Multiresolution Feature Extraction from Unstructured Meshes

Transcription

1 Multirsolution Fatur Extraction from Unstructurd Mshs Andras Hubli, Markus Gross Dpartmnt of Computr Scinc ETH Zurich, Switzrland Abstract W prsnt a framwork to xtract msh faturs from unstructurd two-manifold surfacs. Our mthod computs a collction of picwis linar curvs dscribing th salint faturs of surfacs, such as dgs and ridg lins. W xtnd ths basic tchniqus to a multirsolution stting which improvs th quality of th rsults and acclrats th xtraction procss. Th framwork is smiautomatic, that is, th usr is rquird to input a fw control paramtrs and to slct th oprators to b applid to th input surfac. Our msh fatur xtraction algorithm can b usd as a prprocssor for a varity of applications in gomtric modling including msh fairing and subdivision. CR Dscriptors: Surfac Rprsntations, Gomtric Modling, Triangl Dcimation, Multirsolution Modls, Fatur Extraction. 1 Introduction 1.1 Motivation Rcnt advancs in acquisition systms hav rsultd in th rady cration of vry larg, dnsly sampld surfacs, usually rprsntd as triangl mshs. Th impossibility of ral-tim intraction with ths larg modls has motivatd many rsarchrs in th computr graphics community to dsign advancd msh procssing mthods including subsampling, rstructuring, fairing and othrs. Th arly approachs, such as th vrtx rmoval algorithm of W. Schrödr [17] or th progrssiv msh algorithm of H. Hopp [10], us local rror norms to construct multirsolution approximations of mshs by itrativly rmoving information from th input msh. Mor rcnt rprsntations ar basd on th gnralization of fairing tchniqus from signal procssing [18], [13], [11], or on subdivision surfacs [14], [19]. Ths approachs combin advancd oprators with multirsolution tchniqus to nabl intraction with larg datasts and provid additional functionality, such as diting. hubli@inf.thz.ch grossm@inf.thz.ch Addrss: Dpartmnt of Computr Scinc ETH Zntrum CH Zurich In this papr w invstigat a rlatd problm: th dtction of msh faturs. Our goal is to xtract picwis linar faturs from mshs which can thn b usd to construct mor sophisticatd multirsolution rprsntations. Th most important advantag of our mthod is that w can forc modling algorithms, such as subdivision and fairing, to rtain fatur information including sharp dgs or ridg lins. 1.2 Prvious Work In his pionring work [2], Canny constructd an optimal filtr to dtct and xtract crtain typs of faturs from imags. Th critria usd to dsign th filtr includ dtction prformanc, sharp localization of th faturs and a uniqu dtction of dgs. Th thory of snaks [12] uss dformabl modls to track faturs. A polygonal curv is assignd an nrgy function and is dformd until its associatd nrgy is minimizd. Th choic of particular nrgy functions nsurs that th polygonal curv tracs a fatur to its nd configuration. Anisotropic diffusion, as usd in [16], is a powrful tool that allows to rmov high-frquncy nois whil prsrving th fatur information. Thus, th tchniqu nabls th construction of robust dg dtction algorithms. This oprator has bn xtndd to th non-paramtrizd stting of triangl mshs with arbitrary connctivity in [3]. A problm rlatd to fatur xtraction is imag sgmntation [9], whr input imags must b subdividd into rgions that possss similar charactristics. A robust sgmntation algorithm was usd to xtract faturs bing dfind as th picwis linar curvs that bound diffrnt rgions in th imag. Most of th tchniqus discussd in this sction apply to imags. Thr ar svral advantags of handling imags ovr triangl mshs: imags hav both a rgular connctivity and a wll known paramtrization, proprtis that triangl mshs do not possss. Ths diffrncs complicat th xtnsion of ths tchniqus to th mor gnral domain of unstructurd triangl mshs. 1.3 Papr Organization Th papr is organizd as follows: in sction 2 w provid som basic dfinitions and giv an ovrviw of th framwork. In sction 3 w prsnt th first major componnt of our framwork, th st of classification oprators, followd in sction 4 by a discussion of th scond componnt, th dtction oprators. In sction 5 w dscrib a multirsolution fatur xtraction tchniqu. In sction 6 w prsnt som of th rsults w obtaind using this framwork and w discuss som application domains. W will conclud th papr with th dscription of som futur challngs. 2 Ovrviw of th Tchniqu Th goal of th framwork prsntd in this papr is to xtract a st of msh faturs from two-manifold polygonal mshs with arbitrary connctivity. To this nd, w first formaliz th notion of a msh fatur and th domain of our oprators: 1 of 8

2 Dfinition: A msh fatur is a picwis linar curv that dscribs an important charactristic of th input msh. W rstrict any msh fatur as bing rprsntd by a collction of dgs of th msh. That is, our xtraction oprators ar not rquird to modify th connctivity of th msh. Dfinition: a surfac is a two-manifold with boundaris if all its points hav an opn nighborhood homomorphic to ithr R 2 or R 2 +. Th approach that w introduc is indpndnt of th fatur smantics, i.. th oprators ar not traind to rcogniz particular pattrns. Instad, w dcidd to us application-nutral oprators which ar basd on notions such as Laplacian or curvatur. Th rsulting framwork is subdividd into two componnts, as shown in figur 1: In a classification phas, vry dg in th modl is assignd a wight proportional to th probability that th dg lis on a msh fatur. Th oprators usd in this stp ar not basd on any assumption on th input msh, xcpt that it has to b a two manifold surfac with boundaris. In a dtction phas, msh faturs ar rconstructd from th information computd in th prvious stp. By our dfinition, this stp constructs picwis linar curvs, dfind as collctions of dgs. Figur 1: Classification Phas Ovrviw of th fatur xtraction framwork. Th distinction btwn th classification and dtction oprators has svral advantags: first, it is possibl to asily swap btwn classification oprators and thus choos th bst suitabl oprator for th particular input msh. Furthrmor, if additional information on th application domain is availabl, nw optimizd oprators can b includd into th framwork mor asily. In th nxt two sctions w will dscrib th two componnts in dtail and prsnt th st of oprators that w constructd and tstd in our framwork. W illustrat th rsults by applying thm to surfacs with diffrnt proprtis. 3 Classification Phas Dtction Phas Edg slction Patch construction Skltonizing Th goal of th classification phas is to assign a wight to vry dg in th input msh. Idally, th wight should b proportional to th probability that th dg blongs to a msh fatur: dgs clos to msh faturs should b assignd largr wights than dgs that ar farthr away. Th information computd in this first phas will thn b usd to xtract a st of th most important dgs, which in turn will b usd to rconstruct th msh faturs. In th classification stp w ar facd with diffrnt problms. First, th oprators must b capabl to handl mshs with diffrnt rsolutions. Som mshs ar highly optimizd maning that th numbr of triangls is rducd to a minimum, whil othrs ar ovrsampld. In addition, th classification procss is aggravatd by th prsnc of high frquncy nois in th input mshs, which could mistaknly b rcognizd as usful information. To allviat som of ths problms w provid a st of diffrnt oprators, ach with diffrnt proprtis, and bttr suitd to handl crtain classs of mshs. In addition, w can optionally prfiltr th input mshs using standard oprators, such as [13], [8] and [4]. In th rmaindr of this sction w will introduc th oprators w us in our framwork and discuss thir capabilitis and limitations. 3.1 Scond Ordr Diffrnc (SOD) Th SOD is th simplst classification oprator. It assigns a wight to vry dg in th msh proportional to th dihdral angl dfind by th normals of its two adjacnt triangls. Th ida is basd on th scond ordr diffrnc oprator constructd in [8], which was usd to fair mshs of arbitrary connctivity. Th oprator, dscribd by quation (1), is locally bound and can b valuatd fficintly. Th variabls n i and n j corrspond to th normals of th two triangls that shar dg, as shown in figur 2. Figur 2: Support of th SOD classification oprator. This tchniqu is bst suitd for coars, pr-optimizd mshs. Howvr, SOD prforms poorly on highly dtaild or noisy mshs, sinc all computations ar carrid out within a small rgion of support. 3.2 Extndd Scond Ordr Diffrnc (ESOD) Th ESOD oprator xtnds th SOD oprator by using a largr support to valuat th wight of an dg. Instad of dfining n i and n j as th normals of th two nighboring triangls of, w dfin thm as th avrag normals computd from th triangls on th on-ring of th vrtics x i and x j opposit to and apply th thm in quation (1). Th xtndd support of ESOD is illustratd in figur 3. Figur 3: n i n i w ( ) x i Support of th ESOD oprator. n i n i Th incrasd support of th oprator has th xpctd consquncs: th influnc of nois on th classification procss is attnuatd. Howvr, as a th support of th oprator is largr and cannot b adaptd to th input msh, ESOD dos not prform wll on vry coars mshs. 3.3 Bst Fit Polynomial (BFP) Th BFP oprator uss a diffrnt philosophy than th prvious oprators, in that it maks us of a paramtr domain associatd with vry dg in th msh. Th wight assignd to an dg is computd as follows: first, a subst of th msh vrtics ar pro- n j = cos n j x j n j n j (1) 2 of 8

3 jctd onto th paramtr domain and a bst fit polynomial pu ( ) of dgr n is computd. Th curvatur of th (planar) polynomial is thn valuatd at th paramtr position of, as dscribd by quation (2): w ( ) = p'' ( ) (2) Th major challngs of this approach ar th dfinition of th paramtr domain and th propr projction of th st of vrtics from 3-spac. An intuitiv dfinition of th paramtr spac is givn in figur 4. Figur 4: b) Th BFP oprator: Paramtr plan. b) Intrsction btwn th paramtr plan and th msh. Bst fit polynomial fittd in paramtr spac. W propos to st th paramtr plan to b prpndicular to th vctor dfind by th dg. A uniqu plan is dfind by rquiring th midpoint of to li on th plan. Th points usd in th bst fit procss ar computd from th intrsction of th plan with a st of nighboring triangl dgs, as shown in figur 4.b. Th most important advantag of this stratgy is that th support of th oprator can b chosn frly and that it can b adaptd locally for ach dg. An additional dgr of frdom is givn by th dgr of th fitting polynomial which can b adjustd to th siz of th support of th oprator. This approach has th advantag of bing vry flxibl, bcaus th support can b adaptd both globally and locally. Thus, it is lss influncd by nois which is filtrd out during th bst fit procss. Furthrmor, it has th potntial to b usd for any typ of msh, providd that an appropriat st of paramtrs is chosn. Th main disadvantag of th approach is that its computational cost is highr than th cost of th prvious oprators. 3.4 Angl Btwn Bst Fit Polynomials (ABBFP) Th ABBFP oprator is a variation of th BFP oprator which is also basd on bst fit polynomials. As for th prvious approach, polynomials ar fittd in th paramtr domain of vry dg (figur 5). Th ABBFP oprator fits two polynomials: on for th vrtics that li on on sid of in th paramtr domain and anothr for th vrtics lying on th othr, as illustratd in figur 5.c. Th wight assignd to is chosn to b proportional to th angl btwn tangnts of th two curv valuatd at th paramtr position of. Th wight is thn computd according to quation (3): w ( ) ( 1, p l '( ) ) ( 1, p r '( ) ) = cos ( 1, p l '( ) ) ( 1, p r '( ) ) As for th BFP oprator, th usr is allowd to spcify both th siz of th support and th dgr n of th polynomial usd in th bst fit procss. A potntial advantag of this variant is that it is adpt in capturing discontinuitis at th paramtr position of, which hints at th prsnc of a msh fatur in th nighborhood of. pu ( ) (3) Figur 5: 3.5 Comparison of th Oprators In ordr to compar th diffrnt oprators that wr dscribd in this sction, w apply thm to two diffrnt typs of mshs. Figur 6.a dpicts a coars msh that rprsnts a motor part using fw triangls. Figur 6.b-c illustrat th rsult computd using th SOD and BFP oprators rspctivly. In this xampl th most fficint oprator is SOD, sinc it can captur all th important information fficintly and it can b valuatd mor ffctivly than BFF. Th BFP oprator is capabl of gnrating good rsults, but its paramtrs must b tund carfully. Figur 6: p l ( u) Th ABBFP oprator: Paramtr plan. b) Intrsction btwn th paramtr plan and th msh. Two polynomials ar fittd on both sids of and th angl btwn thir tangnts at is masurd. Figur 7.a shows a highly dtaild msh that rprsnts a gological surfac using a larg numbr of triangls. Th rsults computd using th SOD and BFP oprators ar dpictd in figur 7.bc rspctivly. Th SOD oprator rcognizd important rgions in th msh, but noisy rgions wr also rcognizd as a rsult of th limitd support of th oprator. Th BFP oprator prformd bttr, sinc its support has bn tund to filtr out most of th nois prsnt in th surfac. For both xampls prsntd in figur 6 and figur 7 th sam paramtrs wr chosn for th hystrsis which was usd to slct th st of most important dgs. A dscription of th hystrsis function is givn in sction Dtction Phas b) Classification oprators applid to an optimizd msh: Input msh. b) Rsult computd using th SOD oprator. Rsult computd using th BFP oprator. p r ( u) b) Th classification stp discussd in th prvious sction assigns a wight to vry dg in th input msh, which is proportional to th probability that th dg blongs to a msh fatur. Th dtction phas analyzs ths valus and rconstructs a st of important msh faturs in four stps: 3 of 8

4 accpt rjct 0 Bl Bu 1 w() Figur 8: Hystrsis function for th thrshold valus Bl and Bu. An hystrsis has svral advantags vrsus standard thrsholding stratgis: first, thrsholding can b simulatd by th hystrsis function by stting th uppr and lowr bounds to th sam valu: (6) Bl = Bu b) Furthrmor, th hystrsis clustrs dgs bttr than thrsholding. As a rsult, th patching algorithm is capabl of gnrating lss patchs with largr ara, which in turn allows us to xtract msh faturs mor robustly. Figur 9 illustrats th st of patchs obtaind by using diffrnt bound valus for th sam st of input wights. If th uppr and lowr bounds ar st to th sam valu, as in figur 9.b, th hystrsis procss is rducd to standard thrsholding and th slctd dgs hav not bn clustrd wll. Th us of diffrnt bounds, as illustratd in figur 9.c, nabls us to rmov som of th isolatd dgs and gnrat bttr clustrs. Th choic of a vry larg uppr bound and a small lowr bound allow us to slct dgs clos to th most important faturs, as shown in figur 9.d. Figur 7: Classification oprators applid to a highly dtaild msh: Input msh. b) Rsult computd using th SOD oprator. Rsult computd using th BFP oprator. First, th st of th most important dgs is computd. Th importanc of an dg is dfind both by its wight and by th wight of th nighboring dgs. Th st of dgs is analyzd to construct a st of patchs. A patch is a msh rgion that is likly to contain on or mor msh faturs. Th msh faturs ar xtractd from th patchs using a skltonizing algorithm which itrativly simplifis vry patch to a collction of picwis linar curvs. Optionally, unimportant msh faturs can b rmovd from th rsult. Th importanc of a msh fatur is computd from its lngth and from th wights associatd with its dgs. In th nxt four subsctions w will dscrib ach of ths stps in dtail and illustrat th rsults with an xampl. 4.1 Slction of Important Edgs b) d) Th first stp in th dtction phas is crucial, sinc it idntifis th dgs clos to th msh faturs. This procss is havily msh dpndnt: mshs contain a varying numbr of msh faturs and thir dg dnsity is variabl. Furthrmor, th procss is also influncd by th usr who might b intrstd in only a subst of th msh faturs. Hnc, w rquir th usr to spcify th bounds of th hystrsis function usd to mark th st of important dgs. Hystrsis Thrsholding This approach rquirs two usr-dfind valus which srv as a lowr bound B l and as an uppr bound B u of th hystrsis. An dg in th msh is slctd if it satisfis on of two conditions: Figur 9: Edgs slctd by th hystrsis function: Input msh. b) Rsult gnratd using th paramtrs Bu = 0.92, Bl = Rsult gnratd using th paramtrs Bu = 0.96, Bl = d) Rsult gnratd using th paramtrs Bu = 0.995, Bl = 0.9. Condition 1: th wight w ( ) of is largr than th uppr bound B u of th hystrsis function: w ( ) Bu 4.2 Patch Construction Th construction of th patchs is actually not rquird in th dtction phas. Howvr, th us of patchs allows us to apply xtnsions of standard skltonizing algorithms from th fild of computr vision [6]. This is advantagous, sinc th problm has bn wll studid in that fild and many robust skltonizing algorithms ar availabl. As mntiond in sction 4.1, th hystrsis oprator has good clustring capabilitis. Th goal of th patching algorithm is to transform ths clustrs into uniform patchs, whr all dgs ar markd. Additionally, th algorithm is not allowd to incras th siz of th clustrs, but only fill th intrior. Th rsult is a simpl condition that is chckd to slct additional dgs and insrt thm into th st cratd in sction 4.1: (4) Condition 2: Th wight w ( ) of is largr than th lowr bound B l of th hystrsis function and th st of nighboring dgs N ( ) contains a slctd dg ' : w ( ) B l ' ; ' N ( ) ; is slctd (5) If nithr condition is satisfid, th dg is discardd. Figur 8 illustrats th binary hystrsis function with uppr bound B u and lowr bound B l and as implmntd in our framwork: 4 of 8

5 Condition: An dg is insrtd into a patch if it was markd in th prvious stp, or if both of its ndpoints blong to othr dgs prsnt in th patch, rgardlss of th wight w ( ) of. This condition is similar to th scond condition usd by th hystrsis function. Again, dgs with a small wight ar insrtd into th st of patchs if thy ar clos to important dgs. Th rquirmnt that important dgs ar prsnt on both sids of nsurs that patchs do not grow unncssarily larg. Th absnc of rquirmnts for th wight w ( ) of guarants that all unmarkd dgs in th intrior of a clustr will b addd to th clustr thus rsulting in mor uniform patchs. Figur 10 dpicts th rsult obtaind by our patching algorithm. Th algorithm is initializd with th st of dgs displayd in figur 10.a. Th insrtion of dgs according to th condition spcifid in this sction yilds th st of patchs illustratd in figur 10.b. b) Figur 10: Patch construction: St of dgs slctd by th hystrsis function. b) Patchs gnratd by th framwork. Not that th hols in th patchs dpictd in figur 10.b ar a consqunc of th rquirmnt that patchs should not grow mor than ndd. 4.3 Skltonizing Th most important stp in th dtction phas consists in th xtraction of th final msh faturs from th patchs computd in th prvious stp. W accomplish this goal using a skltonizing algorithm similar to th ons usd computr vision. In particular, w constructd a thinning tchniqu: patchs ar thinnd to a st of msh faturs by burning th dgs from th boundary of th patchs down to a st of picwis linar curvs. W do not us xisting algorithms from computr vision, sinc th skltonizing algorithm is xpctd to handl mshs with arbitrary connctivity and not only hight fild data. Th thinning algorithm is dscribd by th following psudo-cod fragmnt: void patchthinning(list< int > patch) { for all dgs in patch if (isboundaryedg() == tru) dglist.insrt(); whil dglist is not mpty do { = dglist.front(); // Rtriv th first dg dglist.pop_front(); // Rmov it from th list if(blongstomshfatur() == fals) { rmovfrompatch(); dglist.insrt(nwboundaryedgs); } } } Th thinning opration is initializd by insrting all th dgs that li on th boundary of a patch into a linkd list. Th function isboundaryedg chcks th following condition for th dg : Condition: considr an dg that blongs to a patch and its two adjacnt triangls t 1 and t 2. If any of th othr four dgs of t 1 and dos not blong to th patch, thn is a boundary dg. t 2 Th boundary dgs ar xtractd from th list on at a tim, and thy ar analyzd. Th function blongstomshfatur inspcts two conditions to dcid whthr an dg blongs to a fatur: Condition 1: If only on of th ndpoints of blongs to anothr dg in th patch, is only prsrvd if th ndpoint blongs to only on othr markd dg. If th ndpoint blongs to multipl dgs in th patch, thn it must b rmovd. This condition nabls us to rmov dgs that ar prpndicular to th msh fatur bing xtractd. t 1 t 2 t 2 b) Figur 11: Thinning oprator; slctd dgs markd as a bold polylin: Edg is rmovabl. b) Edg is not rmovabl, sinc it would disconnct th fatur locally. Condition 2: If both ndpoints of blong to othr dgs in th patch, thn can only b rmovd if it dos not disconnct th patch locally. Th patch is not disconnctd by th rmoval opration if and only if on of th two adjacnt triangls t 1 and t 2 of is dfind by thr markd dgs. Considr th xampl in figur 11: th dg in th configuration displayd in figur 11.a can b rmovd safly, sinc all th dgs in t 1 ar markd. Convrsly, th dg in figur 11.b cannot b rmovd without disconncting th patch locally. If an dg is rmovd from a patch, two nw dgs will bcom boundary dgs and thy must b insrtd into th list. Aftr th dg has bn analyzd, th nxt dg is xtractd from th list. This procss continus until th list is mpty. Th nd configuration is guarantd to b a collction of picwis linar curvs. Sinc th patchs wr not disconnctd in th thinning procss, th msh faturs ar also guarantd to b connctd. Th msh faturs can intrsct, i.. two or mor dgs that blong to msh faturs can intrsct at a vrtx. Thrfor, complx faturs can b rcognizd and xtractd by th algorithm. Figur 12 illustrats th msh faturs xtractd from th st of patchs computd in th prvious sction. Th input data is visualizd in figur 12.a and th st of output msh faturs in figur 12.b. By analyzing th rsult thoroughly, on can notic that on of th patchs has bn thinnd into two msh faturs implying that th patch has bn disconnctd. Th two faturs wr connctd by a third fatur that has bn rmovd from th rsult. b) Figur 12: Msh faturs gnratd by th thinning algorithm: Th st of patchs usd to initializ th algorithm. b) Th st of rsulting msh faturs. 4.4 Importanc Function Th st of msh faturs can b optionally rducd so that only important faturs ar rturnd to th usr. In our opinion, th importanc of a msh fatur is dtrmind by two factors: its lngth, i.. th numbr of dgs prsnt in th msh fatur and th avrag wight of th dgs. Th limination of unimportant fa- t 1 5 of 8

6 turs is prformd by ranking th msh faturs according to an importanc function such as F i 1 IF ( i ) = F i -- w ( ) n F i Th trm dscribs th lngth of th msh fatur and w ( ) th wight of an dg that blongs to F i. Nxt, th msh faturs with smallst importanc ar rmovd from th rsult and th rmaining faturs ar rturnd to th usr. F i (7) b) 5 Multirsolution Fatur Extraction Th classification and dtction tchniqus that w discussd in th prvious two subsctions ar fficint and gnrat maningful msh faturs. Howvr, as alrady shown in imag procssing [15], th us of multirsolution tchniqus can improv th ovrall framwork, both in trms of fficincy and in trms of quality of th rsults. In particular, th us of a multirsolution rprsntation of th input msh nabls us to bttr captur low frquncy msh faturs than by xtnding th support of th classification oprators. 5.1 Multirsolution Rprsntation Th litratur offrs many diffrnt multirsolution rprsntations for triangulatd two-manifold surfacs, basd on oprators such as vrtx rmoval [17] and dg collaps [10], [5]. W slctd th progrssiv msh algorithm in our framwork, sinc th dg collaps and vrtx split oprators naturally complmnt our own oprators. In addition, w obtain th following advantagous proprtis: Th coars approximation of th input msh only contains th structural information on th msh, i.. th collaps oprator prsrvs th salint faturs of th msh during th simplification procss. During th rfinmnt procss, th vrtx split oprations can b analyzd on th fly to updat th st of msh faturs. Thus w obtain th full-rsolution faturs aftr all th vrtics hav bn r-insrtd into th msh. A multirsolution approach has th potntial of acclrating th xtraction procss. This can b accomplishd by computing th wight of a subst of th dgs of th input surfac that ar clos to msh faturs. 5.2 Multirsolution Fatur Extraction As mntiond in th prvious subsction, multirsolution and in particular th dg collaps oprator can b usd ffctivly to xtract faturs incrmntally from a full rsolution msh. This procss is implmntd in thr stps: I. Givn an input msh, its progrssiv msh rprsntation is first constructd. Th rsult consists of a coars approximation of th msh and a st of vrtx split oprations that allow us to rconstruct th original surfac. II. Th tchniqus dscribd in sction 3 and sction 4 ar applid to th coars rprsntation of th msh to xtract th most important msh faturs. This opration can b prformd fficintly, sinc both th classification and th dtction stps ar applid only to a subst of th dgs of th input msh. III. Finally, th input msh is rconstructd from th coars msh by r-insrting th vrtics and dgs using th st of vrtx split oprations. During this procss, it is crucial to updat th msh fatur information which nds to adapt to th changs of th undrlying msh. Th st of masks ndd to prform ths updats is illustratd in figur 13. Figur 13: Fatur updat masks; fatur displayd as a thickr polylin: Th nw dg is insrtd into th fatur. b) Th nw dg is not insrtd into th fatur. Th nw dg changs th shap of th fatur locally. During th insrtion of a vrtx x j which is split from a vrtx x i, th nighborhood of th vrtx split is analyzd. If x i blongs to a msh fatur, w nd to chck whthr th fatur nds to b modifid. In principl, many diffrnt configurations could b invstigatd in ordr to comput th bst possibl updat. In practic howvr, th masks displayd in figur 13 provd to b adquat and only thr configurations must b considrd: Th cas dpictd in figur 13.a is unambiguous: th nw dg must b includd into th msh fatur, othrwis th insrtion opration would split th fatur into two componnts. Th scond cas shown in figur 13.b is also straightforward: th insrtion of th nw dg dos not affct th fatur. Thrfor is not insrtd into th fatur. Th third cas shown in figur 13.c is mor complx. Th two dgs that ar split by th vrtx split opration blong to th msh fatur. Thrfor, th msh fatur can b updatd in four diffrnt ways. Th choic of th bst path for th msh fatur is dtrmind by analyzing th wights of th dgs locally. Th msh fatur is updatd using th path with th largst avrag wight. b) d) Figur 14: Multirsolution msh fatur xtraction: Full-rsolution input msh ( vrtics). b) Msh faturs xtractd from th bas domain (3 483 vrtics). Msh faturs rconstructd from an intrmdiat rprsntation ( vrtics). d) Full-rsolution msh faturs. 6 of 8

7 Of cours, thr ar spcial cass that must b analyzd, such as th split of vrtics at th nd of a msh fatur, or th split of vrtics whr two or mor msh faturs mt. Ths configurations can also b handld using straightforward variations of th masks discussd prviously and will not b discussd furthr. Considr figur 14 which illustrats our multirsolution approach. A coars msh is computd from th input surfac (figur 14. and its msh faturs ar computd using th tchniqus discussd in sction 3 and 4 (figur 14.b). Th st of msh faturs is thn continuously updatd during th rfinmnt procss. An intrmdiat stat is shown in figur 14.c, whr 50% of th vrtics wr alrady r-insrtd. Th nd configuration is displayd in figur 14.d. This approach could b furthr improvd, sinc th coars approximation of th input surfac dos not ncssarily contain all th msh faturs. Thrfor, th algorithm should support th cration of nw msh faturs in th rconstruction procss. This rsults in a multilvl fatur dtction stratgy whr th oprators discussd in sction 3 and sction 4 ar applid at diffrnt lvls of rsolutions. Figur 16: Th faturs of th Stanford bunny. fold fairing framwork [11]. Using th tchniqus prsntd in this papr, high frquncy nois can b rmovd from complx modls without rmoving important faturs. As an xampl considr figur 17 dpicting a gological modl. Th nois of th input surfac can b liminatd ithr using standard fairing tchniqus (figur 17.b) or using fatur-prsrving fairing (figur Rsults and Applications In this sction w prsnt som of th rsults gnratd by our framwork. W us both wll known mshs, such as th Stanford bunny, th mannquin and th golf club, as wll as modls from th domain of goscinc. Most of ths mshs hav irrgular connctivity and possss a st of faturs radily idntifiabl. Th xcption is th gological surfac displayd in figur 17 which dos not contain any prominnt fatur. In figur 15 w applid th framwork to th mannquin had. To tst th capabilitis of th framwork w first applid thr Loop subdivision stps to th input data, so that th algorithm had to work on a vry smooth domain. Th rsults illustratd in figur 15 hav bn computd using th BFP oprator and a larg support. Th most important componnts of th fac, such as th ys, nos, ars and mouth, hav bn proprly rcognizd. Th localization of th msh faturs is good givn th amount of information prsnt in th msh. b) Figur 17: Fatur-prsrving fairing: Th input modl. b) Smooth modl gnratd by standard fairing. Smooth modl gnratd by fatur prsrving fairing. Msh faturs can b usd in conjunction with othr oprators as wll in ordr to prsrv information about th input modl. Any subdivision oprator can b xtndd to prsrv faturs. This is accomplishd by handling th msh faturs as boundaris [1] and by applying th on-dimnsional subdivision oprators on thm. An xampl is displayd in figur 18. Th input msh (figur 18. is first analyzd and its most important msh faturs ar xtractd. Th us of standard subdivision gnrats rsults of dcnt quality, but th magnitud of faturs is attnuatd, as shown in figur 18.b. Th us of a fatur-prsrving subdivision oprator nabls us to prsrv th important msh faturs much bttr, as illustratd in figur 18.c. Finally, msh faturs could also b usd to govrn a simplification algorithm: an dg collaps oprator should not collaps dgs prpndicular to a msh fatur, but rathr paralll dgs. Of cours, rror norms that control th simplification alrady striv to prsrv faturs, but xplicit fatur prsrvation would guarant that important information will not b rmovd from any of th approximations constructd by th algorithm. Figur 15: Th faturs of th mannquin. Nxt, w prsnt th msh faturs xtractd from th Stanford bunny in figur 16. For this msh w usd th ABBFP oprator and slctd a support that filtrd out most of th nois prsnt in th fur of th bunny. Th important componnts, such as th ars, tail, nck, lgs and vn som of th paws of th bunny, hav bn proprly rcognizd. Th hystrsis function rmovd othr important faturs, such as th ys and mouth. Howvr, it should b notd that th magnitud of ths faturs is almost th sam as th magnitud of th nois prsnt in th fur. W bliv that automatic fatur xtraction algorithms for mshs with arbitrary connctivity can b applid in diffrnt procsss of gomtric modling. Th first application w discuss is fatur prsrving fairing that has bn proposd in our non-mani7 of 8

8 thanks to Kuno Myr and Philipp Zürchr who implmntd parts of this framwork. 7 Conclusions and Futur Work In this papr w prsntd a framwork for th dtction of faturs in mshs with arbitrary connctivity. W proposd a twostag procss consisting of a classification phas and a dtction phas. In ordr to handl a varity of diffrnt mshs, w provid a st of oprators with diffrnt proprtis. W xtndd our framwork to a multirsolution stting, which clarly improvs th quality of th rsults and th prformanc of th algorithms. Th usr must slct th oprators as wll as a fw paramtrs for th classification and dtction stps. As such, th procss is not fully automatic and rquirs manual assistanc and tuning for optimal prformanc. W do not considr this as a major drawback, sinc all algorithms can b computd fficintly. W bliv that th rsults producd by our framwork ar of usful quality. W nvision th following optimizations and xtnsion on ths tchniqus: Improvd classification oprators: on of th major difficultis ncountrd in th classification phas is to distinguish btwn high frquncy nois and fatur information, a wll known problm in computr vision. W addrssd this problm by constructing oprators with adjustabl support, which workd wll in practic. Howvr, w bliv that furthr improvmnts could by obtaind by msh dcomposition [7]. Improvd skltonizing oprators: th quality of th rsults gnratd by th thinning algorithm could b improvd by using th wights computd in th classification phas mor aggrssivly. Currntly, only a topological masur is usd. Th advantag of a topology-drivn thinning algorithm is improvd robustnss at th cost of an infrior localization of th msh faturs. Improvd intraction: th final goal of this projct is, of cours, to provid a sophisticatd msh analysis tool abl to dtrmin th optimal oprators and most of th paramtrs autonomously, rquiring only a fw intuitiv paramtrs from th usrs. Acknowldgmnts b) Figur 18: Fatur-prsrving subdivision: Th input msh. b) Th st of msh faturs gnratd by our framwork. Smooth surfac gnratd by standard subdivision. d) Smooth surfac gnratd by fatur prsrving subdivision. This rsarch was mad possibl by grants from Schlumbrgr ATC, Austin, TX. Th authors would lik to thank Richard Hammrlsy and Karn Lu for many hlpful discussions. Furthr Litratur [1] H. Birmann, A. Lvin, and D. Zorin. Picwis smooth subdivision surfacs with normal control. In K. Akly, ditor, Siggraph 2000, Computr Graphics Procdings, pags ACM Prss / ACM SIGGRAPH / Addison Wsly Longman, [2] J. Canny. A computational approach to dg dtction. IEEE Transactions on Pattrn Analysis and Machin Intllignc (PAMI), 8: , [3] U. Clarnz, U. Diwald, and M. Rumpf. Anisotropic gomtric diffusion in surfac procssing. In Proc. of th 11th Ann. IEEE Visualization Confrnc (Vis) 2000, [4] M. Dsbrun, M. Myr, P. Schrödr, and A. H. Barr. Implicit fairing of irrgular mshs using diffusion and curvatur flow. In SIG- GRAPH 99 Procdings, Computr Graphics Procdings, Annual Confrnc Sris. ACM SIGGRAPH, ACM Prss, Aug [5] M. Garland and P. S. Hckbrt. Surfac simplification using quadric rror mtrics. In T. Whittd, ditor, SIGGRAPH 97 Confrnc Procdings, Annual Confrnc Sris, pags ACM SIG- GRAPH, Addison Wsly, Aug ISBN [6] R. C. Gonzalz and R. E. Woods. Digital Imag Procssing. Addison-Wsly, Rading, MA, USA, [7] M. H. Gross and A. Hubli. Eignmshs. Tchnical rport, ETH Zurich, Mar [8] I. Guskov, W. Swldns, and P. Schrödr. Multirsolution signal procssing for mshs. In SIGGRAPH 99 Procdings, Computr Graphics Procdings, Annual Confrnc Sris. ACM SIG- GRAPH, ACM Prss, Aug [9] R. Haralick and L. Shapiro. Imag sgmntation tchniqus. Applications of Artificial Intllignc II, April [10] H. Hopp. Progrssiv mshs. In H. Rushmir, ditor, SIG- GRAPH 96 Confrnc Procdings, Annual Confrnc Sris, pags ACM SIGGRAPH, Addison Wsly, Aug hld in Nw Orlans, Louisiana, August [11] A. Hubli and M. Gross. Fairing of non-manifolds for visualization. In Proc. of th 11th Ann. IEEE Visualization Confrnc (Vis) 2000, [12] M. Kass, A. Witkin, and D. Trzopoulos. Snaks: Activ contour modls. In Proc. of IEEE Confrnc on Computr Vision, pags , London, England, [13] L. Kobblt, S. Campagna, J. Vorsatz, and H.-P. Sidl. Intractiv multi-rsolution modling on arbitrary mshs. In M. F. Cohn, ditor, Computr graphics: procdings: SIGGRAPH 98 Confrnc procdings, July 19 24, 1998, Computr Graphics -procdings- 1998, pags , Nw York, NY 10036, USA and Rading, MA, USA, ACM Prss and Addison Wsly. [14] C. Loop. Smooth Subdivision Surfacs Basd on Triangls. PhD thsis, Utah Univrsity, [15] D. Park, K. Nam, and R. Park. Multirsolution dg-dtction tchniqus. Pattrn Rcognition, 28(2): , Fbruary [16] P. Prona and J. Malik. Scal-spac and dg dtction using anisotropic diffusion, [17] W. J. Schrödr, J. A. Zarg, and W. E. Lornsn. Dcimation of triangl mshs. In E. E. Catmull, ditor, Computr Graphics (SIG- GRAPH 92 Procdings), volum 26, pags 65 70, July [18] G. Taubin. A signal procssing approach to fair surfac dsign. In R. Cook, ditor, SIGGRAPH 95 Confrnc Procdings, Annual Confrnc Sris, pags ACM SIGGRAPH, Addison Wsly, Aug hld in Los Angls, California, August [19] D. Zorin, P. Schrödr, and W. Swldns. Intractiv multirsolution msh diting. In T. Whittd, ditor, SIGGRAPH 97 Confrnc Procdings, Annual Confrnc Sris, pags ACM SIG- GRAPH, Addison Wsly, Aug ISBN of 8