To Do. Resources. Algorithm Outline. Simplifications. Advanced Computer Graphics (Fall 2010) Surface Simplification: Goals (Garland)

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1 danced Compuer Graphcs (Fall 2) CS 283, Lecure 7: Quadrc Error Mercs Ra Ramamoorh o Do ssgnmen, Due Oc 7. Should hae made some serous progress by end of week hs lecure reews quadrc error mercs and some pons regardng mplemenaon hp://ns.eecs.berkeley.edu/~cs283/fa Resources Garland and Heckber SIGGRPH 97 paper Garland webse, mplemenaon noes (n hess) Noes n hs and preous lecures Surface Smplfcaon: Goals (Garland) Effcency (7 o faces n 5s n 997) Hgh qualy, feaure preserng (prmary appearance emphaszed raher han opology) Generaly, non-manfold models, collapse dsjon regons Smplfcaons lgorhm Oulne Par conracons n addon o edge collapses Preously conneced regons may come ogeher

2 Quadrc Error Mercs Based on pon-o-plane dsance Beer qualy han pon-o-pon a d c b d b d a c Background: Compung Planes Each rangle n mesh has assocaed plane ax + by + cz + d For a rangle, fnd s (normalzed) normal usng cross producs B C n n - B C Plane equaon? æaö n b d - ç çèc ø Quadrc Error Mercs Sum of squared dsances from erex o planes: D å p Ds ( p, ) 2 æ xö æ aö y b, z p c ç è ø çè d ø Ds ( p, ) ax + by + cz + d p p Quadrc Error Mercs D å p å p ( p ) 2 pp æ ö ç åpp çè p ø Q Q pp Common mahemacal rck: quadrac form symmerc marx Q mulpled wce by a ecor Quadrc Error Mercs Smply a 4x4 symmerc marx Sorage effcen: floang pon numbers per erex Inally, error s for all erces Quadrc Error Mercs 2 nd degree polynomal n x, y and z Leel surface ( Q k) s a quadrc surface Ellpsod, parabolod, hyperbolod, plane ec. 2

3 Ellpsods: so-error surfaces Quadrc Vsualzaon Smaller ellpsod greaer error for a gen moon Lower error for moon parallel o surface Lower error n fla regons han a corners Elongaed n cylndrcal regons Usng Quadrcs pproxmae error of edge collapses Each erex has assocaed quadrc Q Error of collapsng and 2 o s Q + Q 2 Quadrc for new erex s Q Q +Q 2 Usng Quadrcs Fnd opmal locaon afer collapse: é q q q q ù ê ú q 2 q22 q23 q 24 Q ' q3 q23 q33 q 34 q4 q24 q34 q êë 44 úû mn ' Q ' ' : ' x y z Usng Quadrcs lgorhm Oulne Fnd opmal locaon afer collapse: é q q q q ù é ù ê ú ê ú q 2 q22 q23 q 24 ' q3 q23 q33 q 34 ê ú ê ú ê ú ê ú ë û ë û - é q q2 q3 q ù é 4 ù q 2 q22 q23 q 24 ' q3 q23 q33 q 34 ê ú ê ú ë û ë û 3

4 lgorhm Summary Fnal lgorhm Compue he Q marces for all he nal erces Selec all ald pars Compue he opmal conracon arge or each ald par. he error (Q +Q 2 ) of hs arge erex becomes he cos of conracng ha par Place all pars n a heap keyed on cos wh mnmum cos par on he op Ieraely remoe he leas cos par, conrac hs par, and updae he coss of all ald pars of neres Resuls Resuls Orgnal Quadrcs Orgnal Quadrcs k rs rs 25 rs 25 rs,, edge collapses only ddonal Deals Preserng boundares/dsconnues (wegh quadrcs by approprae penaly facors) Preenng mesh nerson (flppng of orenaon): compare normal of neghborng faces, before afer Has been modfed for many oher applcaons E.g. n slhouees, wan o make sure olume always ncreases, neer decreases ake color and exure no accoun (followup paper) See paper, oher more recen works for deals Implemenaon ps Incremenal, es, debug, smple cases Fnd good daa srucure for heap ec. May help o sualze error quadrcs f possble Challengng, bu hopefully rewardng assgnmen 4

5 Quesons? Issues or quesons? ll he maeral for assgnmen coered Sar early, work hard Res of un NURBs (oday f me perms) Subdson (nex lecure) Oulne Raonal Splnes Quadrac raonal splnes NURBs (brefly) Paramerc paches (brefly) Paramerc Polynomal Cures paramerc polynomal cure of order n: xu ( ) å au n danages of polynomal cures Easy o compue Infnely dfferenable eerywhere n yu ( ) å bu Raonal Splnes Can represen ceran shapes (e.g. crcles) wh pecewse polynomals Wder class of funcons: raonal funcons Rao of polynomals Can represen any quadrc (e.g. crcles) exacly Mahemacal rck: homogeneous coordnaes Rao of 2 polynomals n 3D equalen o sngle polynomal n 4D Raonal Splnes Sandard decaseljau Example: creang a crcular arc wh 3 conrol pons B (,) (,) (???) (,;) ( ) B (,) Polynomal splne: parabolc arc Raonal splne: crcular arc 5

6 Raonal decaseljau w B w B Homogeneous decaseljau w w w wb B wb wb ( ) wwbb ( ) w w B ( ) w wbb ( ) w w ( w ) wb B ( w ) wb B Non-raonal splnes smply hae all weghs se o nsead Oulne Raonal Splnes Quadrac raonal splnes NURBs (brefly) Paramerc paches (brefly) Ealuaon: Raonal Splnes dd an exra wegh coordnae Mulply sandard coords of conrol pon by wegh (essenally he same as he use of homogeneous coords) pply sandard decaseljau or oher ealuaon alg. Dde by fnal alue of wegh coordnae Essenally he same as perspece dson/dehomogenze Raonal because of fnal dson: Raonal polynomal a he end (rao of wo polynomals) Quadrac Bezer Cure Example: quarer-crcle arc, wegh mdpon Smaller w: pulled away from mddle conrol pon Larger w: pulled owards mddle conrol pon (,;) (,;) (,;w) (,;) Quadrac Bezer Cure General Bezer Formula for quadrac? ( ) w2 ( ) wbb wcc ( ) 2 ( ) wbb C ( ) w 2 ( ) w w ( ) 2 ( ) w B c B (,;) (,;) (,;w) (,;) Polynomal splne: parabolc arc Raonal splne: resul depends on w Polynomal splne: parabolc arc Raonal splne: resul depends on w 6

7 Dependence of cure on w ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb For w ery large, cure pulled oward mddle conrol pon, ge a secon of a hyperbola For w (sandard splne), sandard parabola For w <, pose, cure moes away from mddle conrol pon For w, cure becomes a sragh lne When s cure par of a crcle? (homework) Md-Pon ( ½?) ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb 2 2( w) 2( w) W ery large ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb 2 2( w) 2( w) (,;>>) (,;) W (non-raonal splne) ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb 2 2( w) 2( w) (,;) (,;) Polynomal splne: parabolc arc W < ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb 2 2( w) 2( w) W ( ) w 2 ( ) w B w C ( ) 2 ( ) w B C ( ) 2 ( ) ( ) 2 ( ) B c B w wb wc wb 2 2( w) 2( w) (,;w) (,;) (,;w) (,;) Raonal splne: resul depends on w Sragh Lne 7

8 Oulne Raonal Splnes Quadrac raonal splnes NURBs (brefly) Paramerc paches (brefly) NURBS Non-unform (ary me neral per segmen) Raonal B-Splnes Can model a wde class of cures and surfaces Same conenen properes of B-Splnes Sll wdely used n CD sysems Polar Forms: Cubc Bsplne Cure NURBS For Unform B-splnes, unform kno ecor (below) For non-unform, only requre non-decreasng, no necessarly unform (can be arbrary) 2 Unform kno ecor: -2, -,,, 2,3 Labels correspond o hs Oulne Raonal Splnes Quadrac raonal splnes NURBs (brefly) Paramerc paches (brefly) Paramerc Paches Each pach s defned by blendng conrol pons FDFH Fgure.44 8

9 Paramerc Paches Pon Q(u,) on he pach s he ensor produc of cures defned by he conrol pons Paramerc Bcubc Paches Pon Q(u,) defned by combnng conrol pons wh polynomal blendng funcons: Q(,) Q(,) Q(u,) Q(,) Q(,) Wa Fgure 6.2 P, P,2 P,3 P,4 P 2, P2,2 P2,3 P 2,4 Qu (, ) UM M V P 3, P 3,2 P 3,3 P 3,4 P 4, P 4,2 P 4,3 P 4,4 M V U u u u V Where M s a marx descrbng he blendng funcons for a paramerc cubc cure (e.g., Bezer, B-splne, B ec.) Summary Splnes sll used commonly for modelng Sar wh smple splne cures (Bezer, unform non-raonal B-splnes) Dscussed exenson o raonal cures (add homogeneous coordnae for raonal polynomal) Bref dscusson of NURBs: wdely used Bref dscusson of Paramerc paches for modelng surfaces (raher han cures) 9