Similarity-based denoising of point-sampled surfaces *
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1 Wang et al. / J Zhejang Unv Sc A 008 9(6): Journal of Zhejang Unversty SCIENCE A ISSN X (Prnt); ISSN (Onlne) E-mal: jzus@zju.edu.cn Smlarty-based denosng of pont-sampled surfaces * Ren-fang WANG,, Wen-zh CHEN, San-yuan ZHANG, Yn ZHANG, Xu-z YE ( School of Computer Scence and Technology, Zhejang Unversty, Hangzhou 3007, Chna) ( Faculty of Computer Scence and Informaton Technology, Zhejang Wanl Unversty, Nngbo 3500, Chna) E-mal: wrfwpln@6.com; syzhang@cs.zju.edu.cn Receved Sept., 007; revson accepted Mar. 8, 008 Abstract: A non-local denosng (NLD) algorthm for pont-sampled surfaces (PSSs) s presented based on smlartes, ncludng geometry ntensty and features of sample ponts. By usng the trlateral flterng operator, the dfferental sgnal of each sample pont s determned and called geometry ntensty. Based on covarance analyss, a regular grd of geometry ntensty of a sample pont s constructed, and the geometry-ntensty smlarty of two ponts s measured accordng to ther grds. Based on mean shft clusterng, the PSSs are clustered n terms of the local geometry-features smlarty. The smoothed geometry ntensty,.e., offset dstance, of the sample pont s estmated accordng to the two smlartes. Usng the resultng ntensty, the nose component from PSSs s fnally removed by adjustng the poston of each sample pont along ts own normal drecton. Expermental results demonstrate that the algorthm s robust and can produce a more accurate denosng result whle havng better feature preservaton. Key words: Pont-sampled surfaces (PSSs), Smlarty, Geometry ntensty, Geometry feature, Non-local flterng do:0.63/jzus.a07465 Document code: A CLC number: TP39.7 INTRODUCTION Pont-sampled models wthout topologcal connectvty are normally generated by samplng the boundary surface of physcal 3D objects wth 3D-scannng devces. Despte the steady mprovement n scannng accuracy, undesrable nose s nevtably ntroduced from varous sources such as local measurements and algorthmc errors. Thus, nosy models need to be denosed or smoothed before performng any subsequent geometry processng such as smplfcaton, reconstructon and parameterzaton. It remans a challenge to remove the nevtable nose whle preservng the underlyng surface features n computer graphcs. In partcular, fne features are often lost f no specal treatment s provded. Correspondng author * Project supported by the H-Tech Research and Development Program (863) of Chna (Nos. 007AA0Z3 and 007AA04ZA5), and the Research Fund for the Doctoral Program of Hgher Educaton of Chna (No ) In recent years, a varety of pont-based denosng and smoothng approaches have been ntroduced, and they can be roughly categorzed nto the followng four groups: () Spectral technques. The technques of spectral flters n mage settng were generalzed to pont-based surfaces, for example, Pauly and Gross (00) created a spectral decomposton of a pont cloud and denosed t by manpulaton of the spectral coeffcents. () Interpolaton or approxmaton approaches. The raw pont clouds are nterpolated or approxmated wth smooth surfaces such as extremal surfaces (Amenta and Kl, 004), mplct surfaces (Carr et al., 00; Samozno et al., 006; Danels II et al., 007), movng least-squares (MLS) surfaces (Mederos et al., 003; Weyrch et al., 004; Dey and Sun, 005; Lpman et al., 006; Danels II et al., 007) and local parametrc surfaces (Pauly et al., 003), etc. (3) Statstcal technques. Based on robust statstcs, the smoothng technques for pont-sampled surfaces (PSSs) were ntroduced. Pauly et al.(004)
2 808 Wang et al. / J Zhejang Unv Sc A 008 9(6): proposed a framework for analyzng shape uncertanty and varablty n pont-sampled geometry based on statstcal data analyss, whch can be appled to reconstruct surfaces n the presence of nose. Schall et al.(005) developed a method for robust flterng of a gven nosy pont set usng a mean shft based clusterng procedure. Postons on a smooth surface were found by movng every sample to maxmum lkelhood postons. Jenke et al.(006) showed how to generate a smooth pont cloud from a gven nosy one usng Bayesan statstcs. (4) Extenson of D flters to 3D ones. The D flters were drectly extended nto 3D settngs to smooth PSSs by applyng local poston estmatng teratvely or non-teratvely, sotropcally or ansotropcally, based on statstcs, dfferental geometry theory, approxmaton theory, etc. The last group s more attractve snce t s smple and straghtforward. Technques for mage smoothng such as Laplacan, blateral and trlateral flterng, moreover, commonly act as foundatons for 3D surface denosng algorthms. In ths paper, we ntroduce a non-local denosng (NLD) method for PSSs nspred by a non-local algorthm for mage denosng (Buades et al., 005) whch presents remarkable results. Unlke the non-local mage flter, our flter computes the denosed poston of a vertex as a weghted average of the vertces n ts vcnty wth smlar geometry features whch are determned by mean shft clusterng. The non-local approach defnes the ntensty smlarty of two ponts by comparng regons of the surface around the vertces rather than usng only ther postons and sometmes normals locally. Ths yelds a more accurate denosng result of the surface and mproves the removal of hgher-level noses compared to prevous state-of-the- art flterng technques. Moreover, fne geometry features are better preserved. An example of the effectveness of our approach s presented n Fg.. In ths paper all the pont models are rendered by usng a pont-based renderng technque. The rest of ths paper s organzed as follows. In Secton, we brefly revew the related works. In Secton 3, we brefly overvew the non-local means approach of Buades, Coll, and Morel. We present our smlarty-based method for denosng PSSs n Secton 4. We compare our method wth two denosng technques n Secton 5. Secton 6 concludes the paper. (a) (c) Fg. Denosng Buste model wth our method. (a) Nosy model; (b) Model denosed by our method; (c) Nosy model colored by mean curvature; (d) Denosed model colored by mean curvature. Notce that hgh-frequency nose s properly removed, whle fne detals n har regon are accurately preserved RELATED WORKS Earler methods such as Laplacan (Pauly et al., 00b) for denosng PSSs are sotropc, whch result commonly n pont drftng and oversmoothng. So the ansotropc methods were ntroduced. Clarenz et al.(004) presented a PDE-based surface farng applcaton wthn the framework of processng pontbased surface va PDEs. Lange and Polther (005) proposed a new method for ansotropc farng of a pont-sampled surface based on the concept of ansotropc geometrc mean curvature flow. Based on dynamc balanced flow, Xao et al.(006) presented a novel approach for farng pont-sampled geometry. Other methods have also been proposed for denosng the PSSs. Algorthms that recently attracted the nterest of many researchers are MLS approaches. Based on MLS or ts varants, the algorthms for denosng PSSs have also been ntroduced (Mederos et al., 003; Weyrch et al., 004; Dey and Sun, 005; (b) (d) Max Mn
3 Wang et al. / J Zhejang Unv Sc A 008 9(6): Lpman et al., 006; Danels II et al., 007). The man problem of MLS-based methods s that promnent shape features are blurred whle smoothng PSSs. All of the above methods dd not take nto consderaton the smlarty of geometry features of sample ponts. Hu et al.(006) ntroduced a mean shft based denosng algorthm for PSSs. Durng the course of denosng PSSs, they respected the knd of smlarty va a 3D mean shft procedure. However, all the above-mentoned methods determne the ntensty smlarty of two ponts locally usng only ther postons and sometmes normals. Concernng these ssues, ths paper defnes the ntensty smlarty n a non-local fashon and takes nto account the smlarty of geometry features whle denosng PSSs. We ntroduce the followng technques to acheve the non-local denosng algorthm: () By usng the trlateral flterng operator, the geometry ntensty of each sample pont s determned; () Based on covarance analyss, a regular grd of geometry ntensty s constructed for each pont and the geometry-ntensty smlarty between two ponts s measured; (3) Based on mean shft clusterng, the PSSs are clustered accordng to the surface-features smlarty. NON-LOCAL IMAGE DENOISING The non-local algorthm for mage denosng was proposed by Buades et al.(005). The basc dea behnd the non-local method s very smple: for a gven pxel, ts denosed ntensty value s estmated as a weghted average of the other mage pxels wth weghts reflectng the smlarty between local neghborhoods of the pxel beng processed and the other pxels. More precsely, f an mage Ω={I(u) u P} s gven, where u=(x, y) s a pxel and I(u) s the ntensty value at u, the smoothed pxel ntensty I'(u) can be computed as the average of all pxel ntenstes n the mage weghted by a smlarty factor Φ(u,v), Φ( uv, ) I( v) P I ( u)=, Φ( uv, ) v v P () where Φ(u,v)=exp( S u,v /h ). The parameter h acts as a degree of flterng, and the smlarty S u,v between u and v s measured as Suv, = Ga ( o ) I( u+ o) I( v+ o ), () o whch depends on the pxel-wse ntensty dfference of two square neghborhoods centered at the pxels u and v. The vector o denotes the offset between the center pxel and an arbtrary neghborhood pxel. The nfluence of a pxel par on the smlarty falls wth ncreasng Eucldean dstance to the center of the neghborhoods. For the dstance weghtng a Gaussan kernel G a ( ) wth a standard devaton a>0 s used. NON-LOCAL DENOISING ALGORITHM FOR PSSs For the reason that mage pxels are usually algned on a regular and equspaced grd, whch s n general not true for a pont-sampled model, the man dffculty of extendng the non-local method to PSSs conssts of how to determne the ntensty smlarty of two ponts. Our strategy for solvng ths ssue s as follows. By usng the trlateral flterng operator, the dfferental value of each sample pont s determned and called geometry ntensty as a counterpart to the ntensty value of an mage. Based on covarance analyss, the local reference plane s defned on whch a regular grd of geometry ntensty s then constructed for each pont, and accordng to geometry ntenstes on the correspondng grds, the geometryntensty smlarty of two ponts s fnally measured. Computaton of geometry ntensty Although the trlateral flter presented by Choudhury and Tumbln (003) for hgh contrast mages and meshes can be extended to PSSs, t does not consder the curvature-related nformaton. We desgn the followng trlateral flter wth the curvature-related term to compute the geometry ntensty δ of each pont p δ = w <, >, ( ) j n qj p w qj N p qj N( p ) j wj = wc( qj p ) ws( < n, qj p > ) wh( Hj H ), (3) where n s the surface normal at the pont p, N(p ) s the neghborhood of p, H s the mean curvature and w(x) s a Gaussan kernel: w ( x)=exp[ x /( σ )], c c
4 80 Wang et al. / J Zhejang Unv Sc A 008 9(6): w ( x)=exp[ x /( σ )] and w h (x)=exp[ exp( H j s s H /)]. The term w h (x) denotes that the nfluence on the weght w j ncreases wth an ncrease n the curvature dfference (H j H ) so that the hgh gradent regons can be effcently smoothed. In ths paper we take the parameter σ c as σ c =r/, where r s the radus of the enclosng sphere of N(p ), and σ s as the standard devaton of the projectons of the vector (q j p ) onto n. Because our trlateral flterng operator consders not only the pont postons and normals but curvatures, the geometry ntensty computed by t can reflect the local geometry feature and descrbe the dfferental property at each pont more effcently. Fg.b demonstrates the geometry-ntensty value vsualzaton of the ponts for the Face model. Fg.e llustrates the palette for vsualzaton of mean curvature, geometry ntensty and the dstance between two ponts. (a) (b) Measurng the geometry-ntensty smlarty of two ponts By covarance analyss, a local frame can be constructed based on the tangent plane and normal for each sample pont. The covarance matrx for the pont set P s defned as p p p p p p p p C =, p p k p p k where p k s the kth neghborng pont around p and p= p / k s the centrod of the neghborhood. k Snce C s symmetrc and postve sem-defnte, all egenvalues λ (=0,, ) are of real value and the egenvectors v (=0,, ) form an orthogonal bass. The egenvalue λ measures the varaton of the local pont set along the drecton of the correspondng egenvector. Assumng λ 0 λ λ, let the plane (x p ) v 0 =0 through p mnmze the sum of the squared dstance to the neghborng ponts of p, then the normal v 0 of ths plane can be regarded as the normal of the local surface at p. In ths partcular case, λ 0 expresses the varaton of the surface along the normal v 0. Pauly et al.(00a) defned T σ = σ ( p ) = λ ( λ + λ + λ ) (4) k k 0 0 (c) Mn (e) Fg. (a) Nosy Face model; (b) Geometry-ntensty value vsualzaton of the ponts; (c) Our mean-shft clusterng; (d) The pont set of local modes; (e) Palette for vsualzaton of geometry ntensty (d) Max as the surface varaton at p assumng a neghborhood of sze k. It s also observed that σ k (p ) s closely related to the local curvature. To ensure a consstent orentaton of the normal vectors, we use a method based on the mnmum spannng tree (Hoppe et al., 99). For each pont, we construct the local frame (v, v, v 0 ) wth the local orgn p and on the plane v p v, a regular geometry-ntensty grd G wth the sze of M M centered at p can be bult by an nterpolaton method. Let N R (p )={q j q j P n, q j p M E } be the set of the neghbors of p whose elements are wthn a fxed radus R bound centered at p, where E = n r mn n s the average edge length of the sample =0
5 Wang et al. / J Zhejang Unv Sc A 008 9(6): pont set P n, n= P n and r mn s the dstance between p and ts nearest pont. Assumng that the vector v g s the projecton of the vector (q j p ) onto the plane v p v and δ g s the geometry ntensty of the pont q j, the geometry ntensty δ m of each node g m (0 m M M ) among G s estmated as δ m = vg N ( gm ) vg N ( gm ) wg( vg) δg, w ( v ) g g s an adaptve nonparametrc estmator of the densty at locaton x n the feature space. The functon k(x) (0 x ) s called the profle of the kernel, and the normalzaton constant c k,d assures that K(x) ntegrates to one. The functon g(x)= k'(x) can always be defned when the dervatve of the kernel profle k(x) exsts. Usng g(x) as the profle, the kernel G(x) s defned as G(x)=c g,d g( x ). By takng the gradent of Eq.(5) the followng property can be proven where N(g m ) s the neghborhood of g m and w g (x) s a Gaussan kernel: w g (v g )=exp[ v g g m /(σ g )]. In ths paper, we take agan the parameter σ g as σ g =r/, where r s the radus of the enclosng crcle of N(g m ). When the regular geometry-ntensty grd G of p s regarded as a counterpart to the square neghborhood of the pxel u, we can, as a result, measure the geometry-ntensty smlarty of two ponts by means of Eq.(). m ( x)= C ˆ f ( x) fˆ ( x ), G K G where C s a postve constant and = d + ( ( x x )/ h ) ( ( x x)/ h ) n x g d + = h mg ( x) = x n g h (6) Mean shft clusterng for PSSs Unlke the non-local mage flterng algorthm, we do not sum over all pont postons to flter a pont but over a local neghborhood of ths pont whch s determned by a mean shft clusterng method. The mean shft algorthm s a nonparametrc clusterng technque for the analyss of a complex multmodal feature space and the delneaton of arbtrarly shaped clusters (Comancu and Meer, 00), and t has a wde varety of applcatons n the felds of computer vson and pattern recognton. Recently t has been extended to the feld of dgtal geometry processng (Yamauch et al., 005; Hu et al., 006; Shamr et al., 006). In the followng we frst present a short revew of the adaptve mean shft technque and then descrbe how to apply t to the pont model. Assume that each data pont x ú d (=,,,n) s assocated wth a bandwdth value h >0. The sample pont estmator (Georgescu et al., 003; Yamauch et al., 005) n f ˆ ( x K )= K( ( )/ h d ) nh x x (5) = based on a sphercally symmetrc kernel K wth bounded support satsfyng ( x)= kd, ( x ) > 0, x K c k s called the mean shft vector pontng toward the drecton of the maxmum ncrease n the densty. A gradent-ascent process wth an adaptve step sze y = m ( y ), j=0,,,... (7) [ j+ ] [ j] G consttutes the core of the mean shft clusterng procedure. For clusterng S={x, x,, x n } wth mean shft, the followng two steps are performed on each x S: () Intalze y [0] wth x ; () Compute y [] j accordng to Eq.(7) untl convergence. It s shown n (Comancu and Meer, 00) that under some general assumptons the sequences { [] j y } converge to the ponts where fˆ K ( x ) defned by Eq.(5) attans ts local maxma (mode). Accordngly, the ponts that converge to the same mode are assocated wth the same cluster. One smple extenson of the above clusterng procedure conssts of dealng wth a set S, each element of whch has two components of a dfferent nature, S={x =(c,q ) c C, q Q}. In such a stuaton, t s convenent to use the mean shft clusterng procedure wth separable kernels ( ) ( ) n f ˆ ( x )= k ( c c )/ h k ( q q )/ h. K d d nh h =
6 8 Wang et al. / J Zhejang Unv Sc A 008 9(6): In ths paper, we consder the sample ponts {p } equpped wth the normals {n } and the mean curvature {H } as scattered data S={x =(c,q ) c p, q (n, H )} n ú 7. For both c and q, we use the normal kernel. For the bandwdth values h, there are numerous methods to defne them, most of whch use a plot densty estmate. The smplest way to obtan the plot densty estmate s by nearest neghbors. To accelerate the mean shft computaton, we construct a k-d tree for the pont set {c }. Accordng to the k-nearest neghbors N k (c ) of c, we can adaptvely take h = c c,k, where c,k s the k-nearest neghbor of c, and h =max{ q q,, q q,,, q q,k }. After clusterng for PSSs by usng ths mean shft technque, the geometry features of the ponts n the same cluster, whch contan the pont postons, normals and mean curvatures, are locally smlar, respectvely. Fg.c demonstrates the mean shft clusterng of the Face model, and ts pont set of local modes s llustrated n Fg.d. RESULTS AND DISCUSSION In our experments, we use Mcrosoft Vsual C++ programmng language on a personal computer wth a Pentum IV.8 GHz CPU and GB man memory. We have mplemented our non-local denosng (NLD) and two state-of-the-art denosng technques: the Blateral denosng (BIL) (Fleshman et al., 003) and the Mean Shft denosng (MST) (Hu et al., 006) to compare ther denosng results. We use three models n our comparson: a nosy Buste model wth 5 83 sample ponts (Fg.a), a nosy Face model wth sample ponts (Fg.a) and a nosy Dragon-head model wth sample ponts (Fg.3). For these models, Table presents the related statstcs and parameter settngs used for our method and our mplementatons of BIL and MST. For BIL and MST, we tred to choose the parameter settngs that produce the best results. Non-local denosng of PSSs From the geometry-ntensty smlartes of two ponts and Eq.(), we can compute the denosed geometry ntensty δ of p as C ( Φ δ ) δ = ( p, q ) Φ( p, q ), qj ( p ) j j qj C( p ) j where C(p ) s the cluster to whch p belongs,.e., a local neghborhood of p determned by our mean shft clusterng method. The smlarty factor Φ(p, q j ) s computed by Φ S h ( p, qj )=exp( p, / ), qj where the geometry-ntensty smlarty S, between p and q j s measured by the method descrbed p q j n the subsecton Measurng the geometry-ntensty smlarty of two ponts. Accordng to the offset dstance δ ', the smoothed poston p ' s gven by p '=p +δ 'n. Snce the pont p s moved along ts normal drecton, our denosed method wll not ntroduce undesrable ponts drftng over the surface. Moreover, ths method s more effectve and robust than the local denosng methods as t consders not only the geometry-features smlarty between two ponts but also the geometry-ntensty smlarty between them. (a) (b) Fg.3 (a) Nosy Dragon-head model colored by mean curvature; (b) Closer vew of ts upper jaw Table Parameter settng and the related statstcs Fg. Method Iters. Sm. k Max. error ( 0 4 ) Avg. error ( 0 4 ) T MS (s) T D (s) BIL MST NLD BIL MST NLD BIL MST NLD Iters. stands for the number of teratons. Sm. s the sze of the regular grd consdered to measure the geometry-ntensty smlarty of two ponts. For BIL, k s the number of the neghbors, and for NLD and MST, k s the number of the sample ponts n bandwdth wndow for mean shft clusterng. Max. error s the maxmum of dstances between the orgnal (nosy) ponts and ther correspondng denosed ponts, and Avg. error s the average of dstances. T MS ndcates the tme of mplementng the mean shft clusterng and T D denotes the pont estmatng tme for one teraton
7 Wang et al. / J Zhejang Unv Sc A 008 9(6): We use two vsualzaton schemes to compare the technques wth our method. The frst scheme conssts of colorng by the mean curvature. The second one measures the dfference between the orgnal and denosed pont model,.e., we vsualze the dfferences n the postons of the correspondng sample nosy denosed ponts of the models p p. In Fg.4, we demonstrate a comparson of the denosed Face models by BIL, MST and NLD. The denosed models are llustrated n the top row of Fg.4, and ther correspondng mean curvature vsualzatons n the bottom row. As seen n Fg.4, our NLD removes the hgh-frequency nose properly and acheves a more accurate result than BIL or MST does. Fg.5 shows a comparson of BIL, MST and NLD concernng feature preservaton. Note that our NLD preserves sharp features more accurately than BIL or MST does whle producng a smooth result, as shown n the closer vews of the upper jaw of the denosed model. In Fg., we show the denosng effcency of our approach on the nosy Buste model (Fg.a), whch s produced by addng zero-mean Gaussan nose wth σ nose =0.3 E to the orgnal model. It can be notced that the hgh-frequency nose s properly removed, whle fne detals n har, mouth and ear regons are accurately preserved. At the same tme, we demonstrate that our NLD presents the best performance accordng to the entropy of the dfferences between the nosy and denosed models, as shown n the bottom row of Fg.6. From the Max. and Avg. errors n Table we can also notce that our method outperforms ts two rvals. As a result, our method produces the lowest oversmoothng when compared wth the other two denosng technques. Due to our regon-based defnton of the geometry-ntensty smlarty measure whch adds more geometrc nformaton nto the denosng process, our algorthm removes the hgh-frequency nose properly and acheves a more accurate result than BIL or MST does. Furthermore, our method has a better feature preservaton than BIL or MST does whle producng a smooth result, manly because the method utlzes the trlateral flter n the denosng process. From the executon tme lsted n Table, we notce that our method s slower than BIL or MST snce our method needs to construct the regular geometry-ntensty grd for each sample pont and measure the geometry-ntensty smlarty of two ponts. Max Mn (a) (b) (c) Fg.4 Denosng the nosy Face model (Fg.a). Top: the denosed models. Bottom: the correspondng denosed model colored by mean curvature. Mean curvature colorng helps us to compare ther correspondng fne detals. (a) BIL; (b) MST; (c) NLD
8 84 Wang et al. / J Zhejang Unv Sc A 008 9(6): (a) (b) (c) Fg.5 Denosng the nosy Dragon-head model (Fg.3). Top: the denosed model colored by mean curvature. Bottom: a closer vew of the upper jaw of the correspondng denosed model. (a) BIL; (b) MST; (c) NLD Max (a) (b) (c) Mn Fg.6 Denosng the nosy Buste model (Fg.a). Top: the denosed model colored by mean curvature. Bottom: the correspondng denosed model colored accordng to the entropy of the dfferences between the nosy and denosed models. (a) BIL; (b) MST; (c) NLD CONCLUSION In ths paper, we presented an NLD algorthm for PSSs by extendng the non-local technque for mage denosng to a pont-sampled model. Snce the orgnal non-local method reles heavly on the mage structure regularty, the man dffculty n extendng t to PSSs s how to determne the ntensty smlarty of two sample ponts. We frst compute the geometry ntensty of the sample pont by usng the trlateral flterng operator. Based on covarance analyss, a regular geometry-ntensty grd as a counterpart to the
9 Wang et al. / J Zhejang Unv Sc A 008 9(6): square neghborhood of the pxel s then constructed. Fnally, the geometry-ntensty smlarty of two ponts s measured accordng to ther grds. Furthermore, the neghborhood of the sample pont s adaptvely selected n the denosng process by means of our mean shft method so as to produce a more accurate denosng result. Our expermental results demonstrate that ths proposed algorthm s robust, and can produce more accurate denosng results than the two state-of-theart smoothng technques, the Blateral denosng and the Mean Shft denosng, whle havng better feature preservaton. ACKNOWLEDGEMENTS Models are courtesy of Yutake Ohtake (Nosy Face), Thous Jones (Nosy Dragon Head) and EU AIM@Shape project (Buste). References Amenta, N., Kl, Y.J., 004. Defnng pont set surfaces. ACM Trans. on Graph., 3(3): [do:0.45/ ] Buades, A., Coll, B., Morel, J.M., 005. A Non-Local Algorthm for Image Denosng. Proc. IEEE Computer Socety Int. Conf. on Computer Vson and Pattern Recognton, p [do:0.09/cvpr ] Carr, J.C., Beatson, R.K., Cherre, J.B., Mtchell, T.J., Frght, W.R., McCallum, B.C., Evans, T.R., 00. Reconstructon and Representaton of 3D Objects wth Radal Bass Functons. Proc. ACM SIGGRAPH, p [do:0. 45/ ] Choudhury, P., Tumbln, J., 003. The Trlateral Flter for Hgh Contrast Images and Meshes. Int. Conf. on Computer Graphcs and Interactve Technques, p [do:0. 45/ ] Clarenz, U., Rumpf, M., Telea, A., 004. Farng of Pont Based Surfaces. Proc. Computer Graphcs Internatonal, p [do:0.09/cgi ] Comancu, D., Meer, P., 00. Mean shft: a robust approach toward feature space analyss. IEEE Trans. on Pattern Anal. Machne Intell., 4(5): [do:0.09/ ] Danels II, J., Ha, L.K., Ochotta, T., Slva, C.T., 007. Robust Smooth Feature Extracton from Pont Clouds. Proc. Shape Modelng Internatonal, p [do:0.09/ SMI.007.3] Dey, T.K., Sun, J., 005. An Adaptve MLS Surface for Reconstructon wth Guarantees. Proc. Symp. on Geometry Processng, p Fleshman, S., Dror, I., Cohen-Or, D., 003. Blateral mesh denosng. ACM Trans. on Graph., (3): [do:0.45/ ] Georgescu, B., Shmshon, I., Meer, P., 003. Mean Shft Based Clusterng n Hgh Dmensons: A Texture Classfcaton Example. ICCV, p Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W., 99. Surface Reconstructon from Unorganzed Ponts. Proc. 9th Annual Conf. on Computer Graphcs and Interactve Technques, p [do:0.45/ ] Hu, G.F., Peng, Q.S., Forrest, A.R., 006. Mean shft denosng of pont-sampled surfaces. The Vsual Computer, (3): [do:0.007/s ] Jenke, P., Wand, M., Bokeloh, M., Schllng, A., Strasser, W., 006. Bayesan pont cloud reconstructon. Computer Graphcs Forum, 5(3): [do:0./j x] Lange, C., Polther, K., 005. Ansotropc smoothng of pont sets. Computer Aded Geometrc Desgn, (7): [do:0.06/j.cagd ] Lpman, Y., Cohen-Or, D., Levn, D., 006. Error Bounds and Optmal Neghborhoods for MLS Approxmaton. Proc. Eurographcs, p [do:0.3/sgp/sgp06/07-080] Mederos, B., Velho, L., de Fgueredo, L.H., 003. Robust Smoothng of Nosy Pont Clouds. Proc. SIAM Conf. on Geometrc Desgn and Computng, p.-3. Pauly, M., Gross, M., 00. Spectral Processng of Pont- Sampled Geometry. Proc. ACM SIGGRAPH, p [do:0.45/ ] Pauly, M., Gross, M., Kobbelt, L.P., 00a. Effcent Smplfcaton of Pont-Sampled Surfaces. Proc. IEEE Vsualzaton, p Pauly, M., Kobbelt, L.P., Gross, M., 00b. Multresoluton Modelng of Pont-Sampled Geometry. Techncal Report, CS #379, ETH, Zurch. Pauly, M., Keser, R., Kobblet, L.P., Gross, M., 003. Shape modelng wth pont-sampled geometry. ACM Trans. on Graph., (3): [do:0.45/ ] Pauly, M., Mtra, N.J., Gubas, L.J., 004. Uncertanty and Varablty n Pont Cloud Surface Data. Proc. Eurographcs Symp. on Pont-Based Graphcs, p Samozno, M., Alexa, M., Allez, P., Yvnec, M., 006. Reconstructon wth Vorono Centered Radal Bass Functons. Proc. Eurographcs, p [do:0.3/sgp/ SGP06/05-060] Schall, O., Belyaev, A., Sedel, H.P., 005. Robust Flterng of Nosy Scattered Pont Data. Eurographcs Symp. on Pont-Based Graphcs, p [do:0.3/spbg/ SPBG05/07-077] Shamr, A., Shapra, L., Cohen-Or, D., 006. Mesh analyss usng geodesc mean-shft. The Vsual Computer, (): [do:0.007/s ] Weyrch, T., Pauly, M., Keser, R., Henzle, S., Scandella, S., Gross, M., 004. Post-processng of Scanned 3D Surface Data. Proc. Eurographcs, p Xao, C.X., Mao, Y.W., Lu, S., Peng, Q.S., 006. A dynamc balanced flow for flterng pont-sampled geometry. The Vsual Computer, (3):0-9. [do:0.007/s ] Yamauch, H., Lee, S., Lee, Y., Ohtake, Y., Belyaev, A., Sedel, H.P., 005. Feature Senstve Mesh Segmentaton wth Mean Shft. Proc. Shape Modelng Internatonal, p [do:0.09/smi.005.]
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