Fairing of Polygon Meshes Via Bayesian Discriminant Analysis

Transcription

1 Farng of Polygon Meshes Va Bayesan Dscrmnan Analyss Chun-Yen Chen Insue of Informaon Scence, Academa Snca. Deparmen of Compuer Scence and Informaon Engneerng, Naonal Tawan Unversy. 5, Tawan, Tape, Nankang Kuo-Young Cheng Insue of Informaon Scence, Academa Snca. Deparmen of Compuer Scence and Informaon Engneerng, Naonal Tawan Unversy. 5, Tawan, Tape, Nankang Hong-Yuan Mark Lao Insue of Informaon Scence, Academa Snca. 5, Tawan, Tape, Nankang ABSTRACT Desgn of an ansoropc dffuson-based fler ha performs Bayesan classfcaon for auomac selecon of a proper wegh for farng polygon meshes s proposed. The daa analyss based on Bayesan classfcaon s adoped o deermne he decson boundary for separang poenal edge and non-edge verces n he curvaure space. The adapve dffuson fler s governed by a double-degenerae ansoropc equaon ha deermnes how each polygon verex s moved along s normal drecon n he curvaure space eravely unl a seady sae s reached. The deermnaon of how much a polygon verex should be moved depends on wheher s a poenal edge or a non-edge verex. A each farng sep, concepually, a b-dreconal curvaure map whose boundary lne whle couched by he wegh value can be ploed o undersand he ype of a verex. Expermenal resuls show ha he proposed dffuson-based approach could effecvely smooh ou noses whle reanng useful daa o a very good degree. Keywords Polygon Mesh Smoohng, Bayesan Dscrmnan Analyss, Feaure Deecon.. INTRODUCTION When one res o acqure a 3D objec model by a 3D laser scanner, he scanned objec shape s, usually, represened n he form of rangular polygon meshes. However, such sampled pons are always nhered wh noses and he resulan polygon surface looks faceed due o he exsence of noses. A echnque ha can remove nose whle reanng polygon surface s always preferable. Such a echnque falls no he caegory of specal fler desgn, n whch a degraded polygon shape s smoohed by gradually farng ou nose eravely. The nformaon ha can be obaned from a scanned 3D objec model ncludes he poson of a verex, he passng angen planes, and he curvaures of surfaces ha are formed by angen planes. A Permsson o make dgal or hard copes of all or par of hs work for personal or classroom use s graned whou fee provded ha copes are no made or dsrbued for prof or commercal advanage and ha copes bear hs noce and he full caon on he frs page. To copy oherwse, or republsh, o pos on servers or o redsrbue o lss, requres pror specfc permsson and/or a fee. Journal of WSCG, Vol., No.-3, ISSN WSCG 004, February -6, 004, Plzen, Czech Republc. Copyrgh UNION Agency Scence Press nosy verex causes sgnfcaon of local face effecs and hus produces low qualy surfaces. I s well known ha he fler desgn based upon a consraned opmzaon echnque s able o generae sasfacory surface wh hgh qualy. In hs paper, we le he smoohng of he daa of a scanned objec governed by a hn shell hea equaon. In he smoohng process, each verex s consdered as a pon emperaure and he curvaure of a verex s consdered as he energy assocaed wh. Therefore, he fler desgn problem s convered no an opmzaon problem. In he opmzaon process, he proposed algorhm res o locae a new poson for each verex so ha he assocaed energy can be mnmzed. Because dfferen verex ypes have dfferen curvaure values, hey wll be governed by he hea equaon wh dfferen hea ransfer raes. In he leraure, he mos smlar exsng approach s he dscree farng echnque proposed by Kobbel e al. [Kob98]. In ha work, hey appled a so-called umbrella algorhm ogeher wh a hn plae energy funcon o derve a recursve equaon for desgnng her fler. However, hey used a fxed wegh value for relocang each verex durng he erave process. I s well know ha he polygon smoohng process based on he fler desgn concep

2 reles heavly on he local fne srucure as well as he nose ha assocaed wh he gven 3-D polygon model. Whou consderng he verex ype and dfferen weghs for dfferen verces, he desgned smoohng algorhm may produce some unwaned faces n he fnal oupu resuls. In hs paper, we propose a mehod for polygon smoohng usng a fler desgn approach. The proposed scheme s able o perform nellgen daa analyss and o classfy each verex as eher a feaure verex or a non-feaure verex. Ths classfcaon resul can be used as he bass for selecng approprae wegh for each verex a a farng sep. To dfferenae our mehod wh ohers, we call he fler desgn based on our proposed mehod he adapve dffuson fler desgn. The expermenal resuls show ha he proposed adapve dffuson fler can mprove he resulan surface qualy sgnfcanly as compared o some prevous polygon fler desgns.. FILTER DESIGN USING DIFFU- SION EQUATIONS: A SURVEY In an soropc dffuson-based mehod, he desgn of a low pass or a band pass fler s o use a dffuson-syle paral dfferenal equaon o conrol he process of even operaon. The soropc dffuson equaon ha wll f n he polygon mesh farng process can be defned as follows: M = M, () where s a Laplacan operaor and M represens a gven polygon mesh model. An analyc soluon for equaon () can be expressed as a form of a lnear nvaran Gaussan kernel convolued wh M [Tau96]. However, an analyc mehod ha can be used o solve equaon () n 3D space s raher complcaed. Therefore, s usually solved by a fne dfference approach usng he followng erave equaon: M = M λ M, () where λ s a user-defned consan. For speedng up he process descrbed n Equaon (), an umbrella operaor s adoped o lnearly approxmae he Laplacan operaor. In [Tau96], an umbrella operaor proposed by Taubn s defned as follows: ( x ) = x x # N( x ), (3) j x j N ( x ) where N ( x ) ndcaes he neghborng verces of x, and # N ( x ) ndcaes he number of verces beng conaned n N x ). ( λ M M Fgure. The Laplacan fler. Equaon () can be expressed as an soropc fler as shown n Fgure. I s bascally a Gaussan dffuson fler whch conans wo man componens: he Laplacan operaor and he weghng consan λ. The Laplacan operaor s used for calculang he normal dfference beween a verex and s neghborng verces. The value of he weghng consan deermnes he degree of dffuson. A fler can be called an soropc dffuson fler f he defned wegh s mananed consan n he whole process; oherwse, we call an ansoropc one. Taubn [Tau96] found ha a Gaussan-ype dffuson process can suppress nose very effecvely bu also creae a model-shrnkng problem. In order o solve he model-shrnkng problem, he desgned a muually compensaed dffuson fler called Taubn dffuson fler as shown n Fgure. The process of a Taubn fler can be defned by he followng erave procedure: M M = M λ M. (4) = M µ M λ µ M M Fgure. The Taubn dffuson fler. Fgure llusraes how a Taubn fler works. In Fgure, λ and µ are wo prese consans wh one posve and he oher negave. The λ fler s a Gaussan dffuson fler whch enforces he dffuson drecon o make he model shrnk a lle b. The µ fler, on he oher hand, s a Gaussan dffuson fler whch enforces he dffuson drecon o make he model expand a lle b. Wh a proper selecon of λ and µ, he Taubn dffuson fler can funcon effcenly o remove noses whle reanng he orgnal mesh daa as much as possble. In order o acheve beer convergence resul, Desbrun e al. [Des99] proposed an mplc negraon fler desgn approach. The erave equaons of her approach are as follows: M = M λ M or ( λ ) M = M.(5) M

3 λ M M Fgure 3. The mplc fler. Fgure 3 llusraes how Desbrun e al. s approach funcons. Equaon (5) s bascally a represenaon of mplc sysem equaons. Under hese crcumsances, f one would lke o derve M from M, a numercal mehod usng sparse marx nverson or pre-condonal b-conjugae graden s requred o solve he equaons. I s known ha he above menoned numercal mehods wll evenually brng o a sable soluon, bu hey are me-consumng. Desbrun e al. [Des99] poned ou ha he umbrella operaor s napproprae n dealng wh nonsymmerc polygon mesh. Ths s because he verex of a non-symmerc polygon mesh wll shf away from s orgnal poson when encounerng a dffuson operaon. Therefore, hey proposed o use a mean curvaure flow operaor o replace an umbrella operaor. The mean curvaure flow operaor s defned as follows: c ( x ) = κ H n, (6) where s he mean curvaure flow operaor, c x s he verex subjeced o he mean curvaure flow operaon, κ s he mean curvaure on he verex H x, and n s he un normal vecor on x. Usually, he verex x of a sragh plane wll say seady when encounerng a mean curvaure flow operaon. As a consequence, Equaon (5) can be rewren as: ( λ c ) M = M, (7) and he fler desgn of he mplc curvaure flow sysem can be modfed and shown n Fgure 4. λ c M M Fgure 4. The mplc curvaure flow sysem. The flers usng Gaussan dffuson, Taubn dffuson, mean curvaure flow dffuson, or her combnaons are classfed as soropc dffuson-ype flers. The facor λ remans consan n an soropc dffuson operaon, regardless of dffuson drecon. An soropc dffuson operaon can elmnae nose very effecvely bu also smooh ou useful daa. In [Des99], Desbrun e al. proposed o execue fler desgn based on ansoropc dffuson. An ansoropc fler uses a funcon of wo prncpal curvaures, κ and κ, as he weghs for each dffuson drecon. The operaon of an ansoropc fler can be represened graphcally as shown n Fgure 5, and he equaon ha descrbes hs operaon s as follows: M = M c κ, κ ) c M, (8) w( w κ, κ ) ( M M Fgure 5. An ansoropc fler. where w ( κ, κ ) s an ansoropc dffuson weghng funcon. Wh he proper desgn of w ( κ, κ ) n accordance wh dffuson drecons, he desgned ansoropc fler can effecvely remove he nose and a he same me preserve he shape of corners and edges. Accordng o he fler desgn rule descrbed n [Mey0], f he values of κ and κ of a verex are boh smaller han a prese hreshold, hen he verex s regarded as a nosy verex. Under hese crcumsances, he nose assocaed wh he verex can be removed effecvely by a Gaussan dffuson fler or an ansoropc fler f w ( κ, κ ) s se o. If he absolue values of boh prncpal curvaures are larger han a prese hreshold value, hen he verex s regarded as a corner verex and s remaned nac by seng w ( κ, κ ) equal o 0. A verex s consdered an edge verex f s mnmum curvaure s very small and s mean curvaure very large. To deermne he value of w ( κ, κ ), he followng rules wll be used: w( κ, κ ) = f κ T and κ T 0 f κ > T and κ > T and κ κ > 0. (9) κ κ H f κ = mn( κ, κ, κ H ) κ κ H f κ = mn( κ, κ, κ H ) f κ H = mn( κ, κ, κ H ) In addon o he defnon of κ and κ whch has been made prevously, κ H here represens he mean curvaure of κ and κ. Fleshman e al. [Fle03] exended he blaeral flerng mehod from D mage o 3D polygon meshes. Ther mehod adoped wo sandard Gaussan flers, one s smoohng funcon and he oher s smlary funcon. The smlary funcon denfes he degree

4 of smlary. If a verex has hgh smlary wh s neghborng verces, hen wll be smoohed whou any hesaon; oherwse, wll be denfed as a feaure verex and wll no be smoohed. By unng up smoohng and smlary funcons, he blaeral flerng mehod wll produce a feaure preservng resul for sem-regular polygon meshes. The underlyng concep of ansoropc dffuson s from he hea ransfer heory, where a zero wegh corresponds o an nsulaor and a full wegh corresponds o a perfec conducor. In wha follows, we shall propose a new ansoropc dffuson-based fler o execue he farng process of 3D polygon models. 3. FILTER DESIGN BASED ON ADAPTIVE ANISOTROPIC DIFFU- SION In he prevous secon, we have surveyed a number of soropc and ansoropc dffuson flers whch can be appled o smooh 3D polygon meshes. In hs paper, we propose a b-dreconal curvaure mappng funcon, w ( κ, κ ), based on he hea ransfer heory. A hea ransfer expresson s commonly used n nonlnear flerng [We97]. Le κ be he maxmum curvaure and κ be he mnmum curvaure. Assume he relaon κ κ always holds. The weghng funcon w ( κ, κ ) of a b-dreconal curvaure mappng can be expressed as follows: w( κ, κ ) = 0 ( k / k ) e e ((T k ) / k ) e e f κ f κ f κ f κ (0) T and κ κ > T > T and κ T T > T and T < κ T Fgure 6. The b-dreconal curvaure map wh T=5. In addon o he new weghng funcon, we also propose a double degenerae hea equaon [Kob98] o govern he ansoropc dffuson-based fler. The equaon showng how hs operaon works s as follows: M = M c c [ w( κ, κ ) M ] () c w κ, κ ) ( c M M Fgure 7. The double degenerae ansoropc dffuson-based fler. Fgure 7 llusraes graphcally how he wo degenerae hea equaons are ncorporaed no he desgn of he fler. An ansoropc dffuson fler usng he b-dreconal curvaure mappng funcon defned n Equaon (0) works well f, a each farng sep, he hreshold value T can be properly deermned. The hreshold T can be used o judge he verex ype n he farng process. Havng he ype of a verex, he farng algorhm rggers an approprae process o smooh ou he verex. Therefore, he selecon of an approprae T s of grea mporance o he success of a farng process. To he bes of our knowledge, mos exsng ansoropc dffuson mehods requre user o provde a hreshold value for a farng process. However, hs hreshold value s daa dependen (ll-posed) and dfferen hreshold values may resul n dfferen farng resuls. Therefore, an auomac hreshold selecon procedure ha can adapvely decde an approprae hreshold for dfferen mesh models s always preferable. In hs paper, we propose a mehod whch can adapvely selec an approprae hreshold value ha wll f n any gven 3D polygon model. The mehod s based on Bayesan classfcaon whch s commonly used n he feld of paern recognon [Sch9]. In Secon 3., we shall descrbe how o sysemacally separae he feaure verces and nonfeaure verces of a 3D polygon model. Secon 3. wll dscuss how o dsngush he edge and corner verces from he se of feaure verces. The convergence es descrbed n Secon 3.3 dscusses how o sablze our algorhms. 3. Verex Classfcaon For a 3D polygon model, edge and corner verces are boh classfed as feaure verces [Mey0]. The dfference beween an edge verex and a corner verex can be judged by he magnudes of her wo prncpal curvaures. For an edge verex, he magnude of one of s prncpal curvaures s small and he oher s que large. For a corner verex, he magnudes of s boh prncpal curvaures are large. Therefore, f one would lke o dsngush feaure verces from non-feaure verces, he magnude of he maxmum prncpal curvaure can be used as a good ndcaor. For hose non-feaure verces, he magnude of her maxmum prncpal curvaure s small.

5 For dsngushng feaure verces from nonfeaure verces, we assume ha non-feaure verces and feaure verces of a gven 3-D model are normal dsrbued along he maxmum prncpal curvaure axs. Le µ and µ be he mean values for he wo classes, σ and σ be wo correspondng sandard devaons. Then, by applyng Bayesan classfcaon rule [Sch9], he bes hreshold value can be deermned by he boundary lne ha bes dvdes he wo classes wh he followng propery: p x w ) = p( x ), () ( w where w and w are he wo classes and x s he bes hreshold value. Under hese crcumsances, he bes hreshold value can be derved by solvng equaons (), (3), and (4), where equaons (3) and (4) are as follows: x µ p ( x w ) = exp( ( ) ), (3) πσ σ x µ p ( x w ) = exp( ( ) ). (4) π σ σ The bes soluon for he hreshold value s hus obaned as follows: b c x =, (5) a where a = σ σ, b = ( σ µ σ µ ), c = b 4ad, and σ d = σ µ σ µ σ σ ln( ). σ Fgure 8. The dashed lne shows where he bes hreshold s locaed by a Bayesan classfer. Snce he classfcaon process s auomacally performed on every gven 3D model, he number of classes s an mporan ssue. For a 3D shape such as a sphere, here s no edge or corner on s surface. As a resul, here s only one class for a sphere. Therefore, before he classfcaon ask s execued, we have o check every 3D model o see f has a leas wo separable classes. In order o do so, we defne he followng funcon: D( µ, σ, h) = ( µ h) / σ, (6) where h s he hreshold value derved from equaon (5). If boh values of D( µ, σ, h) and D( µ, σ, h) are less han a prese δ value, hen he verces of a gven 3D polygon model are no separable and all of hem are regarded as non-feaure verces. For he res of he paper, we use δ = n all examples. Tha s, here exss only one non-separable class f he dsance beween wo calculaed means, µ and µ, s less han σ σ. 3. Edge and Corner verces The reason why we have o perform feaure and non-feaure verex classfcaon s because hey wll be processed dfferenly n he subsequen seps of our algorhm. A verex s regarded as a poenal edge verex f has a leas wo neghborng feaure verces n s -rng, where -rng of a verex s he collecon of common planar verces of he verex. If a feaure verex has a mnmum curvaure, whch s larger han he hreshold value obaned from equaon (5), and has a leas hree neghborng poenal feaure verces n s -rng, hen s regarded as a poenal corner verex. If a verex s no of one of he above ypes, hen s classfed as a non-feaure verex. The formal algorhm ha can be appled o deec edge, corner, or non-feaure verces s as follows: Corner verex: If k > hreshold and k > hreshold and here are a leas 3 verces n -rng neghbor wh her k > hreshold. Edge verex: If k > hreshold and here are a leas verces n -rng neghbor wh her k > hreshold. Non-feaure verex: None of he above wo ypes. Here k and k represen he maxmum and he mnmum prncpal curvaure, respecvely, of an arbrary verex on a 3D model. Fgure 9 llusraes an example showng how he feaure verex classfcaon algorhm works. The lef hand sde of Fgure 9 shows a fandsk model. On s rgh s he resul afer performng feaure verex classfcaon. Fgure 9. On he lef s a gven fandsk model; On he rgh shows he calculaed feaure verces (n dark color).

6 3.3 Convergence Tes A verex s sad o reach a seady sae f he change of s curvaure values n wo consecuve eraons becomes neglgble. Whenever a verex reaches s seady sae, s poson should be remaned nac n he subsequen eraons. Thus, we need a convergence es for each verex o deermne wheher a verex has reached s seady sae. Therefore, we conduc a convergence es n whch we calculae he oal curvaure and hen use o es he convergence of non-feaure verex. On he oher hand, we use he mnmum prncpal curvaure of every verex o es he convergence of feaure verex. The funcon ha s adaped o he convergence es for a verex a he p-h eraon s defned as follows: p C ( v) ( c ( k p c < ε ) ( c p = p p k < ε) ( p k p < ε ) < ε ) f v { F} (7) f v { F} where he superscrp p ndcaes he p-h eraon and he oal curvaure c can be calculaed by k k, { F } s he se of feaure verces, and ε s a prese hreshold value. The value of ε has o be small and we se 0.0 n our expermens. Wh he eraon goes on, we say a verex s converged or has reached o s seady sae f passes he convergence es. When a verex reaches s seady sae, he correspondng feaure value remans nac n he subsequen eraon seps. I s noed ha he convergence rae of each verex s dfferen. Therefore, he number of verces ha reaches o seady sae wll decrease gradually as he erave process goes on. The smoohng process for polygon meshes s compleed when all verces reach her seady sae. Fgure 0 llusraes he flow char of he proposed adapve ansoropc dffuson process. 4. EXPERIMENTAL RESULTS A seres of expermens was conduced o verfy he effecveness of he proposed mehod. In he frs se of expermens, we conduced expermens on wo ses of non-separable models. Fgure (a) shows a nosy sphere and s correspondng prncpal curvaure dsrbuon. Snce he dsrbuon was non-separable, our algorhm rggered an soropc dffuson process o smooh ou he sphere. The resul s shown n Fgure (b). Fgure s anoher example. The dsrbuons of he model aken from dfferen vews are shown n Fgure (a) and (c) were non-separable. Afer applyng an soropc dffuson process, he resuls are shown n Fgure (b) and (d), respecvely. As o he case when our algorhm s facng a separable 3D model, we have obaned he followng resuls. Fgure 3(a) shows a nosy 3D polygon model. The resuls obaned by applyng he proposed adapve dffuson process, he ansoropc mean curvaure mehod, he blaeral flerng mehod and he Gaussan smoohng, are shown n Fgure 3(b), (c), (d) and (e), respecvely. I s obvous ha our algorhm was powerful n erms of preservng edges and corners. The ansoropc mean curvaure mehod was able preserve edges bu fal o rean corners. The reason ha led o hs falure was due o he use of fxed hreshold value. The blaeral mehod was able o preserve edge feaures bu also damage some corner and edge verces. Ths s due o he use of a fxed coeffcen value o deermne he smlary degree for all verces. As o he Gaussan flerng, faled o preserve boh he edges and corners. Ths oucome s predcable because a Gaussan fler always flers ou boh nose and useful daa. In Fgure 4, we show several prncpal curvaure dsrbuons correspondng o dfferen eraon numbers. Fgure 4(a) shows he orgnal prncpal curvaure dsrbuon before our algorhm was appled o he 3D model shown n Fgure 3(a). Fgure 4(b) shows he dsrbuons afer our algorhm was appled 5 eraons. Fnally, Fgure 4(c) shows he fnal dsrbuon afer our algorhm compleed s job (6 eraons). In he las se of expermens, we esed our algorhm agans some nosy 3D models. Fgure 5(a) shows a fandsk model. Fgure 5(b) shows a corruped model wh 0.3% Gaussan nose. Fgure 5(c) and (d) show, respecvely, he resuls afer applyng our adapve smoohng algorhm and he Gaussan model. I s apparen ha our algorhm could well preserve boh edges and corners n hs case. Fgure 6(a) s a golf mesh model. Fgure 6(b) shows % of Gaussan nose was added o he orgnal model. The resul obaned afer applyng our algorhm s shown n fgure 6(c). 5. CONCLUSIONS In hs paper, we have proposed an ansoropc dffuson-based mehod o smooh 3D polygon meshes. We used a b-dreconal curvaure mappng funcon whch s based on he hea ransfer heory o deal wh dfferen ypes of verces. In order o dsngush he verex ype, we perform Bayesan classfcaon o separae all verces of a 3D model no feaure and non-feaure verces. Then an energybased convergence es funcon was used o check wheher each verex has reached s seady sae or no. Expermenal resuls show ha our approach was ndeed powerful n smoohng dfferen 3D polygon models.

7 6. REFERENCES [Des99] M. Desbrun, M. Meyer, P. Schroder, and A. H. Barr. Implc Farng of Irregular Meshes usng Dffuson and Curvaure Flow. SIG- GRAPH 99 Conference Proceengs, 37 34, 999. [Fle03] S. Fleshman, I. Dror, and D. Cohen-Or. Blaeral Mesh Denosng. SIGGRAPH 03 Conference Proceedngs, , 003. [Kob98] L. Kobbel, S. Campagna, T. Vorsaz, and H. P. Sedel. Ineracve mulresoluon modelng on arbary meshes. SIGGRAPH 98 Conference Proceengs, 05 4, 998. [Mey0] M. Meyer, M. Desbrun, P. Schroder, and A. H. Barr. Dscree Dfferenal-Geomery Operaors for Trangulaed -Manfolds, Vsualzaon and Mahemacs Proceedngs, May, 00. [Sch9] R. J. Schalkoff, Paern Recognon: Sascal, Srucural and Neural Approaches. John Wley & Sons. press 99. [Tau96] G. Taubn. Opmal Surface Smoohng as Fler Desgn. Research Repor RC-0404, IBM Research,. March 996. [We97] J. Wecker, A Revew of Nonlnear Dffuson Flerng, n Scale-space Theory for Compuer Vson, Lecure Noes n Compuer Scence, Vol. 5, Barer Haar Romeny, Ed., 3-8, Sprnger, New York, 997. No 3D nosy model Curvaure analyss Can verces be separaed no wo classes? Yes Verces classfcaon Ansoropc dffuson Do all verces converge? No Isoropc dffuson Fgure 0. The adapve dffuson flerng process. Yes Dsplay he smoohed 3D model (a) (b) Fgure. (a) A nosy sphere model and s prncpal curvaure dsrbuon, (b) he resul obaned afer applyng our adapve dffuson algorhm. (a) (b) (c) (d) Fgure. Expermenal resuls of anoher model. (a) and (c) are orgnal model aken from dfferen vews, (b) and (d) are he resuls obaned afer smoohng.

8 (a) (b) (c) (d) (e) Fgure3. (a) Orgnal polygon model, and (b) he resul of he proposed adapve dffuson, (c) he resul of he ansoropc mean curvaure flow, (d) he resul of blaeral flerng, and (e) he resul of Gaussan smoohng. (a) (b) (c) Fgure 4. The prncpal curvaure dsrbuons of he nosy model shown n Fgure 3. (a) orgnal daa, (b) applyng adapve dffuson 5 eraons, (c) smoohng procedure fnshed. (a) (b) (c) (d) Fgure 5. Expermens on a Fandsk model. (a) orgnal polygon meshes, (b) he model wh 0.3% Gaussan nose, and (c) (d) he resuls obaned by applyng our adapve dffuson mehod and Gaussan smoohng mehod, respecvely. (a) (b) (c) Fgure 6. Our adapve dffuson s appled o a golf mesh model, (a) orgnal meshes, (b) add % Gaussan nose, (c) he flered resul.