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1 ISSN , CODEN RUXUEW Joural of Software, Vol8, No2, February 2007, pp http//wwwosorgc DOI 0360/os80442 Tel/Fax by Joural of Software All rights reserved,2+,,, (, 006) 2 (,00049) A Fittig Algorithm of Subdivisio Surface from Noisig ad Dese Triagular Meshes WU Jia-Huag,2+, LIU Wei-Ju, WANG Tia-Ra, ZHAO Ji-Bi (Laboratory of Advaced Maufacture Techology, Sheyag Istitute of Automatio, The Chiese Academy of Scieces, Sheyag 006, Chia) 2 (Graduate School, The Chiese Academy of Scieces, Beiig 00049, Chia) + Correspodig author wuh@siac Wu JH, Liu WJ, Wag TR, Zhao JB A fittig algorithm of subdivisio surface from oisig ad dese triagular meshes Joural of Software, 2007,8(2) http//wwwosorgc/ /8/442htm Abstract A fittig system is developed to fit subdivisio surface with sharp feature from oisig ad dese triagular meshes of arbitrary topology The system icludes a improved mesh deoisig method based o bilateral filterig of images, sharp features extractio, feature-preservig mesh simplificatio ad topological optimizatio A estimatig method for Loop subdivisio surface is itroduced to predict how may subdivisio iteratios are ecessary to meet a user-defied tolerace The method of adaptive subdivisio is proposed durig the fittig process to hadle local detailed surface features Both experimetal results ad practical applicatios i egieerig demostrate that the system ca effectively achieve a good quality of the fittig subdivisio surface with ice details while usig few facets i the approximatio Key words subdivisio surface; mesh deoisig; adaptive subdivisio; depth estimatig; surface fittig,,,, 3D ; ; ; TP39 A [] Catmull-Clar Supported by the Natural Sciece Foudatio of Liaoig Provice of Chia uder Grat NosL05020, L0502 ( ) Received ; Accepted

2 443 Doo-Sabi B, B [], Loop [2],Butterfly,Modified Butterfly,Kobbelt,, MPEG4 AFX(aimatio framewor extesio), CT,, [,3] Hoppe [4] Loop [2] Hoppe Loop,,,, Suzui [5],,,,, Ma Zhao [6] [7] Catmull-Clar CT,, Catmull-Clar,Ma [8,9] Hoppe [4] Loop, (quadric error metrics, QEM) [0,],,,,,, [2] [8,9],, [4 6,8,9],,,,,,,,,,,, ), [8,9,2],, ;2),, ;3),, [8,9] ;4),,,,,, ; ;,,, (mas), Loop

3 444 Joural of Software Vol8, No2, February 2007 Dese triagular meshes Feature extractio, deoisig Simplificatio, optimizatio Subdivisio surfaces Subdividig, fittig Fig Overview of our method Loop Loop, Charles Loop [2], (box splie),, Loop,,, 4,, 4 ;,,Hoppe [4] Schweitzer [3] Loop, [8,9,2], Hoppe Loop,,, [4] ;,,,,, Loop [,4,8,3], 2,,,,, [5,6] Taubi [7] Desbru [8], Joes [9] Fleishma [20],,Fleishma, Fleishma,, () v ew v old +(v old )λ (),(v) ( ),λ (), Fleishma, - ( ),, -,, -,,,, [6] -, Fleishma m(t),, ξ(t,s) m( T ) = S N ( T ) ξ ( T, S) ( S) ξ(t,s)=a(s)exp{ (/θ) 2 },=ϕ/d (3) (2)

4 445,A(S) S,θ,N(T) T -,d T S 2 Fleishma Taubi Fleishma Fleishma, ;Taubi, ; Fleishma, (a) The Staford buy is artificially corrupted by Gaussia oise (a) Staford (b) Smoothed by 8 iteratios of Taubi λ µ with λ=06307, µ= (b) Taubi λ µ (λ=06307,µ= 06732),8 (c) Smoothed by 5 iteratios by the method of Fleishma (c) Fleishma,5 Fig2 2 (d) Smoothed by 3 iteratios of our improved Fleishma s method (d) Fleishma,3 3, [0],, [8,9,2], Garlad [0] (QEM),, v f Q f (v) Q f (v)=d 2 =( T v+d) 2 =v T ( T )v+2d T v+d 2 (4), f ax+by+cz+d=0, f =(a,b,c) T v Q(v), T T Q( v) = Q f ( v) = v Av + 2b v + λ f N ( T, v),q(v) (A,b,λ),A= T,b=d,λ=d 2, v, v Q(v), (5), Q(v) Q(v)=2Av+2b (6) Q(v) 0, v mi = A b ;A, v e, Q(v), Loop (5)

5 446 Joural of Software Vol8, No2, February 2007,,,,, ;,, ;,, [8,9] Q( v) = Mi ( Q( v )) vi Nv ( v) i Q( v) = Mi ( Q( vi )),,{va,v b } v { v, v } i a b,, ;, ;,,,;, [0], [8,9],,,, [4],, ; 4,,,, [8,9,2,2] [8,9] ~2,, ;, [2], [2,22],,,,,,,,,, 4,, [23,24] Loop, [23,25] Stam [7,26], Loop, - [26], s(u,w)=c T b(u,w), (u,w) Ω (7) Ω,Ω={(u,w) u [0,],w [0, u]};b(u,w) ;C T -,, f s ξ = f ( u, w) s( u, w) (8) f ( u, w) Ω,ξ f,

6 447 ϑ ξ [23] f ϑ = vi vi v i ;V ; [2] 3 v i, a = β +, ϑ = vi vi vi V +, +, ϑ v vi V = ( a) v + a 5 3 β = cos 4 v = 0 2π 2 = + vi ( a) vi a v v V = 0 ϑ v i 3,>3; β =,=3 6 = avi a v i vi V = 0 + vi V + i = 0 + (9) v i, (0) () ϑ = av a v (2), [23,24],,Loop (2), ϑ = avi a v vi V = 0 = 5 β avi a v vi V = 0 β 5, β > 0 = a ( β ) vi + β vi V 5 8 = 0 v a = 0 3 ( vi + v ) + ( v v β ϑ (3) 5, = 3 β = 6 (4) cos(2π / ), 3 8 4, (3) ϑ 5 β ϑ β + ) 5 0 ϑ (5), ε, (5),, 5 0 β ϑ < ε, 5 0, ρ = β, σ = ϑ / ε 42 log ρ σ (6), [8,9,2],,, Loop, 4,, 6,, ;,,,,, ;,

7 448 Joural of Software Vol8, No2, February 2007 ( ),, ( ) ( ),,,,,,, ( ),,,, [27 30] [30],, [30],,, ;,,,,, 3(a), 3,, ; [30],, 3(a),, ;,,,, (a) (b) (c) (d) (e) Fig3 Results of adaptive subdivisio o Buy ad David 3 Buy David,,,, [3], ; 3, [30],, ;, ; 3,,,, [30] 3(b) (, ); 3(c) 3(b), 4 (6) ; 3(e) 3(d),, ;,,,

8 449, [8,9,2],, v, [8,9,2] v v v v v v, t,t 2 t t 2 t Loop [2,4,3] 2π v, = cos t = 2 v 0 = 0 t = 2π si (7),v v - [2,4,3],, v, ;, v, v,, [8,9,2] 5 5 Π, Π 2 Π 2, Π,Π 2 X,X 2, Χ 2 Π 2,, M 2 Χ =M2X [2,2,3] 2, M, Χ = 2 Π 2 Χ 2 2 ΜΧ 2 Π 2 Π 2,, Π 2 52 MM 2 X 2 =X (8) Visual C++60 OpeGL, 3M Widows XP,P4-8GHz,256MB,64M Geforec4MX440 3D,,, [8,9,2] (a),, 4(b) 4(c) Fleishma 3, [8,2], ( ), , (d), , 7 654, [8,9] , ; [8,9] 6963%, 805%,,,, [8,2],, 4,,, 4(e) 4(f) [8,9]

9 450 Joural of Software Vol8, No2, February 2007, Desbru [8] [8,9], 4(h) 4(g),4(h) [8,9],4(f) 4(f) 4(g), [8,9], ;, (a) (b) (c) (d) (e) Fig4 (f) Results of our subdivisio surface fittig method (top row) ad comparisos betwee our method ad the method i Ref[8,9] (bottom row) (g) 4 ( ) [8,9] ( ) 5 Mechpart 5(a), ;5(b) 5(a) ;5(c) Loop ;5(d) 5(e) 5(f) [8,9,2] [2],, (5(e)) [8,9](5(d)),, 5(f), (h) (a) (b) (c) (d) (e) (f) Fig5 Results ad comparisos of our fittig method betwee the methods i Ref[8,2] o the Mechpart model 5 [8,2] Mechpart [32] MESH(measurig error betwee surfaces usig the Hausdorff distace), Hausdorff,,,,,,,

10 45 Table Comparisos of our method ad the methods i Ref[8,2] o some parameters for models i Fig4 ad Fig5 [8,2] 4 5 Model Methods Vertices Faces Ratio (%) Mi Max Mea RMS Fig4 Ref[8,9] Our method Fig5 Ref[2] Our method Note Mi,, mea ad RMS mea miimum error, imum error, mea error ad root mea square error respectively 6, Fleishma,,,,,,,,,, Loop,,3, Refereces [] Zori D, Schroder P Subdivisio for modelig ad aimatio I SIGGRAPH 2000 Course Notes, Course 23 New Orleas ACM SIGGRAPH, http//wwwmultirescaltechedu [2] Loop C Smooth subdivisio surfaces based o triagles [MS Thesis] Utah Uiversity of Utah, 987 [3] Wag GJ, Wag GZ, Zheg JM Computer Aided Geometric Desig Beiig Chia Higher Educatio Press; Berli Spriger-Verlag, (i Chiese) [4] Hoppe H, DeRose T, Duchamp T, Halstead M, Ji H, McDoald J, Schweitzer J, Stueltzle W Piecewise smooth surface recostructio I Proc of the SIGGRAPH 994 New Yor ACM Press, [5] Suzui H, Taeuchi S, Kaai T Subdivisio surface fittig to a rage of poits I Proc of the 7th Pacific Cof o Graphics ad Applicatios (Pacific Graphics 99) Washigto IEEE Computer Society, [6] Ma W, Zhao N Catmull-Clar surface fittig for reverse egieerig applicatios I Proc of the Geometric Modelig ad Processig 2000 Washigto IEEE Computer Society, [7] Stam J Exact evaluatio of Catmull-Clar subdivisio surfaces at arbitrary parameter values I Proc of SIGGRAPH 998, Aual Cof Series New Yor ACM Press, [8] Ma W, Ma X, Tso S, Pa Z A direct approach for subdivisio surface fittig from a dese triagular mesh Computer Aided Desig, 2004,36(6) [9] Ma W, Ma X, Tso S, Pa Z Subdivisio surface fittig from a dese triagle mesh I Proc of the Geometric Modelig ad Processig -Theory ad Applicatios (GMP 2002), Washigto IEEE Computer Society, [0] Garlad M, Hecbert PS Surface simplificatio usig quadric error metrics I Proc of the SIGGRAPH 97 New Yor ACM Press, [] Taeuchi S, Kaai T, Suzui H, Shimada K, Kimura F Subdivisio surface fittig with QEM-based mesh simplificatio ad recostructio of approximated B-splie surfaces I Proc of the 8th Pacific Cof o Graphics ad Applicatios (Pacific Graphics 2000) Washigto IEEE Computer Society, [2] Li G Ma W, Bao H, Fittig system usig loop subdivisio surfaces with sharp features Joural of Computer-Aided Desig ad Computer Graphics, 2005,7(6)79 85 (i Chiese with Eglish abstract) [3] Schweitzer JE Aalysis ad applicatio of subdivisio surface [PhD Thesis] Washigto Uiversity of Washigto, 996 [4] Halstead M, Kass M, DeRose T Efficiet, fair iterpolatio usig Catmull-Clar surfaces I Cuigham S, ed Proc of the

11 452 Joural of Software Vol8, No2, February 2007 SIGGRAPH 93 New Yor ACM Press, [5] Vollmer J, Mecl R, Muller H Improved Laplacia smoothig of oisy surface meshes Computer Graphics Forum, 999,8(3) 3 38 [6] Yu O, Belyaev AG, Bogaevsi I Mesh regularizatio ad adaptive smoothig Computer-Aided Desig, 200,33() [7] Taubi G A sigal processig approach to fair surface desig I Proc of the SIGGRAPH 95 New Yor ACM Press, [8] Desbru M, Meyer M, Schroder P, Barr AH Implicit fairig of irregular meshes usig diffusio ad curvature flow I Rocwood A, ed Proc of the SIGGRAPH 99 New Yor ACM Press, [9] Joes T, Durad F, Desbru M No-Iterative, feature preservig mesh smoothig I Proc of the SIGGRAPH 2003 New Yor ACM Press, [20] Fleishma S, Drori ID, Cohe-Or D Bilateral mesh deoisig I Proc of the SIGGRAPH 2003 New Yor ACM Press, [2] Lee A, Moreto H, Hoppe H Displaced subdivisio surfaces I Aeley K, ed Proc of the SIGGRAPH 2000 New Yor ACM Press, [22] Mogolam P, Razda A, Fari G Lossy 3D mesh compressio usig loop scheme I Hamza MH, ed Proc of Computers Graphics ad Imagig 2003 Calgary ACTA Press, [23] Cheg F, Yog J Subdivisio depth computatio for Catmull-Clar subdivisio surfaces 2003 http//csegruyedu/~cheg/ publhtml [24] Cheg F Estimatig subdivisio depths for ratioal curves ad surfaces ACM Tras o Graphics, 992,(2)40 5 [25] Laqueti S, Neveu M A priori computatio of the umber of surface subdivisio levels 2004 http//wwwgraphicoru/2004/ Proceedigs/Techical/3%5B3%5Dpdf [26] Stam J Evaluatio of loop subdivisio surfaces 999 http//wwwdgptorotoedu/people/stam/reality/research/pdf/sig98pdf [27] Li S, He J, Guo Y, Kog B Butterfly adaptive subdivisio algorithm Joural of Computatioal Iformatio Systems, 2005,() [28] Yog JH, Cheg FH Adaptive subdivisio of Catmull-Clar subdivisio surfaces Computer-Aided Desig ad Applicatios, 2005, 2(-4) [29] Settgast V, Muller K, Fufzig C, Feller D Adaptive tesselatio of subdivisio surfaces Computers ad Graphics, 2004,28() [30] Amresh A, Fari G, Razda A Adaptive subdivisio schemes for triagular meshes I Fari G, Hage H, Hama B, eds Hierarchical ad Geometric Methods i Scietific Visualizatio Berli Spriger-Verlag, [3] Seeger S, Horma K, Hausler G, Greier G A sub-atomic subdivisio approach I Ertl T, Girod B, eds Proc of the Visio Modelig ad Visualizatio Cof 200 (VMV 200) Berli Aa GmbH, [32] Aspert N, Sata-Cruz D, Ebrahimi T MESH Measurig error betwee surfaces usig the Hausdorff distace 2002 http//meshberliosde/ [3],, ;, [2],, Loop,2005,7(6)79 85 (980),,,,,CAD/ CAM (943),,,,,,, (969),,,,, CAD/CAM,, (970),,,,,CAD/CAM

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