Feature-Preservng Denosng of Pont-Sampled Surfaces Jfang L College of Computer Scence and Informaton Technology Zhejang Wanl Unversty Nngbo 315100 Chna Abstract: Based on samplng lkelhood and feature ntensty, n ths paper, a feature-preservng denosng algorthm for pont-sampled surfaces s proposed. In terms of movng least squares surface, the samplng lkelhood for each pont on pont-sampled surfaces s computed, whch measures the probablty that a 3D pont s located on the sampled surface. Based on the normal tensor votng, the feature ntensty of sample pont s evaluated. By applyng the modfed blateral flterng to each normal, and n combnaton wth samplng lkelhood and feature ntensty, the fltered pont-sampled surfaces are obtaned. Expermental results demonstrate that the algorthm s robust, and can denose the nose effcently whle preservng the surface features. Key-Words: samplng lkelhood; feature ntensty; blateral flterng; pont-sampled surfaces denosng 1 Introducton 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. Consequently, nosy models need to be denosed or smoothed before performng any subsequent geometry processng such as smplfcaton, reconstructon and parameterzaton. It remans a challengng task to remove the nevtable nose whle preservng the underlyng surface features n computer graphcs. Earler methods such as Laplacan [1] for denosng pont-sampled surfaces (PSS) are sotropc, whch result commonly n pont drftng and oversmoothng. So the ansotropc methods were ntroduced. Clarenz et al. [] presented a PDE-based surface farng applcaton wthn the framework of processng pont-based surface va PDEs. Lange and Polther [3] 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. [4] presented a novel approach for farng PSS. Other methods have also been proposed for denosng PSS. Algorthms that recently attracted the nterest of many researchers are movng-least squares (MLS) approaches [5-7] to ft a pont set wth a local polynomal approxmaton; the pont set surface can be smoothed by shftng pont postons towards the correspondng MLS surface. The man problem of MLS-based methods s that promnent shape features are blurred whle smoothng PSS. Concernng the above problem of MLS approaches, ths paper puts forward a feature-preservng denosng algorthm for PSS. Based on MLS surface, the samplng lkelhood of sample pont s frst computed. In order to more effcently preserve the surface features whle denosng PSS, we adopt normal votng tensor to evaluate the feature ntensty of sample pont and apply the modfed blateral flterng to flter the normal of each sample pont. The smoothed model s fnally obtaned va the combnaton of samplng lkelhood and feature ntensty. Computng the samplng lkelhood In ths paper, we consder the probablty that a 3D pont s located on the sampled surface as the samplng lkelhood. The sample pont closer to the sampled surface should be characterstc of hgher samplng lkelhood than one beng more dstant. We approxmately take the MLS surface approxmatng the k nearest neghbors Nk(p) of sample pont p as the sampled surface. In the followng we wll brefly revew the MLS method and then descrbe how to compute the samplng lkelhood. Alexa et al.[8] proposed a representaton of pont-sampled model by fttng a local polynomal approxmaton to the pont set usng a MLS method. The result of the MLS-fttng s a smooth, -manfold surface for any pont set. Gven a pont set P = { p }, the contnuous MLS surface S s defned mplctly as ISSN: 1790-5117 1 ISBN: 978-960-474-41-3
the statonary set of a projecton operator ψ ( r ) that projects a pont onto the MLS surface. To evaluateψ, a local reference plane H = { x 3 n x D =0} s frst computed by mnmzng the weghted sum of squared dstances,.e., arg mn ( n p n q) θ ( p q ), where q s the nq, p P projecton of r onto H and θ s the MLS kernel functon θ ( d) = exp(- d /h ), where h s a global scale factor. Accordngly, the local reference doman s gven by an orthonormal coordnate system on H so that q s the orgn of ths system. Then a bvarate polynomal g( uv, ) s ftted to the ponts projected onto the reference plane H usng a smlar weghted least squares optmzaton. Here ( u, v) s the representaton of q n the local coordnate system n H, where q s the projecton of p onto H. So, the projecton of r onto S s gven asψ ( r) = q + g00 (, ) n. More detals on the MLS method can be found n[9]. We take a thrd degree polynomal to approxmate the cluster of N k (p ): 3 3 y gx (, y)= ax 9 + ay 8 + axy 7 + ax 6 +ax 5. (1) + a y + a xy+ a x+ a y+ a 4 3 1 0 Let q j be the projecton of p j N k ( p ) onto the above MLS surface gx (, y) ( q 0,.e. q, s the projecton of p ) and defne the samplng lkelhood l of p as l = 1-d / dmax k 1 θ ( pj p ) pj qj j = 0 d = k 1, () θ ( pj p ) j= 0 dmax = max N{ d} where N s the sze of pont-sampled model and d s the weghted-average dstance of p j to ts projecton onto the MLS surface. Obvously, the nfluence of p j on d decreases exponentally wth Eucldean dstance to p. On the other hand, the smaller d s, the greater d. Consequently, the samplng lkelhood l can effectvely denote the value of probablty that p s located on the sampled surface. Fg.1b demonstrates mean curvature vsualzaton of nosy Igea model as shown n Fg.la and samplng lkelhood vsualzaton of nosy model s llustrated n Fg.1c. In ths paper, all the pont-sampled models are rendered by usng a pont-based renderng technque. (a) (b) (c) (d) Fg.1 (a) Nosy Igea Model; (b) Mean curvature vsualzaton of (a); (c) Samplng kelhood vsualzaton of (a); (d) Feature ntensty vsualzaton of (a). 3 Measurng the feature ntensty We measure the feature ntensty of sample pont by extendng normal votng tensor appled to the extracton of sharp edge on 3D mesh [10] to pont-sampled surfaces. A normal votng tensor T for a sample pont p s defned T by T = u n n, where N () s the ndex set of j N () j j j p j belongng to N k (p ), u j s a weght defned as uj = exp(- pj p / σ e ). We take standard devaton σ e as σ e = r / 3, r s the radus of the enclosng sphere of N k (p ). n j s p j s normal ( n j = 1) and n j s determned as nj = ( nj wj ) wj nj, where wj = ( p pj ) nj ( p pj )( wj =1). From the defnton, T s symmetrc and postve sem-defnte. Accordngly, ts egenvalues are real-valued and non-negatve: v1 v v3 0. Furthermore, the correspondng egenvectors e1, e and e 3 form an orthonormals bass. So we defne the feature ntensty of sample pont p as ISSN: 1790-5117 13 ISBN: 978-960-474-41-3
1 n e 1 < δ s = 1 v3 > α( v1-v) v3 > β( v- v3), ( v v3)/ v1 otherwse (3) n = u n s a weghted sum of votng where j N () j j normal n j, andδ,α, β are postve real numbers. In ths paper, we expermentally setδ,α, β to 0., 0.3 and 0.3, respectvely. The equaton classfes each sample pont nto three types whch correspond to the type of feature that the pont belongs to,.e. face, sharp edge, or corner. The feature ntensty becomes about 1.0 f sample pont les on a sharp edge or a corner, or about 0.0 f t les on a face. As a result, s ndcates the geometrc feature of surface at p. The feature ntensty vsualzaton of nosy Igea model (Fg.la) s llustrated n Fg.1d. where wx () s a Gaussan kernel: wc()=exp(- x x / σ c) and w ()=exp(- / s x x σ s ). The normal varaton α j s defned as α j = acos( n nj )( n = nj =1). Here, we take the parameterσ c as σ c = r/ and σ s as the standard devaton of the normal varatonα j. Accordng to the equaton, n s the weghted average of p j s normal where the weght of each normal s computed usng a standard Gaussan functon w c n the spatal doman multpled by an nfluence functon ws n the ntensty doman that decreases the weght of normals wth large normal varaton. Therefore, n s nfluenced manly by the smaplng ponts n N k (p ) that have a smlar ntensty. 4 Denosng of PSS The man dea of ths paper s denosng algorthm s as follows: p s normal s frst fltered by usng the modfed blateral flterng. In combnaton wth the samplng lkelhood and feature ntensty, the dstance weght m of p s then determned when denosng PSS. p s moved n the fltered normal drecton wth an offset D so as to smooth PSS. 4.1 Normal flterng If p s moved n the vector p q drecton when smoothng PSS, the surface features are blurred. In order to preservng those features more effectvely, we frst smooth p s normal accordng to the modfed blateral flterng. Surface normals play an mportant role n surface denosng as surface features are best descrbed wth the frst-order surface normals. It also s well-known that normal varatons offer more ntutve geometrc meanng than pont poston varatons. A smooth surface can be descrbed as one havng smoothly varyng normals whereas features such as sharp edges and corners appear as dscontnutes n the normals. Thus unlke the blateral flterng n [11], we desgn the followng blateral flterng wth the normal-varaton term to compute the fltered normal n w ( - ) ( ) () c pj p ws α n j N j j n, (4) w ( p - p ) w ( α ) = j N () c j s j 4. Sample pont flterng Next, for each sample pont p we fnd ts smoothed poston p by movng t along n wth an offset D,.e., p = p+d n, where D s determned as D = m q - p. In terms of the samplng lkelhood and feature ntensty, defne the dstance weght m as -l - m (-1) [ e +(1- )e s = ], where λ(0 λ 1) s a user-adjustable parameter; τ s set to 0 when n p q > 0, whch ndcates that p s moved along n, otherwse τ s set to 1 and p s moved along. n 5 Expermental results and dscusson In our experments, we use Mcrosoft Vsual C++ programmng language on a personal computer wth a Pentum IV.8 GHz CPU and 1 GB man memory. We have mplemented our denosng algorthm and another two denosng technques: the Blateral denosng (BIL) and the MLS-based denosng to compare ther denosng results. We use two models n our comparson: a nosy Fandsk model wth 97580 sample ponts (Fg.a) and a nosy Igea model wth 134345 sample ponts (Fg.1a). In ths paper, we use the vsualzaton scheme of mean curvature to compare these two technques wth our method. In Fg., we demonstrate a comparson of the denosed Fandsk models by MLS, BIL and our method. The denosed models are llustrated n the top row of Fg., and ther correspondng mean curvature vsualzatons n the bottom row. As seen n ISSN: 1790-5117 14 ISBN: 978-960-474-41-3
Fg., our algorthm removes the hgh-frequency nose properly and acheves a more accurate result than MLS or BIL does. Fg.3 shows a comparson of MLS, BIL and our algorthm concernng feature preservaton. Note that our alogrthm preserves sharp features more accurately than MLS or BIL does whle producng a smooth result. Snce the MLS-based denosng shfts sample ponts to ts projecton onto the correspondng MLS surface, sharp features are sgnfcantly smoothed out. Although the blateral denosng performs well n general, the sharp features are not able to be effcently preserved as t actually uses a statc wndow/kernel n the two domans. Due to take not only nto account the samplng lkelhoods of sample ponts but also the feature ntenstes whle denosng PSS, our algorthm can delver qualty smoothng whle preservng the surface features more effcently than BIL or MLS. (a) Nose model (b) MLS (c) BIL (d) Ours Fg. Denosng nosy Fandsk model. Top: the denosed models. Bottom: the correspondng denosed model colored by mean curvature (The colorng helps us to compare ther correspondng fne detals). (a) MLS (b) BIL (c) Ours Fg.3 Denosng the nosy Igea model. Top: the denosed models. Bottom: the correspondng denosed model colored by mean curvature. ISSN: 1790-5117 15 ISBN: 978-960-474-41-3
6 Concluson In ths paper, we presented a denosng algorthm for PSS. In terms of the MLS surfaces, the samplng lkelhood of sample pont s computed and the feature ntensty of sample pont s evaluated based on normal votng tensor. The pont s normal s fltered by usng the modfed blateral flterng. The PSS s smoothed by movng each sample pont along ts own fltered normal wth an offset determned accordng to the combnaton of the samplng lkelhood and feature ntensty. Our expermental results demonstrate that the proposed algorthm s robust, and can denose the nose effcently whle preservng the surface features. defned by pont sets. In: Proc. of the Eurographcs Symposum on Pont-Based Graphcs. Eurographcs Assocaton 004.149-155. [10] Shmzu T, Date H, Kana S, Kshnam T. A New Blateral Mesh Smoothng Method by Recognzng Features In: Proc. of Nnth Internatonal Conference on CAD/CG05 005.81-86. [11] Fleshman S, Dror I, Cohen-Or D. Blateral mesh denosng. In: Proc. of the Computer Graphcs Annual Conf. Seres, ACM SIGGRAPH 003. 950-953. References: [1] Pauly M, Kobbelt, L.P., Gross, M. Multresoluton Modelng of Pont-Sampled Geometry. Techncal Report, CS #379, ETH, Zurch 00. [] Clarenz U, Rumpf M, Telea A. Farng of pont based surfaces. In: Computer Graphcs Internatonal(CGI04) 004:600-603. [3] Lange C, Polther K. Ansotropc smoothng of pont sets. Comput.Ad Geomet.Des. 005, (7):680-69. [4] Xao C.X., Mao Y.W., lu S., Peng Q.S. A dynamc balanced flow for flterng pont-sampled geometry. The Vsual Computer 006, (3):10-19. [5] Mederos B, Velho L, de Fgueredo L.H. Robust smoothng of nosy pont clouds. In: Proc. of the SIAM Conf. on Geometrc Desgn and Computng. Seattle: Nashboro Press 003. [6] Weyrch T, Pauly M, Keser R, Henzle S, Scandella S, Gross M. Post-processng of scanned 3D surface data. In: Proc. of Symp. on Pont-Based Graphcs 04, 004. 85-94. [7] Danels II J, Ha,L.K., Ochotta, T., Slva, C.T. Robust Smooth Feature Extracton from Pont Clouds. Proceedngs of Shape Modelng Internatonal 007.13-136. [8] Alexa M BJ, Cohen-Or D, Fleshman S, Levn D, Slva T. Pont set surfaces. In: Proceedngs of IEEE Vsualzaton, San Dego, Calforna 001:1-8. [9] Alexa M, Adamson A. On normals and projecton operators for surfaces ISSN: 1790-5117 16 ISBN: 978-960-474-41-3