be two partcles wth assocated drectons v and v j (ther orentatons do not play a role n the followng analyss), respectvely. Consder four ponts x, x, x j and x j that are unt dstance away from and along these drectons, as shown n Fgure 8. Let m rs be the mdpont of the lne segment x r xs j, r, s {, }, and o rs the perpendcular projecton of m rs onto the estmated tangent lne or ts extensons. Note that f and concdes ( s undefned), we smply let o rs =. We defne the normalzed dstance to prortze normal propagaton by D j = v, v j max r,s {,} m rs o rs. () + Note that D j [0, ]; t combnes Eucldean dstance (the denomnator), angular dstance v, v j, and a thrd term d j = max r,s {,} m rs o rs, whch s desgned to wegh n propagaton drecton. See Fgure 8 for an analyss of d j and D j and Fgure 9 for a comparson to the tradtonal scheme. We remark that our dstance D s not a metrc and nether should t be. From Fgure 3(b), we see that propagaton from the black (a) x m o d j (b) x (c) d j x d j x Fgure 8: Propagaton dstance D j () between partcles and wth unsgned normal drectons. (a) Maxmal projected dstance d j from mdpont m rs, r, s {, }, to captures propagaton drecton nformaton. (b) When normal drectons x x and x j x j concde wth xxj, sgnfyng a propagaton along normal drecton, we have d j = 0 and D j = at ts maxmum. (c) As normal drectons become more algned and perpendcular to, d j ncreases. It attans maxmum value when these condtons hold exactly, sgnfyng a propagaton along tangental drecton. Further, f and are concdent then D j s mnmzed at 0. Fgure 9: Steps of our normal propagaton scheme. (a) Raw data wth thn features. (b) Result of tradtonal scheme. (c) Wth new dstance measure and awareness of thn features (green), better results but errors stll reman. (d) Addng normal flppng fxes some errors (under the pvot). (e) Three correctve teratons wth orentaton-aware PCA lead to fnal successful orentaton. partcle to each blue partcle s encouraged (small D), but not between the blue partcles (large D) as they belong to opposte surfaces; such a dstance confguraton volates the trangle nequalty. Fnally, takng drecton nformaton nto account when measurng dstances s not new. The Mahalanobs dstance s defned between a pont and an orented pont wth a stretch factor characterzng the ellptcal feld around the orented pont. Our dstance avods such a free parameter and s an ntegrated measure defned on two unsgned drectons assocated wth partcles. Thn surface features and normal flppng Although D can by and large avod propagaton between partcles resdng on close-by and opposte surfaces, t does not prevent propagaton through a thn surface feature, one whch separates two such surfaces, as shown n Fgure 3(b). We desgn a smple and conservatve method to detect such features. The knns of partcle are projected onto ts tangent plane, whch s determned by the current unsgned normal at. If the projecton of les outsde the convex hull of ts knn projectons, then s deemed to be at a thn feature; see Fgure 0(a). Partcles at a thn surface feature can be orented, but are not allowed to propagate ther orentatons. Note that the above detecton mechansm s only specalzed to handle thn surface features: t s not a generc sharp feature detector. Moreover, t cannot dstngush between a flat neghborhood and a thn feature whose crease s a concave curve; see Fg-

5 (a) (b) (c) Fgure 0: Partcle projecton (red) lyng outsde the convex hull of projecton of knns mples a thn surface feature (a). Ths test does not dstngush between a flat neghborhood (c) and a case where the partcle les on a concave crease curve (b). Trangles are used only to ad vsualzaton; they are not part of the data. ure 0(b-c). As a remedy, we execute a check durng the normal propagaton to detect and reverse orentaton between close-by surface sheets. Specfcally, for a propagated partcle par and, f both cos( (n, x )) and cos( (n j, x )) exceed a threshold (set to 0.8 throughout), sgnfyng a potental propagaton along normal drecton, we flp the normal orentaton at partcle. Then, the prorty-drven propagaton contnues. Fgure : Effect of pont consoldaton on up-samplng. (a) Nosy nput data wth 84,398 ponts. (b) Result of up-samplng, to 95,863 ponts, from a down-sampled (,84 ponts or 3.3%) pont set obtaned from (a), after data cleanng by LOP and normal estmaton va classcal PCA. (c) Up-samplng to 9,438 ponts after the same down-sampled pont set s consoldated usng our algorthm. 4. Orentaton-aware PCA The normal propagaton scheme descrbed above works on a fxed set of unsgned normal drectons. Despte all the care taken so far, orentaton errors may occasonally persst due to error n the normal drectons computed by the classcal, orentaton-oblvous PCA. By makng PCA orentaton-aware, unsgned normal drectons and orentaton estmatons can complement each other and fx errors wthn a corrector loop. In our mplementaton, when performng local weghted PCA at a partcle, we exclude from a Eucldean h-ball centered at those partcles whose orented normals are opposte (negatve dot product) to the normal at. In other words, the consdered neghbors are now all those facng the same way as. The unsgned normals recomputed n ths way would not be expected to change much on flat, correctly orented regons, but they may well vary and become more accurate near thn structures or areas of surface nterference. Thus, the errors n subsequent orentaton sweeps va normal propagaton may be reduced. We apply such corrector teratons untl normal orentatons no longer change. Fgures 9 and 5 show orentaton errors corrected va orentaton-aware PCA. 5 Results and applcatons Pont cloud consoldaton cleans up raw nput, removes a varety of data artfacts, and provdes essental geometrc attrbutes, n our case pont normals, to facltate subsequent processng. In ths secton, we demonstrate how a well consoldated pont set va WLOP and normal estmaton usng our teratve framework can beneft such processng tasks as up-samplng and surface reconstructon. Vsualzaton of pont sets s best acheved usng splattng, based on ponts wth normals or surfels. A frequently encountered operaton durng splattng s pont cloud up-samplng, e.g., for a zoomed-n vew or when the gven pont cloud was under-sampled durng data acquston or subsampled for effcent processng. We employ the fast dynamc algorthm of Guenebaud et al. [004] for real-tme pont cloud refnement n our experment. Accurate normals and regular pont dstrbutons are typcal requrements to acheve qualty for such up-samplng, and Fgure shows the knd of postve dfference pont consoldaton can make. Let us now show the necessty of a dscplned pont consoldaton step for qualty surface reconstructon. Nosy nput must frst be Fgure : Effect of WLOP on RBF surface reconstructon. Raw scan (a) of an Inukshuk s cleaned by the orgnal LOP (b) wth the pont regularty measure takng on value σ = and our WLOP (d) wth σ = Observe the unformty of the resultng partcle dstrbutons. After normal consoldaton and upsamplng, RBF constructons, (c) from (b) and (e) from (d), show qualtatve dfference n hole fllng, e.g., around the neck. cleaned before surface generaton, snce mperfect pont dstrbuton or orentaton, whch occurs wth exstng schemes, can result n vsble reconstructon error, as frst shown n Fgure. We manly draw comparsons wth the use of classcal PCA and the tradtonal normal propagaton scheme due to Hoppe et al. [99]. In addton, we also provde an example comparng our consoldaton framework wth a Delaunay-based one: NormFet+AMLS. These steps, combned wth the well-known Cocone mesh generaton [Dey and Gesen 00], are a seres of technques developed by Dey and co-authors. In partcular, NormFet [Dey and Gesen 005] performs normal estmaton n the presence of nose usng the Delaunay ball technque and AMLS [Dey and Sun 005] employs adaptve MLS for smoothng nosy pont clouds based on normals and detected features from NormFet. In other cases, we choose RBF [Carr et al. 00] and Posson [Kazhdan et al. 006] surface reconstructon for demonstraton. The mplementatons are due to FarFeld Technology (FastRBF) and M. Kazhdan, respectvely. Frst, we show the effect of WLOP n Fgure. The nput s a raw scan wth mssng data. After WLOP, up-samplng and robust normal consoldaton, RBF s able to successfully close holes and construct a qualty surface. In contrast, wth the orgnal LOP operator, although nose and outlers are removed as well, the resultng rregular partcle dstrbuton (quantfed by σ) may cause some defects on hole closure durng surface generaton. Next, we show the effect of normal orentaton, where all the nput pont clouds were frst cleaned va WLOP and then subsequently up-sampled. For a Delaunay-based approach, we consder the (NormFet, AMLS, Cocone) combnaton. Our experments show that errors arsng from the tradtonal scheme or Delaunay-based

6 Orgnal. RBF. RBF. Orgnal. Posson. Posson. Fgure 3: Effect of normal consoldaton on surface reconstructon for models wth thn structures. In each seres, followng the orgnal, we show reconstructon results after normals are computed va the tradtonal scheme, and then results after our normal estmaton algorthm. Hghlghted areas show dfferences made by the latter. Fgure 5: Effect of normal consoldaton, n partcular, the corrector teraton and orentaton-aware PCA. (a) Raw scan. (b) RBF result after normal estmaton va the tradtonal scheme. (c) RBF result based on normals orented by one pass of our propagaton scheme. (d) RBF result after further correcton of orentaton errors va teraton and orentaton-aware PCA. Table : CPU runtme for consoldaton of several raw datasets. O-No: number of orgnal ponts; P-No: number of projected partcles; W-T: tme for WLOP; N-T: tme for normal estmaton; U-T: tme for up-samplng. Only the Face model n Fgure s upsampled twce. All examples were run on an Intel Pentum 4, 3. GHz CPU wth GB RAM and tmes are reported n seconds. Fgure Fgure Fgure Fgure 4 Fgure 5 (a) (b) (c) O-No 634,386 84,398 06,00 04,068 6,0 P-No,47 8,440 0,30 0,407 3,05 W-T N-T U-T (d) Fgure 4: Effect of normal consoldaton on surface reconstructon for a raw pont cloud (a) wth close-by surfaces and mssng data. (b) Result from (NormFet, AMLS, Cocone). (c) RBF result after normal estmaton va the tradtonal scheme. (d) RBF result after our normal estmaton scheme. approach may lead to varous topologcal artfacts n the reconstructons. Such errors typcally occur near thn surface structures (Fgure 3) or close-by surface sheets (Fgures 4 and 5), where our pont cloud consoldaton method succeeds. In partcular, Fgures 4 and show that accurate normals can effectvely compensate for mssng data n a pont cloud, allowng reconstructons, such as RBF, to nfer the underlyng shape correctly. Fnally, we provde tmng results for our algorthm n Table. Although the strengths of our method le n ts handlng of thn surface structures, falure cases can stll occur n cases under extreme condtons such as severe nose or undersamplng. For example, n Fgure 6, we see that the ears of the horse are thn structures havng extremely low samplng rate. Our algorthm treats each ear as a sngle sheet and the resultng reconstructon has obvous defects. Another such example leadng to topologcal error can be observed between the feet of the mannequn n Fgure 5. Ideally, we would lke to obtan a theoretcal guarantee for the correctness of our normal estmates under approprate Lmtatons Fgure 6: A falure case n the presence of extreme undersamplng. (a) A horse pont set consoldated usng our algorthm; note severe undersamplng near the ears. (b) Back face cullng vew. (c) Front face cullng vew. (d) Posson surface reconstructon. samplng condtons. Also on the theory front, we do not have a convergence proof for the teratve predctor-corrector scheme for normal estmaton. In practce, we have not encountered a case of oscllaton ether. Lke LOP, our pont consoldaton framework does not address the mssng data problem. However, numerous examples hghlght the mportance of havng accurate normals for surface completon schemes such as RBF and Posson to succeed. 6 Concluson and future work Accurate estmaton of normals s crucal to obtanng a correct nterpretaton of the nput data. We show that the ncorporaton of propagaton drecton nformaton nto prorty settng, as well as a coupled and teratve approach on normal orentaton and orentaton-aware PCA, provdes consoldaton of the data ponts

7 n varous dffcult settngs. The prelude to all these s a necessary step for data clean-up, for whch we develop WLOP, an mproved locally optmal projector wth weghtng opton for denosng and outler removal from mperfect pont data and producng an evenly dstrbuted set of partcles whch fathfully adheres to the captured shape. Wth our pont cloud consoldaton, conventonal surface reconstructon schemes can better nfer the topology and geometry of the shape from raw nput data n challengng stuatons. We beleve that such consoldaton of ponts should be a routne procedure appled to raw data smlarly to common denosng procedures. Whle our current consoldaton algorthm has been shown to perform robustly and effcently through numerous experments, we next would lke to seek a rgorous theoretcal analyss of the predctor-corrector teraton. Also possble as future work s better handlng of mssng data, takng advantage of the relable orentaton nformaton we can extract from the raw nput. Fnally, we would lke to ncorporate recovery and enhancement of sharp features nto our pont consoldaton framework. Acknowledgments The authors would lke to thank all the revewers for ther valuable comments. Ths work s supported n part by grants from NSERC (No and No. 6370), the Israel Mnstry of Scence, and the Israel Scence Foundaton. The hands, horse, dancer, and scssors data are from the AIM@SHAPE shape repostory. The face model n Fgure s courtesy of Yaron Lpman. Our code s based on the VCG lbrary from the Vsual Computng Lab n Psa, Italy. Thanks go to Federco Poncho for the orgnal LOP mplementaton and consultaton on VCG. References ALEXA, M., BEHR, J., COHEN-OR, D., FLEISHMAN, S., LEVIN, D., AND SILVA, C. T Computng and renderng pont set surfaces. IEEE Trans. Vs. & Comp. Graphcs 9,, 3 5. AMENTA, N., AND BERN, M. W Surface reconstructon by Vorono flterng. In Symp. on Comp. Geom., AMENTA, N., AND KIL, Y. J Defnng pont-set surfaces. ACM Trans. on Graphcs 3, 3, AMENTA, N., CHOI, S., AND KOLLURI, R. K. 00. The power crust. In ACM Symp. on Sold Modelng and Appl., ASCHER, U., AND PETZOLD, L Computer Methods for Ordnary Dfferental Equatons and Dfferental-Algebrac Equatons. SIAM, Phladelpha, PA. CARR, J. C., BEATSON, R. K., CHERRIE, J. B., MITCHELL, T. J., FRIGHT, W. R., MCCALLUM, B. C., AND EVANS, T. R. 00. Reconstructon and representaton of 3D objects wth radal bass functons. In Proc. of ACM SIGGRAPH, CAZALS, F., AND GIESEN, J Delaunay trangulaton based surface reconstructon. In Effectve Computatonal Geometry for Curves and Surfaces. Sprnger, DEY, T. K., AND GIESEN, J. 00. Detectng undersamplng n surface reconstructon. In Symp. on Comp. Geom., DEY, T. K., AND GIESEN, J Normal estmaton for pont clouds: a comparson study for a Vorono based method. In Eurographcs Symp. on Pont-Based Graphcs, DEY, T. K., AND GOSWAMI, S Provable surface reconstructon from nosy samples. Comp. Geom.: Theory & Appl. 35,, 4 4. DEY, T. K., AND SUN, J An adaptve MLS surface for reconstructon wth guarantees. In Symp. on Geom. Proc. (SGP), FLEISHMAN, S., COHEN-OR, D., AND SILVA, C. T Robust movng least-squares fttng wth sharp features. ACM Trans. Graph. 4, 3, GUENNEBAUD, G., BARTHE, L., AND PAULIN, M Realtme pont cloud refnement. In Eurographcs Symp. on Pont- Based Graphcs, HOPPE, H., DEROSE, T., DUCHAMP, T., MCDONALD, J., AND STUETZLE, W. 99. Surface reconstructon from unorganzed ponts. In Proc. of ACM SIGGRAPH, KAZHDAN, M., BOLITHO, M., AND HOPPE, H Posson surface reconstructon. In Symp. on Geom. Proc. (SGP), LANGE, C., AND POLTHIER, K Ansotropc smoothng of pont sets. Comput. Aded Geom. Des., 7, LEHTINEN, J., ZWICKER, M., TURQUIN, E., KONTKANEN, J., DURAND, F., SILLION, F., AND AILA, T A meshless herarchcal representaton for lght transport. ACM Trans. on Graphcs 7, 3, 37: 37:9. LIPMAN, Y., COHEN-OR, D., LEVIN, D., AND TAL-EZER, H Parameterzaton-free projecton for geometry reconstructon. ACM Trans. on Graphcs 6, 3, : :6. MA, W.-C., HAWKINS, T., PEERS, P., CHABERT, C.-F., WEISS, M., AND DEBEVEC, P Rapd acquston of specular and dffuse normal maps from polarzed sphercal gradent llumnaton. In Eurographcs Symp. on Renderng, MELLO, V., VELHO, L., AND TAUBIN, G Estmatng the n/out functon of a surface represented by ponts. In ACM Symp. on Sold Modelng and Appl., MITRA, N. J., NGUYEN, A., AND GUIBAS, L Estmatng surface normals n nosy pont cloud data. Int. J. Comput. Geom. and Appl. 4, NEHAB, D., RUSINKIEWICZ, S., DAVIS, J., AND RAMAMOOR- THI, R Effcently combnng postons and normals for precse 3D geometry. ACM Trans. on Graphcs 4, 3, OHTAKE, Y., BELYAEV, A., ALEXA, M., TURK, G., AND SEI- DEL, H.-P Mult-level partton of unty mplcts. ACM Trans. on Graphcs, 3, PAGE, D. L., SUN, Y., KOSCHAN, A., PAIK, J., AND ABIDI, M. A. 00. Normal vector votng: Crease detecton and curvature estmaton on large nosy meshes. Graphcal Models 64, PAULY, M., GROSS, M., AND KOBBELT, L. P. 00. Effcent smplfcaton of pont-sampled surfaces. In Proc. of IEEE Vsualzaton, SUN, J., SMITH, M., SMITH, L., AND FAROOQ, A Examnng the uncertanty of the recovered surface normal n three lght photometrc stereo. Image Vs. Comput. 5, 7, TURK, G., AND LEVOY, M Zppered polygon meshes from range mages. In Proc. of ACM SIGGRAPH, WOODHAM, R. J Photometrc method for determnng surface orentaton from multple mages. Optcal Engneerng 9,,