Shape controllable geometry completion for point cloud models

Transcription

1 Noname manuscrpt No. (wll be nserted by the edtor) Shape controllable geometry completon for pont cloud models Long Yang 1, 2 Qngan Yan 1 Chunxa Xao 1 * Receved: date / Accepted: date Abstract Geometry completon s an mportant operaton for generatng a complete model. In ths paper, we present a novel geometry completon algorthm for pont cloud models, whch s capable of fllng holes on ether smooth models or surfaces wth sharp features. Our method s bult on the physcal dffuson pattern. We frst decompose each pass hole-boundary contracton nto two steps, namely normal propagaton and poston samplng. Then the normal dssmlarty constrant s ncorporated nto these two steps to fll holes wth sharp features. Our algorthm mplements these two steps alternately and termnates untl generatng no new hole-boundary. Expermental results demonstrate ts feasblty and valdty of recoverng the potental geometry shapes. Keywords pont cloud model geometry completon sharp features normal propagaton poston samplng 1 Introducton Beneftng from ts smple representaton, pont cloud model has been wdely used n the last two decades [5, 13, 21]. Although capture devces have been mproved substantally, the scanned data stll contan defcent holes n certan stuatons. Moreover, we often confront abraded surfaces and damaged models wth dfferent defcences. All these holes need to be completed approprately. Many technques have been proposed to deal wth ths ll-posed problem. The exstng methods, such as [3, 9, 15, Long Yang, Qngan Yan, Chunxa Xao(Correspondng author) E-mal:{yanglong,yanqngan,cxxao}@whu.edu.cn 1 Computer school, Wuhan Unversty, Chna 2 College of nformaton engneerng, Northwest A&F Unversty, Chna 23, 24, 28, 33], are desgned to fll holes for polygonal mesh models. Most of these methods defne geometry completon operators by utlzng connecton topology and generate robust hole-fllng results. More detals about mesh completon please refer to the surveys of [2, 6, 18]. Fllng holes drectly on pont cloud models currently turns to be an essental requrement for many practcal applcatons. Early work [8] presents an overall ppelne of geometry completon for pont cloud models. Although many technques [7, 19, 20, 25, 29, 30, 32] work on pont cloud models, most of them only generate smooth holefllng results. In contrast to smooth hole-fllng, sometmes t makes specal sense to complete a hole by preservng sharp features (.e. edge / corner / apex) or usng the least materals, see fgure 1(e) and 1(b). Snce general smooth hole-fllng methods cannot complete the protrudng features plausbly, t s stll a dffcult task to recover the potental sharp features on a defcent pont cloud surface. Our work s nspred by the observaton that geometry completon should reasonably provde potental shape optons for defcent regons. Therefore, n ths paper, we propose a novel hole-fllng approach for pont cloud models. Our algorthm smulates energy dffuson process and progressvely contracts a hole-boundary untl the hole s closed. Unlke the exstng boundary propagatng method [10], t could control the propagatng process effectvely. The man contrbutons of ths paper are as follows: Presentng a unfed geometry completon algorthm that recovers both smooth and feature preserved holes for pont cloud models. Sharp features are reproduced by controllng the hole-boundary contractng process. Developng a new poston samplng operator based on elastc force to generate the fllng ponts. It avods local

2 Long Yang et al. (a) a defcent sphere (b) r=0.4 BBL (c) r=0.25, r=r/1.1 (d) r=0.11 BBL (e) r=0.05 BBL Fg. 1 Our shape controllable geometry completon algorthm recovers a large defcent sphere (a) wth dfferent neghborhood raduses r. (b), (d) and (e) use constant sze of r whle (c) employs a set of decreasng values of r. All these results use the dentcal elastc force parameter σ r =0.22 BBL (the boundng box dagonal length of the nput model) and are generated wthout the normal dssmlarty constrant. and global reconstructons so that ponts on non-hole regons keep unchanged. Before elaboratng our algorthm, we brefly revew the related technques of geometry completon and feature reconstructon for pont cloud models n next secton. 2 Related Work Many surface reconstructon methods, whether global or local fttng of the scattered pont cloud data, could fll holes to a certan extent. Carr et al. [7] employ radal bass functon (RBF) to construct a global sgned dstance feld (SDF) for orgnal pont set. It could fll the defcent holes and generate smooth surface. The MPU method [25] uses pecewse quadratc fttngs to construct the feature preserved SDF. Kazhdan et al. [20] present a Posson reconstructon method whch uses pecewse constant ndcator gradents to construct potental surface for the nput pont cloud. A new enhanced verson s Screened Posson reconstructon [19]. It could reconstruct feature preserved surface fathfully even f the nput model contans small defcences. The Volfll method [10] smulates the heat dffuson process to propagate SDF from known parts to the adjacent hole-regon. Once the dffuson completed, a hole s flled. Ths method could fll holes wth complex shapes due to ts local shape propagaton n a progressve way. However, smply propagated SDF cannot descrbe the sharp turn of a surface exactly. It also needs the support of local space subdvson. Weyrch et al. [30] employ movng least square (MLS) projecton to replace the dstance feld estmaton n [10]. Snce lackng effectve constrants for the dffuson process, t stll cannot fathfully recover the holes wth large defcency or holes wth extreme feature, especally for the sharp (edge / corner) regons. Another type of hole-fllng technque, example-based geometry completon, recovers defcent regons by fndng smlar patches from ether the nput model [14, 27, 29, 31] or other models belongng to the same category [22]. Example-based hole-fllng method suts the hole whose boundary regon has smlar shape on the known model. Hence, t does not effectvely handle a hole whose shape cannot be learned from non-hole regons. In order to produce sharp features, Fleshman et al. [12] segment a pont model nto pecewse smooth regons based on a robust statstcs. Öztrel et al. [26] present a RIMLS method. It combnes the robust local kernel regresson wth the mplct MLS to descrbe sharp features. Recently, Huang et al. present an edge-aware resamplng (EAR) method [16] for pont cloud models. It generates sharp features by progressvely resamplng a surface to approach ts feature edges. Although these methods could generate appealng sharp features, they need suffcent samples near or on the feature regons. Therefore, they cannot fll holes wth large data defcency. To recover the mssed features, state-of-the-art technques [15, 24, 32] resort to nteractve method. [15] and [24] are desgned for meshes and adopt the strategy whch frst recovers the feature curve under user nteractons then flls the dvded smooth sub-holes. Morft [32] can recover some complex surfaces by nteractvely manpulatng the curve skeleton and profle curve of the nput pont cloud model. Through feature edtng, t can reconstruct sharp edges. Morft requres ntal skeleton and apples to the generalzed cylnder objects whose topology can be descrbed by curve skeleton. Dfferng from these technques, our method recovers sharp features n an automatc hole-fllng process. 3 Formulaton of our geometry completon algorthm Our geometry completon algorthm takes advantage of the local propagaton propertes [10]. To repar sharp features, t goes a step further to decompose the boundary contractng 2

3 Shape controllable geometry completon for pont cloud models process nto two prmary steps: normal propagaton and poston samplng. The former controls orentatons of flled ponts as well as the shape of recovered surface, whle the latter practcally generates a new boundary for one pass hole-boundary contracton. These two steps are mplemented alternately durng the hole-fllng process so that they could beneft from each other. Ths decomposton gves shape-controllng chance to our algorthm. We ncorporate the normal dssmlarty constrant nto both steps of normal propagaton and poston samplng for recoverng sharp features n hole-regons. 3.1 Algorthm overvew Gven the orented pont cloud our geometry completon algorthm frst mplements a preprocessng to denose the nput pont cloud surface. After determnng a hole-boundary, t contracts the hole-regon by propagatng ts boundary teratvely untl no new boundary s generated. The man procedure of our method can be concsely nterpreted as algorthm 1. Algorthm 1 Shape controllable geometry completon for pont cloud models. Input: a pont cloud model wth normals; Output: the completed pont cloud model. 1. nput a model P (has a hole Ω ) wth normalsn; 2. preprocessng: N = f 1 (N) and P= f 2 (P) ( 3.2); 3. determne the hole-boundary Ω ( 3.3); 4. Do / teratvely hole-boundary contracton. / 5. propagate the normals of ponts near Ω ( 3.4); 6. If generated new boundary Ω ( 4.1) 7. generate B by samplng new boundary Ω ( 4.2); 8. Ω B; 9. EndIf 10. Untl no new boundary Ω s generated 11. End Fg. 2 Dfferent hole-flled results, from left to rght correspondng to fgure 1(b), 1(c) and 1(e) respectvely, are guded by dstnct processes of normal propagaton. 3.3 Determnng hole-boundary Although there are some hole-boundary detectng operators [1, 4, 8] for pont cloud model, they are not robust enough for the hole near a sharp regon. In ths paper, for a hole Ω on the fltered pont set P R 3, we present a dvde and conquer approach to determne ts boundary Ω effectvely. It frst selects a small number of feature ponts f ( = 1,,m f and m f s the number of the selected feature ponts) sequentally along the boundary of a specfed hole so that each boundary segment between f and f +1 approxmates a local lnearty. We nsert these feature ponts nto a boundary sequence B. For a specfed segment f f +1, the pont b m on the holeboundary correspondng to the mddle pont M of f f +1 s determned by choosng the nearest neghbor from nput pont cloud to M. If the chosen b m s a new boundary pont (has not entered B ), we nsert t nto B between f and f +1. Wth the chosen b m, our algorthm recursvely mplements the same operaton on segments f b m and b m f +1 respectvely. Ths process termnates when no new boundary pont correspondng to the mddle pont of each new segment s found. We teratvely mplement the above recursve operatons for all segments. Fnally, the constructed pont sequence B P composes the dscrete representaton of ntal holeboundary Ω (see an example n the accompanyng vdeo). 3.2 Preprocessng Our algorthm takes as nput a set of ponts P R 3 and ther normals N R 3. To fnd fathful hole-boundary, t frst mplements a two-stage flterng preprocessng for the nput pont cloud model. Specfcally, we enforce blateral flterng [11] on both orentatons and postons of the nput ponts respectvely. For a certan pont p wth ts normal n ( = 1,,m p and m p s the number of nput ponts), the fltered normal and poston can be expressed as n = f 1 (n ) and p = f 2 (p ) correspondngly. We denote the fltered normals and pont cloud as N and P respectvely. 3.4 Normal propagaton In order to compute the contracted boundary Ω, our algorthm needs to construct a normal feld for those samplng ponts on the new boundary n advance. We assgn a new pont sequence B as the dscrete representaton of Ω. The normal n of a canddate fllng pont b s calculated by the weghted sum of normals from ts local neghbors N r (b ): n = 1 K(b ) n p g 1 ( p b )g 2 ( n p n ), (1) p N r (b ) 3

4 Long Yang et al. K(b )= g 1 ( p b )g 2 ( n p n ), (2) p N r (b ) where r s the neghborhood radus of b, g 1 s the Gaussan dstance weght between dfferent ponts wth standard devaton σ d and g 2 s the Gaussan normal dssmlarty weght wth standard devaton σ n. Equaton (1) resembles blateral normal flter [17, 26] formally. It manly dffers n the purpose that we ntend to nfer the unknown normal for a canddate poston rather than flter a known normal. In fact, we cannot offer a reference normal n to compute the dfference for g 2 between a neghborng normal n p and the normal of canddate b. Instead, we take the normal n, correspondng to a pont b (on the former hole-boundary B ) whose poston s most close to the canddate poston b, as the reference normal n equaton (1). Note that we use the normal dssmlarty weght g 2 to constran our normal propagaton process and further to control the hole-fllng shape. If a canddate b locates near a sharp feature, the neghborng normals on the other sde wll hold large values of normal dssmlarty and contrbute less to n whle the neghborng normals on the same sde contrbute more. Therefore, the recovered hole-boundary could preserve sharp features. Moreover, σ n s an adjustable parameter n our normal propagaton process. A large value leads to smooth orentaton whle a small value results n normal propagaton wth orentaton preservaton. Ths constraned normal propagaton combnng wth progressvely boundary contracton contrbutes to the shape controllable capablty of our algorthm (see fgure 2). 3.5 Poston samplng Guded by the propagated normal, poston samplng for one pass boundary contracton should concern two objectves. Frst, the new generated boundary must match ts surroundng surface. Second, the new flled ponts should hold a reasonable dstrbuton. The latter requres a sequental and practcal contracton of the hole-boundary n one pass teraton and guarantees overall decrease of the hole-regon. We defne the dscrepancy value of a flled pont b, denoted as E 1 (b ), to measure the matchng degree wth ts neghborng surface. The smaller value E 1 has, the better matchng degree b gets. Meanwhle our algorthm ntroduces elastc force to control the dstrbuton of new flled ponts. A canddate pont b s deemed to be a good samplng only f t locates n the equlbrum poston and receves the mnmum force, denoted as E 2 (b ), from ts neghbors on the former and the current boundares. Combnng these two objectves of E 1 and E 2 (both wll be defned specfcally n secton 4), we formulate our poston samplng of one pass boundary contracton as a mnmzng problem of objectve functon (3), E(B)=argmn B b B {E 1 (b )+E 2 (b )}, (3) where B, formed by the latest one pass flled ponts, represents the dscrete new hole-boundary. The soluton of equaton (3) should mnmze the total elastc forces of the flled ponts and the dscrepancy between new generated hole-boundary B and the exstng surface. 4 Generatng new hole-boundary Although we have bult objectve functon for hole-boundary contractng process, the optmal new boundary curve mght not exst for equaton (3). It s because there are countless samplng patterns and the trval soluton makes the objectve functon mnmum. Moreover, determnng a new boundary n hole-regon s also an underdetermned problem snce we do not have suffcent condtons to constran our samplng operaton. Hence the ntenton to solve equaton (3) precsely s unadvsable. Instead, we resort to an approxmate strategy to address ths samplng problem. We propose a new ndrect samplng operator, also ncludng two sequental operatons both based on elastc force, to construct a new hole-boundary B approxmately. It frst computes a control curve C for the new hole-boundary B accordng to those samples on the former pass boundary B. Then under the constrant of the control curve C, one pass poston samplng on a 2D manfold s reduced to a lnear 1D samplng along C. The ntroduced control curve restrcts new samplng ponts n a lmted band and makes samplng problem well-posed. Thus, the new sampled boundary B offers a sound approxmate soluton for objectve functon (3). The control curve C derved from the former pass boundary B should respect the local shape of the exstng surface. We optmze the poston of each control pont relyng on both ts local neghbors and the new normal feld (defned n secton 3.4). Thereafter, we sample along control curve C and mplement poston optmzaton for each sampled pont as well. From these sampled ponts our algorthm constructs the next pass boundary control curve f t does not reach convergence. 4.1 Constructng hole-boundary control curve Defnton of elastc force: Our algorthm leverages Gaussan functon to smulate elastc force and control the dstance of a samplng pont from ts neghbors. Gven a new canddate control pont c, as shown n fgure 3, t 4

5 Shape controllable geometry completon for pont cloud models represents the equlbrum poston O b a former pass boundary pont b drecton. O b wth respect to along ts propagatng receves elastc forces from ts neghborng ponts on the former pass hole-boundary. We defne the elastc force from a neghbor q as (( ) ) r q (O b )=1.0 exp O b q O q q /σr 2, (4) here O q s the equlbrum poston of neghbor q along the drecton of vector qo b. Actually, once the neghbor q s gven, the elastc force receved by O b be determned by the dstance from O q to O b from q can along the drecton of repulsve force. Note that, for a system of elastc force, we assgn the repulsve drecton as the postve drecton and the attractve one as the negatve drecton. Therefore, the defnton of r q (O b ) can be smplfed as equaton (5): r q (O b )=1.0 exp( O b where A l the drecton l. q 1 ) / O q qo σ 2 r, (5) b denotes the sgned dstance of vector A along r O ) 0 b ( b ( c)o b O q1 r ( O q ) 0 1 b r ( O q ) 0 2 b O q2 exp b O b b 1 b 2 b O b b 3 / σ 2 r =10 4. (6) O b1 O b2 O b3 r b O b ) 0 2 ( 3 O b4 O b5 Fg. 4 A 2D llustraton of the equlbrum postons O b, O 1 b 2 for dfferent boundary ponts b 1, b 2, b 3, b 4 and b 5 O b 4 and O b 5 b 4 b 5, O b, 3 along ther contractng drectons respectvely. The control ponts (red crcles) are those equlbrum postons whch receve no repulsve forces from any other adjacent boundary ponts. Fgure 4 shows an example and llustrates the equlbrum postons derved from dfferent boundary ponts along ther contractng drectons accordng to equaton (6). Boundary control curve: In our algorthm, the new boundary control curve s constructed from those equlbrum postons correspondng to the former pass hole-boundary ponts. For a specfed boundary pont b, our algorthm computes a vector, whch s the cross product from the normal of b to the orentaton of boundary curve. We take ths vector as the local contractng drecton of boundary curve B (shown n fgure 5). We search the locaton of c dependng on equaton (6) from the former pass boundary pont b B b q 2 B Fg. 3 Elastc forces of a canddate poston O b receved from ts two dfferent neghbors q 1 and q 2. O b s the equlbrum poston of b along ts contractng drecton. b c B c σ r s another adjustable parameter. Its value can refer to the parameter σ d (n equaton (1)). In our algorthm, σ r determnes the samplng densty for a hole-regon. Specfcally, t controls the equlbrum poston for a gven pont along the specfed drecton. For example, n fgure 3, the equlbrum poston O b can be computed by solvng the followng equaton (see Appendx A), Normals of the former pass boundary ponts Drecton of boundary curve b contractng drecton Propagated normal drecton forc C Fg. 5 Generatng the control ponts and then samplng along the new control curve. Red crcles denote control ponts whle small purple crcles are samplng ponts. 5

6 Long Yang et al. along ts contractng drecton. The calculated canddate of control pont c does not necessarly match well the boundary shape. We use neghborng ponts on the exstng model to optmze ts poston accordng to the dscrepancy defnton n equaton (7). The optmzed control pont c wll be dscarded f t receves any repulsve forces from other boundary ponts. Fnally, by jonng all the reserved control ponts consecutvely, we obtan the new boundary control curve. Examples are shown by red crcles n fgure 4 and fgure Samplng along boundary control curve We ntalze an empty fllng sequence B and push the frst control pont c 0 nto t as the frst samplng pont b 0. Then we teratvely compute the next new samplng pont b +1 (.e. the equlbrum poston of b along control curve C) startng from b 1 untl each control pont has been traversed. Our algorthm also mplements poston optmzaton for each b to match the shape of local surface. The normals of new sampled ponts are computed followng equaton (1). Once our method gets a new boundary samplng pont set B, as the sequence of small purple crcles shown n fgure 5, t fnshed one pass hole-boundary contracton. By executng the man loop between the 4th lne and the 10th lne n algorthm 1, the hole-fllng procedure wll converge f t generates no new control ponts after all ponts on the former boundary B have been traversed. In practce, f the rato between the number of generated control ponts and the number of boundary ponts n B s below a certan threshold, t means the boundary does no longer contract notceably. In our experments, we termnate our algorthm when the rato s lower than 30%. 4.3 Poston optmzaton In order to match local surface shape, a poston canddate (ether a control canddate or a samplng canddate, for the sake of clarty we just explan the samplng canddate) has to be optmzed accordng to ts local neghbors. For a new canddate b, we defne ts dscrepancy E 1 (b ) as the sum of weghted offsets wth respect to ts neghborng ponts along the normal drecton of b. Specfcally, the dscrepancy of b s measured by the total local offsets: Therefore, the optmzed poston can be obtaned by updatng canddate b as: b = b + n b o f f sets(b ). (8) Note that the normal n b probably does not match b after mplementng ths poston optmzaton. The recomputed normal n b wll also cause that the updated b s not the best matchng poston wth respect to the new normal any more. Theoretcally, poston optmzaton s an teratve process and wll be converged fnally when the poston and the normal stop updatng. In practce, the convergence wll be reached quckly. We mplement two teratons of normal updatng and poston optmzaton wthout trggerng notceable artfacts n our experments. Equatons (7) and (8) ndcate that local normals and the normal dssmlarty constrant beneft poston samplng, especally samplng near a sharp regon. In turn, the refned poston samplng combnng wth normal dssmlarty constrant mproves normal propagaton fathfully n feature regon, as explaned n equaton (1). Consequently, normal dssmlarty constrant and the mutual enhancement between normal propagaton and poston samplng enable our method to fll holes wth sharp features. control ponts samplng ponts b 2 overstepped ponts potental surface b 1 b 1 b b 1 b 1 Fg. 6 Combnaton constrant for recoverng sharp features. The samplng canddates overshootng the local potental surface (as b +1 and b +1 ) wll have at least one postve offset value correspondng to a neghborng control pont and wll be rejected by our algorthm. o f f sets(b )= 1 K(b ) g 1 g 2 p b nb, (7) p N r (b ) here, n b denotes the normal of canddate b, g 1 and g 2 are the dstance weght and the normal dssmlarty weght respectvely, K(b ) s defned as the same n equaton (2). Fg. 7 Overshootng samples occur (rght) when we fll the defcent corner of a cube (left) wthout the combnaton constrant. 6

7 Shape controllable geometry completon for pont cloud models Fg. 8 The frst three rows are the geometry completon results of a defcent cube, an ncomplete pyramd and a destroyed fandsk models. The results from the 2nd to the 5th columns correspond to Screened Posson reconstructon, Volfll, MPU and our methods respectvely. The last column s the round surface generated by our approach. The fourth row s a defcent heart shape surface. It s completed by Screened Posson reconstructon, Volfll, MPU and our algorthm successvely. The rghtmost s the top vew of our result. samplng canddate, should always be a negatve value or zero wth respect to the normal drecton of ths control pont. 4.4 Feature constrant To complete a surface contanng sharp features, our poston samplng (stated n secton 4.2) may cause local overshootng samples. An example s shown n the rght of fgure 7. To elmnate ths phenomenon, we ntroduce a samplng constrant for the sharp feature s completon. In fgure 6, the offsets of the canddate sample b+1 contan a postve value. Actually, t overshoots the horzontal potental surface and should be dscarded. Another samplng canddate b+1, whch contans postve offset and overshoots the vertcal surface, s also rejected. In contrast, for a concave boundary such offsets of a canddate should always be non-negatve values. Thus, once a negatve offset appears, the samplng canddate must have overshot the local concave surface and wll be dscarded by our algorthm. Specfcally, durng the boundary contractng process, a few control ponts close to a sharp regon may overshoot the local surface, as the topmost and rghtmost control ponts shown n fgure 6. Our method holds these control ponts for keepng the chance of samplng near sharp regon (as sample b ). However, t mght gve rse to the overshootng samples as well (canddate b+1 and b+1 ). These overshootng samples wll trgger the dvergence of our algorthm. We utlze a combnaton condton to check these overshootng samples. By employng the combnaton constrant, our algorthm elmnates the overshootng samples durng the poston samplng process. The 5th column n fgure 8 exhbts the sharp results of geometry completon under the combnaton constrant. For the control curve locatng n a convex regon, all sampled ponts should le ether on the boundary or n the nner of the polygon f no overshootng occurs. Ths combnaton constrant means that the offset, startng from the tangent plane of a neghborng control pont to the 7

8 Long Yang et al. 5 Results and dscusson We test our geometry completon algorthm on both synthetc and scanned surfaces to explore ts capablty. The dfferent completed results of a defcent sphere n fgure 1 show the shape control capablty of our algorthm. We use the neghborhood radus r of normal propagaton to control the hole-boundary contracton. More neghborng ponts wll be nvolved so that the orentaton of the holeboundary converges quckly f a large r s assgned. Only a few neghbors wll partcpate n the normal propagaton f we set a small r, and the orentaton of hole-boundary wll strctly respect local shape of the exstng model. By takng dfferent r, our algorthm generates 4 dstnct shapes, as shown from fgure 1(b) to 1(e). Fgure 1(e) demonstrates a recovered cone shape wth a small r (0.05) multpled by a default value, the boundng box dagonal length of the nput model. We denote ths value as BBL. We compare our algorthm wth three representatve technques of Volfll [10], MPU [25] and Screened Posson reconstructon [19]. The results produced by three exstng technques on sx defcent pont cloud models are shown from fgure 8 to fgure 10. Note that these three technques reconstructed the nput models and generated mesh results. Our method flls a hole by contractng ts boundary so that non-hole regons keep unchanged. For comparson we dsplay these results n pont cloud pattern. Screened Posson reconstructon method flls all holes robustly but generates smooth results. Volfll algorthm generates feature preserved results for cube, pyramd, fandsk and dhedral models to a certan extent. But t stll fals to recover sharp features. MPU algorthm generates sharp apex for pyramd and recovers the sharp edge for dhedral. But t flls the non-hole regon and expands the boundares of these two open models (see the bottom of pyramd from Fg. 10 Completng a nosed Planck model. Fgures from top-left to bottom-mddle are the nput nosed model, hole-flled results produced by Screened Posson reconstructon, Volfll, MPU and our algorthm respectvely. The last fgure s the ground truth model. a sde vew n fgure 8 and the left part of dhedral n fgure 9). In contrast, due to rgorous constrant of normal dssmlarty (see the small σn n table 1), our method could strctly control the boundary propagaton to recover the sharp features for these models. The results are shown n fgure 8 and fgure 9. Besdes completng sharp features, our method could recover round surfaces by loosenng normal dssmlarty constrant. The completed results on cube, pyramd and fandsk models are shown n the last column of fgure 8. For the heart model wth a large hole, comparng wth Volfll [10], our algorthm can effectvely control the boundary contracton by slghtly decreasng the normal dssmlarty parameter σn. It recovered a desrable surface wth contnuous curvature change, see comparson of the 3rd and the 5th results n the last row of fgure 8. Fg. 9 Preprocessng and completng a dhedral model. Fgures from top-left to bottom-rght are the nput defcent model, denosed dhedral and hole-flled results generated by Screened Posson reconstructon, Volfll, MPU and our method respectvely. Fg. 11 A destroyed sculpture (left) s completed by our method. The detal of the recovered regon s shown n a close-up vew (rght). 8

9 Shape controllable geometry completon for pont cloud models Table 1 Our experments settngs of the core parameters and the statstc data for most hole-fllng cases. Sup means loosenng the normal dssmlarty constrant. Fg.8(1-5) denotes the 5th fgure of the 1st row n fgure 8. Model Fgure r σ n σ r Org. pont Hole-bound. Iteratve Flled pont (BBL) (BBL) (BBL) (num.) Pont(num.) tmes (num.) Sphere Fg.1(c) 0.25; r=r/1.1 Sup Sphere Fg.1(e) 0.05 Sup Cube Fg.8(1-5) Pyramd Fg.8(2-5) Dhedral Fg.9(2-3) Planck nose Fg Sculpture Fg Sup Prnter Fg Hand (11 holes) Fg Sweepng surface (crcle) Fg Sup For the real scanned models (fgure 11, 12 and 13) and the nose contamnated models (fgure 9 and 10), our method mplements an ansotropc flterng preprocessng (secton 3.2) to get a relatvely neat model. We detect a holeboundary on the denosed model and then mplement the geometry completon. Fgure 10 s the Planck model wth the destroyed nose. Snce we want to generate the straght nose brdge and the round nose tp, we separate ths hole-boundary nto two parts. The straght nose brdge s frst generated wth a relatvely small σ n, as lsted n table 1. Fnally we fll the round nose tp wth a lttle bt bgger σ n, shown n fgure 10. A scanned sculpture model, n fgure 11, contans a large defcency whch s composed of two connected holes. Durng the boundary contractng process, our method marks the encountered boundary parts as the non-updatable boundary control ponts and skps poston samplng n these regons (see the accompanyng vdeo). The trajectores of boundary propagaton demonstrate that the combnaton of control curve and elastc force fulflls our poston samplng approprately. Fgure 12 shows a scanned prnter model wth coarse nput normals. Our algorthm s not senstve to the accuracy of ntal normals. The defcent corner (contanng both concave and convex features) s recovered by our method. A scanned hand wth many holes s dsplayed n fgure 13. Too close dstance between adjacent fngers leads to mutual nterference when Screened Posson reconstructon s mplemented. Our method avods ths nfluence by takng the constrant of normal dssmlarty. We fll the complex hole-regon usng pecewse boundary contracton. Fgure 13 shows our repared hand from dfferent vewponts. (More detals see the accompanyng vdeo.) Our method could natvely treat the smooth holes. We loosen the normal dssmlarty constrant to complete a defcent horse and a sweepng surface wth open boundary. The results are shown n fgure 14 and fgure 15 respectvely. There are three parameters needng to be assgned for our algorthm, ncludng the neghborhood radus r, the parameter of normal dssmlarty σ n and the elastc force parameter σ r. A relatve small σ n, correspondng to a strct normal dssmlarty constrant, s needed for fllng holes around sharp regons. σ r has drect proporton relatonshp wth the repulsve force. Large repulsve force means less samplng ponts whle small repulsve force produces more (a) (b) (c) (d) Fg. 12 Our method handles a scanned prnter (left) and generates the hole-flled result. The close-up vew s also gven (rght). Fg. 13 A scanned hand (a) s frst denosed (b) and then completed by Screened Possson reconstructon method (c). (d) shows the repared results of our method from dfferent vewponts. 9

10 Long Yang et al. Table 2 Evaluaton of the results generated by dfferent methods on fve models. Scr. Vol. MPU Our cube pyramd dhedral Planck fandsk max ave max ave max ave max ave Fg. 14 Completng a horse model. Fgures from top-left to bottomrght are nput model, recovered results wth dfferent elastc force parameters σ r correspondng to 0.14 BBL, 0.15 BBL and 0.16 BBL respectvely. samplng ponts (see fgure 14). The values of these parameters for most examples are gven n table 1. Snce the hghlght of our method s reparng geometry feature, we quanttatvely evaluate our results n terms of recoverng sharp features. Fve models (cube, pyramd, dhedral, Planck and fandsk) are chosen. We normalze each model nto a unt cube and calculate the errors for all ponts on each recovered model. The closest dstance from a pont on a repared model to the Ground Truth surface s taken as the error measurement. Three syntheszed complete surfaces (cube, pyramd and dhedral wth sharp features) and two orgnal models (unbroken Planck and fandsk) are taken as the Ground Truth. Table 2 reports the maxmum and average errors for all results generated by dfferent methods. Note that the errors occurred on the flled non-hole regons, ncludng the bottom of pyramd (Volfll and MPU), the left part of dhedral (MPU) and the bottom of Planck model (Screened Posson, Volfll and MPU), were excluded. In table 2, our algorthm has the least values on both maxmum and average errors for each model. Fg. 15 Recoverng a sweepng surface wth three dfferent holes by usng our method. The left s the nput defcent sweepng surface. The mddle and the rght are our completed results from dfferent vewponts. Fgure 16 shows the colored errors for fve models completed by four methods. Some obvous errors along sharp edges were found on results produced by Screened Posson and Volfll methods. The exstng methods faled to complete sharp corners on several cases. Our method recovered fathful sharp features for these defcent edges and corners. The recovered trajectores on both pyramd and dhedral models show that our algorthm performs each step wth a low reparng error. Lmtatons: The lmtatons of our method manly exst n three aspects. Frst, t needs to select a few feature ponts on a hole-boundary for generatng the ntal boundary. Thus t s a sem-automatc approach. Second, just dependng on the constraned local propagaton our method wll fal f two thrds of a sphere has been cut. For ths knd of hghly llposed case whch has more than half shape mssed, more pror normal varatons should be ntegrated n our normal propagaton to generate the desred result. The last one s that our method currently focuses on shape recovery and does not treat the lost geometry detals of a hole f t contans the hgh-frequency features. 6 Conclusons We devsed a shape controllable geometry completon algorthm for pont cloud models. It provdes potental shape optons for those hole-regons that probably contan sharp features. Our method nherts the merts of local propagaton pattern. It augments the capablty of recoverng sharp features by ncorporatng normal dssmlarty constrant nto the decomposed normal propagaton and poston samplng operatons. By defnng the elastc force and ntroducng the boundary control curve, our method has approprately addressed the samplng problem for pont cloud hole-fllng. Those flled ponts shown n our experments exhbt a reasonable dstrbuton on the hole-regons. The completed pont cloud model wll practcally beneft 3D surface reconstructon and many follow-up applcatons. Wth our hole-flled results, two sharp feature preserved reconstructon methods of EAR [16] and RIMLS [26] generated ntrgung results, see the reconstructed surfaces n fgure

11 Shape controllable geometry completon for pont cloud models Fg. 16 Error plots of the quanttatve evaluaton. From top to bottom, cube, pyramd, dhedral, Planck and fandsk. From left to rght, Screened Posson, Volfll, MPU and our method. Applcatons, Proceedngs, IEEE, pp Attene M, Campen M, Kobbelt L (2013) Polygon mesh reparng: An applcaton perspectve. ACM Computng Surveys (CSUR) 45(2):15 3. Bac A, Tran NV, Danel M (2008) A multstep approach to restoraton of locally undersampled meshes. In: Advances n Geometrc Modelng and Processng, Sprnger, pp Bendels GH, Schnabel R, Klen R (2006) Detectng holes n pont set surfaces. Journal of WSCG In the future, we would lke to develop an automatc hole-boundary recognton technque to enhance our geometry completon approach. Another natural thought of the followng work s to explore the detal recoverng method for those defcent pont cloud surfaces contanng hghfrequency geometry features. References 1. Adamson A, Alexa M (2004) Approxmatng bounded, nonorentable surfaces from ponts. In: Shape Modelng 11

12 Long Yang et al. Fg. 17 Our approach benefts surface reconstructon. We mplement two state-of-the-art algorthms (EAR [16] and RIMLS [26]) for sharp features reconstructon on four defcent pont cloud models. For each group, the mddle and the rght fgures of the frst row show the results of EAR and RIMLS methods drectly workng on the orgnal pont model respectvely. The left fgure of the second row s our hole-flled result. The mddle and the rght fgures of the second row are the correspondng results of EAR and RIMLS methods based on our result. 5. Berger M, Taglasacch A, Seversky LM, Allez P, Levne JA, Sharf A, Slva CT (2014) State of the art n surface reconstructon from pont clouds. In: Eurographcs 2014-State of the Art Reports, The Eurographcs Assocaton, pp Campen M, Attene M, Kobbelt L (2012) A practcal gude to polygon mesh reparng. Proceedngs of the 2012 Eurographcs, Caglar, Italy, pp t4 7. Carr JC, Beatson RK, Cherre JB, Mtchell TJ, Frght WR, McCallum BC, Evans TR (2001) Reconstructon and representaton of 3d objects wth radal bass functons. In: Proceedngs of the 28th annual conference on Computer graphcs and nteractve technques, ACM, pp Chalmovanskỳ P, Jüttler B (2003) Fllng holes n pont clouds. In: Mathematcs of Surfaces, Sprnger, pp Chen CY, Cheng KY (2008) A sharpness-dependent flter for recoverng sharp features n repared 3d mesh models. Vsualzaton and Computer Graphcs, IEEE Transactons on 14(1): Davs J, Marschner SR, Garr M, Levoy M (2002) Fllng holes n complex surfaces usng volumetrc dffuson. In: 3D Data Processng Vsualzaton and Transmsson, Proceedngs. Frst Internatonal Symposum on, IEEE, pp Fleshman S, Dror I, Cohen-Or D (2003) Blateral mesh denosng. In: ACM Transactons on Graphcs (TOG), ACM, vol 22, pp Fleshman S, Cohen-Or D, Slva CT (2005) Robust movng least-squares fttng wth sharp features. In: ACM Transactons on Graphcs (TOG), ACM, vol 24, pp Gross M, Pfster H (2011) Pont-based graphcs. Morgan Kaufmann 14. Harary G, Tal A, Grnspun E (2014) Context-based coherent surface completon. ACM Transactons on Graphcs (TOG) 33(1):5 15. Harary G, Tal A, Grnspun E (2014) Feature-preservng surface completon usng four ponts. In: Computer Graphcs Forum, Wley Onlne Lbrary, vol 33, pp Huang H, Wu S, Gong M, Cohen-Or D, Ascher U, Zhang HR (2013) Edge-aware pont set resamplng. ACM Transactons on Graphcs (TOG) 32(1):9 17. Jones TR, Durand F, Desbrun M (2003) Non-teratve, feature-preservng mesh smoothng. In: ACM Transactons on Graphcs (TOG), ACM, vol 22, pp Ju T (2009) Fxng geometrc errors on polygonal models: a survey. Journal of Computer Scence and Technology 24(1): Kazhdan M, Hoppe H (2013) Screened posson surface reconstructon. ACM Transactons on Graphcs (TOG) 12

13 Shape controllable geometry completon for pont cloud models 32(3): Kazhdan M, Boltho M, Hoppe H (2006) Posson surface reconstructon. In: Proceedngs of the fourth Eurographcs symposum on Geometry processng 21. Kobbelt L, Botsch M (2004) A survey of pontbased technques n computer graphcs. Computers & Graphcs 28(6): Kraevoy V, Sheffer A (2005) Template-based mesh completon. In: Symposum on Geometry Processng, Cteseer, pp Lévy B (2003) Dual doman extrapolaton. ACM Transactons on Graphcs (TOG) 22(3): Ngo HTM, Lee WS (2012) Feature-frst hole fllng strategy for 3d meshes. In: VISIGRAPP, pp Ohtake Y, Belyaev A, Alexa M, Turk G, Sedel HP (2005) Mult-level partton of unty mplcts. In: ACM SIGGRAPH 2005 Courses, ACM, p Öztrel AC, Guennebaud G, Gross M (2009) Feature preservng pont set surfaces based on non-lnear kernel regresson. In: Computer Graphcs Forum, Wley Onlne Lbrary, vol 28, pp Park S, Guo X, Shn H, Qn H (2005) Shape and appearance repar for ncomplete pont surfaces. In: Computer Vson, ICCV Tenth IEEE Internatonal Conference on, IEEE, vol 2, pp Pernot JP, Moraru G, Véron P (2006) Fllng holes n meshes usng a mechancal model to smulate the curvature varaton mnmzaton. Computers & Graphcs 30(6): Sharf A, Alexa M, Cohen-Or D (2004) Context-based surface completon. In: ACM Transactons on Graphcs (TOG), ACM, vol 23, pp Weyrch T, Pauly M, Keser R, Henzle S, Scandella S, Gross M (2004) Post-processng of scanned 3d surface data. In: Proceedngs of the Frst Eurographcs conference on Pont-Based Graphcs, Eurographcs Assocaton, pp Xao C, Zheng W, Mao Y, Zhao Y, Peng Q (2007) A unfed method for appearance and geometry completon of pont set surfaces. The Vsual Computer 23(6): Yn K, Huang H, Zhang H, Gong M, Cohen-Or D, Chen B (2014) Morft: Interactve surface reconstructon from ncomplete pont clouds wth curve-drven topology and geometry control. ACM TRANSACTIONS ON GRAPHICS 33(6) 33. Zhao W, Gao S, Ln H (2007) A robust hole-fllng algorthm for trangular mesh. The Vsual Computer 23(12): A Computng the equlbrum poston. To compute the equlbrum poston O b for a former pass holeboundary pont b, as shown n fgure 3, we ntroduce a pont q 0 whose poston just exactly locates n O b, as depcted n fgure 18. The equlbrum poston of q 0 on the drecton of vector O b b must have the same poston wth pont b, that s to say, O q 0 concdes wth the poston of b. Takng the elastc force receved by O q 0 from b nto account, ts value should be the postve maxmum (equals 1, correspondng to the maxmum repulsve force) accordng to the defnton of the elastc force n equaton (5). The overlap postons can be seen as the extremely close dstance between O q0 and b. Wthout loss of generalty we assgn ths repulsve force along the vector b O b. Therefore, we have r b (O q0 ) = 1, specfcally 1.0 exp Oq0 O b /σr 2 =1. By substtutng O q0 wth b we have exp b O b b b O b b O b /σ 2 r =0. Our purpose s to compute the equlbrum poston O b for pont. So we need to take the logarthm for the above equaton. However, the rght sde of ths equaton equals zero whch cannot be taken logarthm operaton mmedately. For the sake of numercal computng, we use a small constant 10 4 to approxmately nstead zero and make our computaton feasble. Fnally, we can use the followng equaton to compute O b exp b O b for b f the parameter σ r s assgned, b O b /σ 2 r =10 4. O b b r ( O b q 0 ) 1 Fg. 18 Computng the equlbrum poston O b for a pont b. q 0 O q0 13