To Do. Advanced Computer Graphics. Motivation. Mesh Data Structures. Outline. Mesh Data Structures. Desirable Characteristics 1

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1 Advancd Computr Graphics CSE 63 [Spring 207], Lctur 7 Ravi Ramamoorthi To Do Assignmnt, Du Apr 28 Any last minut issus or difficultis? Starting Gomtry Procssing Assignmnt 2 du May 9 This lctur starts discussing rlvant contnt Plas START EARLY. Can do most aftr this wk Contact us for difficultis, hlp finding partnrs tc. Motivation A polygon msh is a collction of triangls W want to do oprations on ths triangls E.g. walk across th msh for simplification Display for rndring Computational gomtry Bst rprsntations (msh data structurs)? Compactnss Gnrality Simplicity for computations Efficincy Dsirabl Charactristics Gnrality from most gnral to last Polygon soup Only triangls 2-manifold: 2 triangls pr dg Orintabl: consistnt CW / CCW winding Closd: no boundary Compact storag Msh Data Structurs Msh Data Structurs Dsirabl charactristics 2 Efficint support for oprations: Givn fac, find its vrtics Givn vrtx, find facs touching it Givn fac, find nighboring facs Givn vrtx, find nighboring vrtics Givn dg, find vrtics and facs it touchs Ths ar adjacncy oprations important in msh simplification (homwork), many othr applications Outlin Indpndnt facs Indxd fac st Adjacncy lists Wingd-dg Half-dg Ovrviw of msh dcimation and simplification

2 Indpndnt Facs Facs list vrtx coordinats Rdundant vrtics No topology information Indxd Fac St Facs list vrtx rfrncs shard vrtics Commonly usd (.g. OFF fil format itslf) Augmntd vrsions simpl for msh procssing F Fac Tabl : (x 0,y 0,z 0 ), (x,y,z ), (x 2,y 2,z 2 ) F : (x 3,y 3,z 3 ), (x 4,y 4,z 4 ), (x 5,y 5,z 5 ) : (x 6,y 6,z 6 ), (x 7,y 7,z 7 ), (x 8,y 8,z 8 ) F F 0 Vrtx Tabl v 0 : (x 0,y 0,z 0 ) v : (x,y,z ) : (x 2,y 2,z 2 ) v 3 : (x 3,y 3,z 3 ) : (x 4,y 4,z 4 ) Fac Tabl : 0,, 2 F :, 4, 2 :, 3, 4 v 0 v v 3 Not CCW ordring Storag fficincy? Indxd Fac St Which oprations supportd in O() tim? F v 0 v v 3 Vrtx Tabl v 0 : (x 0,y 0,z 0 ) v : (x,y,z ) : (x 2,y 2,z 2 ) v 3 : (x 3,y 3,z 3 ) : (x 4,y 4,z 4 ) Fac Tabl : 0,, 2 F :, 4, 2 :, 3, 4 Not CCW ordring Efficint Algorithm Dsign Can somtims dsign algorithms to compnsat for oprations not supportd by data structurs Exampl: pr-vrtx normals Avrag normal of facs touching ach vrtx With indxd fac st, vrtx è fac is O(n) Naiv algorithm for all vrtics: O(n 2 ) Can you think of an O(n) algorithm? Efficint Algorithm Dsign Can somtims dsign algorithms to compnsat for oprations not supportd by data structurs Exampl: pr-vrtx normals Avrag normal of facs touching ach vrtx With indxd fac st, vrtx è fac is O(n) Naiv algorithm for all vrtics: O(n 2 ) Can you think of an O(n) algorithm? Usful to augmnt with vrtx è fac adjacncy For all vrtics, find adjacnt facs as wll Can b implmntd whil simply looping ovr facs Outlin Indpndnt facs Indxd fac st Adjacncy lists Wingd-dg Half-dg Ovrviw of msh dcimation and simplification 2

3 Full Adjacncy Lists Full adjacncy: Issus Stor all vrtx, fac, and dg adjacncis v 0 0 v 3 v 3 Edg Adjacncy Tabl 0 : v 0, v ;,ø; ø, 2,,ø : v, ;,F ; 5, 0, 2, 6 Fac Adjacncy Tabl : v 0,v, ; F,ø,ø; 0, 2, F : v,, ; ø,, ; 6,, 5 : v,v 3, ; ø,f,ø; 4, 5, 3 Vrtx Adjacncy Tabl v 0 : v, ; ; 0, 2 v : v 3,,,v 0 ;,F, ; 3, 5,, 0 Garland and Hckbrt claim thy do this Easy to find stuff Issu is storag And updating vrything onc you do somthing lik an dg collaps for msh simplification I rcommnd you implmnt somthing simplr (lik indxd fac st plus vrtx to fac adjacncy) Partial Adjacncy Lists Partial Adjacncy Lists Stor som adjacncis, us to driv othrs Many possibilitis Edg Adjacncy Tabl 0 : v 0, v ;,ø; ø, 2,,ø : v, ;,F ; 5, 0, 2, 6 Som combinations only mak sns for closd manifolds Edg Adjacncy Tabl 0 : v 0, v ;,ø; ø, 2,,ø : v, ;,F ; 5, 0, 2, v 0 0 v 3 v 3 Fac Adjacncy Tabl : v 0,v, ; F,ø,ø; 0, 2, F : v,, ; ø,, ; 6,, 5 : v,v 3, ; ø,f,ø; 4, 5, 3 Vrtx Adjacncy Tabl v 0 : v, ; ; 0, 2 v : v 3,,,v 0 ;,F, ; 3, 5,, v 0 0 v 3 v 3 Fac Adjacncy Tabl : v 0,v, ; F,ø,ø; 0, 2, F : v,, ; ø,, ; 6,, 5 : v,v 3, ; ø,f,ø; 4, 5, 3 Vrtx Adjacncy Tabl v 0 : v, ; ; 0, 2 v : v 3,,,v 0 ;,F, ; 3, 5,, 0 Outlin Indpndnt facs Indxd fac st Adjacncy lists Wingd-dg Half-dg Ovrviw of msh dcimation and simplification Wingd, Half Edg Rprsntations Ida is to associat information with dgs Compact Storag Many oprations fficint Allow on to walk around msh Usually gnral for arbitrary polygons (not triangls) But implmntations can b complx with spcial cass rlativ to simpl indxd fac st++ or partial adjacncy tabl 3

4 Wingd Edg Wingd Edg Most data stord at dgs Vrtics, facs point to on dg ach v F 0 v 0 0 v 3 v 3 Edg Adjacncy Tabl 0 : v 0, v ;, ø; ø, 2,, ø : v, ;,F ; 5, 0, 2, 6 Fac Adjacncy Tabl : v 0,v, ; F, ø, ø; 0, 2, F : v,, ; ø,, ; 6,, 5 : v,v 3, ; ø,f, ø; 4, 5, 3 Vrtx Adjacncy Tabl v 0 : v, ; ; 0, 2 v : v 3,,,v 0 ;,F, ; 3, 5,, 0 Each dg stors 2 vrtics, 2 facs, 4 dgs fixd siz Enough information to compltly walk around facs or vrtics Think how to implmnt Walking around vrtx Finding nighborhood of fac Othr ops for simplification forw,lft F lft back,lft forw,right v nd F right v bgin back,right Half Edg Outlin Instad of singl dg, 2 dirctd half dgs Maks som oprations mor fficint Walk around fac vry asily (ach fac nd only stor on pointr) h nxt F lft h inv Indpndnt facs Indxd fac st Adjacncy lists Wingd-dg Half-dg v bgin Ovrviw of msh dcimation and simplification Msh Dcimation Msh Dcimation Triangls : 4,855 27,970 20,922 2,939 8,385 4,766 Rduc numbr of polygons Lss storag Fastr rndring Simplr manipulation Dsirabl proprtis Gnrality Efficincy Producs good approximation Division, Viwpoint, Cohn Michlanglo s St. Matthw Original modl: ~400M polygons 4

5 Primitiv Oprations Simplify modl a bit at a tim by rmoving a fw facs Rpatd to simplify whol msh Typs of oprations Vrtx clustr Vrtx rmov Edg collaps (main opration usd in assignmnt) Vrtx Clustr Mthod Mrg vrtics basd on proximity Triangls with rpatd vrtics can collaps to dgs or points Proprtis Gnral and robust Can b unattractiv if rsults in topology chang Vrtx Rmov Mthod Rmov vrtx and adjacnt facs Fill hol with nw triangls (rduction of 2) Proprtis Rquirs manifold surfac, prsrvs topology Typically mor attractiv Filling hol wll not always asy Edg Collaps Mthod Mrg two dg vrtics to on Dlt dgnrat triangls Proprtis Spcial cas of vrtx clustr Allows smooth transition Can chang topology Msh Dcimation/Simplification Typical: grdy algorithm Masur rror of possibl simpl oprations (primarily dg collapss) Plac oprations in quu according to rror Prform oprations in quu succssivly (dpnding on how much you want to simplify modl) Aftr ach opration, r-valuat rror mtrics Gomtric Error Mtrics Motivation Promot accurat 3D shap prsrvation Prsrv scrn-spac silhoutts and pixl covrag Typs Vrtx-Vrtx Distanc Vrtx-Plan Distanc Point-Surfac Distanc Surfac-Surfac Distanc 5

6 Vrtx-Vrtx Distanc E = max( v3 v, v3 v2 ) Appropriat during topology changs Rossignac and Borrl 93 Lubk and Erikson 97 Loos for topology-prsrving collapss Vrtx-Plan Distanc Stor st of plans with ach vrtx Error basd on distanc from vrtx to plans Whn vrtics ar mrgd, mrg sts Ronfard and Rossignac 96 Stor plan sts, comput max distanc Error Quadrics Garland and Hckbrt 97 Stor quadric form, comput sum of squard distancs v 3 d c d a v a b d b c Point-Surfac Distanc For ach original vrtx, find closst point on simplifid surfac Comput sum of squard distancs Surfac-Surfac Distanc Comput or approximat maximum distanc btwn input and simplifid surfacs Tolranc Volums - Guézic 96 Simplification Envlops - Cohn/Varshny 96 Hausdorff Distanc - Klin 96 Mapping Distanc - Bajaj/Schikor 96, Cohn t al. 97 Gomtric Error Obsrvations Vrtx-vrtx and vrtx-plan distanc Fast Low rror in practic, but not guarantd by mtric Surfac-surfac distanc Rquird for guarantd rror bounds Edg swap Msh Simplification Advancd Considrations Typ of input msh? Modifis topology? Continuous LOD? Spd vs. quality? vrtx-vrtx surfac-surfac 6

7 Viw-Dpndnt Simplification Simplify dynamically according to viwpoint Visibility Silhoutts Lighting Apparanc Prsrving 488 tris 975 tris,95 tris 3,905 tris Hopp 7,809 tris Caltch & Stanford Graphics Labs and Jonathan Cohn Summary Many msh data structurs Compact storag vs as, fficincy of us How fast and asy ar ky oprations Msh simplification Rduc siz of msh in fficint quality-prsrving way Basd on dg collapss mainly Choos appropriat msh data structur Efficint to updat, dg-collapss ar local Fairly modrn idas (last 5-20 yars) Think about som of it yourslf, s paprs givn out W will covr simplification, quadric mtrics nxt 7