Introduction. Basic idea of subdivision. History of subdivision schemes. Subdivision Schemes in Interactive Surface Design

Transcription

1 Subdvson Schemes n Interactve Surface Desgn Introducton Hstory of subdvson. What s subdvson? Why subdvson? Hstory of subdvson schemes Stage I: Create smooth curves from arbtrary mesh de Rham, 947. Chan, 974. Stage II: Generalze slnes to arbtrary toology Catmull and Clar,978. Doo and Sabn, 978. Stage III: Aled n hgh end anmaton ndustry Pxar Studo, Ger s Game,998. Stage IV: Aled n engneerng desgn and CAD Basc dea of subdvson Start from an ntal control olygon. Recursvely refne t by some rules. A smooth surface (curve) n the lmt. Chan s corner cuttng scheme Chan s corner cuttng scheme

2 Chan s corner cuttng scheme Chan s corner cuttng scheme Chan s Algorthm A set of control onts to defne a olygon ,, 2,..., n Subdvson rocess (more control vertces) Rules (corner chong) Proertes: quadratc B-slne curve, C contnuous, tangent to each edge at ts md-ont = = , 2,..., 2 n Chan s Algorthm Quadratc Slne Cubc Slne

3 Cubc Slne Subdvson rules C2 cubc B-slne curve Corner-chong No nterolaton = = ( ) Curve Interolaton Control onts Rules: At each stage, e ee all the OLD onts and nsert NEW onts n beteen the OLD ones Interolaton! The behavors and roertes of the lmt curve deend on the arameter Generalze to SIX-ont nterolatory scheme! = 0 = ( + )( 2 2 n 0 0 0,, 0 n + 2 2,..., 2 + n + + ) ( ), Curve Interolaton Other modelng rmtves Slne atches. Polygonal meshes. Slne atches Advantages: Hgh level control. Comact analytcal reresentatons. Dsadvantages: Dffcult to mantan and manage nter-atch smoothness constrants. Exensve trmmng needed to model features. Slo renderng for large models. Polygonal meshes Advantages: Very general. Can descrbe very fne detal accurately. Drect hardare mlementaton. Dsadvantages: Heavy eght reresentaton. A smlfcaton algorthm s alays needed.

4 Subdvson schemes Advantages: Arbtrary toology. Level of detal. Unfed reresentaton. Dsadvantages: Dffcult for analyss of roertes le smoothness and contnuty. Unform/Sem-unform Schemes Catmull-Clar scheme Catmull and Clar, CAD 978 Doo-Sabn scheme Doo and Sabn, CAD 978 Loo scheme Loo, Master s Thess, 987 Butterfly scheme Dyn, Gregory and Levn, ACM TOG 990. Md-edge scheme Habb and Warren, SIAM on Geometrrc Desgn 995 Kobbelt scheme Kobbelt, Eurograhcs 996 Classfcaton By Mesh tye: Trangular (Loo, Butterfly) Quadrlateral (Catmull-Clar, Doo-Sabn, Md-edge, Kobbelt) By Lmt surface: Aroxmatng (Catmull-Clar, Loo, Doo-Sabn, Md-edge) Interolatng (Butterfly, Kobbelt) By Refnement rule: Vertex nserton (Catmull-Clar, Loo, Butterfly, Kobbelt) Corner cuttng (Doo-Sabn, Md-edge) Catmull-Clar Scheme Face ont: the average of all the onts defnng the old face. Edge ont: the average of to old vertces and to ne face onts of the faces adjacent to the edge. Vertex ont: ( F + 2 E + ( n 3) V ) / n F: the average of the ne face onts of all faces adjacent to the old vertex. E: the average of the mdonts of all adjacent edges. V: the old vertex. Catmull-Clar Scheme Catmull-Clar Subdvson Intal mesh Ste Ste 2 Lmt surface

5 Catmull-Clar Subdvson Mdedge scheme Mdedge scheme (a) (b) Loo Scheme Box slnes A rojecton of 6D box onto 2D A quartc olynomal bass functon Trangular doman Non-tensor-roduct slnes Loo scheme results from a generalzaton of box slnes to arbtrary toology (c) (d) Box Slne Overve Bass Functons for Loo s Scheme Based on 2D Box Slne Defned by rojecton of hyercube (n 6D) nto 2D. Satsfes many roertes that B-slne has. Recursve defnton Partton of unty Truncated oer Natural slttng of a cube nto sub-cubes rovdes the subdvson rule. π 2 3 box B(,, 2, 3 ) x fbre π () N,, Bass Functon - Evaluaton Successve Subdvson Assgn unt eght to center, zero otherse, over Z 2 lattce The Lmt N 2,2,2 Bass

6 Loo s Scheme Proertes Loo s Scheme Rules Bass Functon Proertes. Suort 2 neghbors from the center 2. C 4 contnuty thn the suort 3. Pecese olynomal 4. N 2,2,2 ( - j), j Z 2 form a artton of unty.e. Σ N(x - j) = The Rules /8 3/8 /8 3/8 - Loo Scheme Rules Loo Scheme Examle /8 B B B 3/8 3/8 -nb /8 B B B = 3/8, for n>3 B = 3/6, for n=3 Butterfly Subdvson Butterfly Scheme

7 Modfed Butterfly Scheme Modfed Butterfly Examle Intal mesh One refnement ste To refnement stes Modelng Shar Features Corner Dart Crease Non-unform Subdvson Schemes Pecese smooth subdvson schemes Hoe et al. Sggrah 94 Hybrd scheme et al. Sggrah 98 NURSS scheme Sederburg et al. Sggrah 98 Combned scheme Levn Sggrah 99 Edge and vertex nserton scheme Habb et al. CAGD 99 Pecese Smooth Subdvson Hybrd Subdvson Scheme (a) (b) (a) (b) (c) (d) Hoe et al. Sggrah 94 (c) (d) DeRose et al. Sggrah 98

8 Herarchcal Edtng Surface Reconstructon Zorn et al. Sggrah 97 Hoe et al. Sggrah 94 Ger s Game DeRose et al. Sggrah 98 Subdvson Slnes We treat subdvson as a novel method to roduce slne-le models n the lmt Key comonents for slne models Control onts, bass functons over ther arametrc doman, arameterzaton, ecese decomoston Parameterzaton s done naturally va subdvson The ntal control mesh serves as the arametrc doman Bass functons are avalable for regular settngs as ell as rregular settngs Control onts for one atch are n the vcnty of ts arametrc doman from ts ntal control vertces Subdvson-based slne formulaton s fundamental for hyscs-based geometrc modelng and desgn, fnte element analyss, smulaton, and the entre CAD/CAM rocesses Chan Curve Examle Interolaton Curve Examle

9 Parameterzaton Butterfly Surface Examle Control Vertces for Butterfly Surface Control Vertces for Surface Patches Butterfly Patches Butterfly Bass Functon

10 Catmull-Clar Surface Examle Catmull-Clar Patches Catmull-Clar Bass Functon Smle Scultng Examles orgnal object deformaton cuttng extruson fxed regons Char Examle --- Fnte Element Smulaton Scultng Tools carvng extruson detal edtng jonng shar features deformaton Intal control lattce Fnte element structure after a fe subdvsons Deformed object Photo-realstc renderng

11 Scultng Tools nflaton deflaton curve-based desgn Scultng Tools ushng seeng curve-based jon materal mang materal robng hyscal ndo curve-based cuttng feature deformaton mult-face extruson Interactve Scultng More Examles Volume Edtng and Vsualzaton Sculted CAD Models orgnal lattce deformed lattce comressve forces orgnal volume dslacement mang deformed volume

12 Subdvson Solds Scenes and Scultures