Accurate Simplification of Multi-Chart Textured Models

Transcription

1 DOI: /j x COMPUTER GRAPHICS forum Volume 29 (2010), number 6 pp Accurae Simplificaion of Muli-Char Texured Models N. Coll and T. Paradinas Insiu d Informaica i Aplicacions, Universia de Girona, Spain {coll,eresap}@ima.udg.edu Absrac Scanning and acquisiion mehods produce highly deailed surface meshes ha need muli-char parameerizaions o reduce sreching and disorion. From hese complex shape surfaces, high-qualiy approximaions are auomaically generaed by using surface simplificaion echniques. Muli-char exures hinder he qualiy of he simplificaion of hese echniques for wo reasons: eiher he char boundaries canno be simplified leading o a lack of geomeric fideliy; or exure disorions and arefacs appear near he simplified boundaries. In his paper, we presen an edge-collapse based simplificaion mehod ha provides an accurae, low-resoluion approximaion from a muli-char exured model. For each collapse, he model is reparameerized by local bijecive mappings o avoid exure disorions and char boundary arefacs on he simplified mesh due o he geomery changes. To beer apply he appearance aribues and o guaranee geomeric fideliy, we drive he simplificaion process wih he quadric error merics weighed by a local area disorion measure. Keywords: surface simplificaion, geomeric error, exure mapping, exure disorion, muli-char parameerizaion ACM CCS: I.3.5 [Compuer Graphics[: Curve, surface, solid, and objec represenaions; I.3.7 [Compuer Graphics]: Colour, shading, shadowing, and exure 1. Inroducion Many compuer graphics and geomeric modelling applicaions require highly deailed surfaces enhanced wih colour deails involving exure mapping o obain a convincing level of realism. However, he high complexiy of he models combined wih he limied capabiliies of curren hardware makes i difficul o process hem. Because heir full complexiy is no always required, i is useful o have simpler versions of hese complex models. Appearance aribues are mapped o he model wih he aid of a surface parameerizaion. For closed surfaces, surfaces of genus greaer han zero or surfaces exhibiing a complex shape, i is necessary o pariion he surface ino several chars o reduce he disorion. This is done by defining a exure alas, which maps he surface of he hree-dimensional (3D) model ino a non-conneced 2D domain. This is known as muli-char parameerizaion. In recen years, various simplificaion echniques for auomaically generaing high-qualiy, appearance-preserving Journal compilaion c 2010 The Eurographics Associaion and Blackwell Publishing Ld. Published by Blackwell Publishing, 9600 Garsingon Road, Oxford OX4 2DQ, UK and 350 Main Sree, Malden, MA 02148, USA approximaions of original models have been proposed. There are wo differen sraegies in he simplificaion field. Firs, here is level-of-deail modelling, which focuses on he rade-off beween fideliy and rendering performance. The aim is o obain a sequence of geomeric approximaions ha share he same exure map and swich hem during rendering, depending on he disance beween he objec and he viewer. The second sraegy consiss of generaing a single, low-resoluion approximaion ha resembles he original one bu has far fewer faces. In his case, he models are generally acquisiions from real objecs using 3D laser scans. Single approximaions of hese virual models are commonly applied in several fields, such as realisic virual environmens. In his paper, we presen a new approach o his second sraegy Relaed work In recen years, various simplificaion algorihms have been proposed. Face-reducion sraegies are commonly based on he edge collapse operaor. Edges are ordered according o a

2 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models 1843 cos and in each ieraion, he lowes-cos edge is conraced and he cos of is neighbouring edges is updaed. Geomeric simplificaion of meshes was he firs opic sudied in his field. The simplificaion process was guided only by a geomeric error, wihou aking ino accoun he appearance aribues of he model. An exensive review of all exising mehods proposed before 2002 can be found in [LWC 02]. Some of he main references are [HDD 93, Hop96, CVM 96, GH97, EM98, LT98]. More recenly, simplifying meshes wih appearance aribues has become he main area of ineres. In his case, he local simplificaion operaors ry o generae faihful geomeric approximaions while minimizing exure deviaion. The exure deviaion of a mapped aribue is defined as he deviaion of is posiion on he final surface from is posiion on he original surface. In [GH98], Garland and Heckber exended he original Quadric Error Meric (QEM) scheme o accoun for a wide range of verex aribues. Hoppe [Hop99] made several improvemens o Garland and Heckber s approach by compuing he error based on he difference, in posiion and aribue, beween a given poin and is closes projecion on he simplified surface. Cohen e al. [COM98] proposed a simplificaion algorihm guided by he exure deviaion. Lindsrom e al. [LT00] inroduced he image-driven simplificaion concep, where muliple images were used o decide which porions of a model have o be simplified. Sander e al. [SSGH01] presened a muli-char parameerizaion, consrained o convex and sraigh char boundaries, o be used in a progressive mesh. Unforunaely, for a small number of chars, he char boundaries do no usually follow he creases of he model and, consequenly, disored resuls may be obained. These approaches guide he simplificaion process by measures ha ry o balance geomeric fideliy and exure preservaion. Neverheless, significan exure disorions and arefacs can be observed on he simplified surfaces. The fundamenal reason is ha he exure map, even when i is very well parameerized, is accessed hrough linear inerpolaion of exure coordinaes on he simplified riangles. One way o minimize he exure disorion is he exure adapaion echnique, proposed by Chen and Chuang [CC06], which adaps he exure conen for each edge collapse. However, commonly used muli-char parameerizaions are no suppored. The presence of char boundaries hinders he qualiy of he simplificaion due o he disconinuiy in exure space. If boundary edges are no allowed o be collapsed, he resuling approximaion presens a poor qualiy mesh wih a lo of skinny riangles near he seams. Furhermore, exure disorions or arefacs appear near char boundaries. There are wo main sraegies o avoid geing seams. The firs one consiss of simplifying he geomeric model wih any of he surface simplificaion mehods, and hen parameerizing he simplified mesh and finally projecing he aribues, such as exure, of he high polygonal model ono he lower one ([TCS03]). The fideliy of he resul depends on various facors, such as he complexiy of he original mesh, he qualiy of he simplificaion mehod or he poin-o-poin correspondence sraegy used o projec he aribues. The mos commonly used sraegy is normal projecion ha needs he coninuiy of he normal field over each face and does no guaranee ha every aribue of he exure is projeced ono he simplified surface. The second sraegy is o choose a exure domain wih he same opology as he given mesh and a similar shape. The survey by Hormann e al. [HLS07] gives complee, deailed informaion on hese seamless exuring parameerizaion echniques. One of he mos powerful of hese mehods is PolyCube-Maps ([THCM04] and [LJFW08]). A polycube is a 3D shape, composed of many uni-sized cubes aached face-o-face, which is used as he exure domain. However, he main drawback of PolyCube- Maps is he fixed resoluion ha has o be carefully chosen o mach he geomeric feaures, which would give a parameerizaion wih oo many cubes for a surface wih complex geomery or opology. [GP08] proposes a mehod ha allows he use of a seamless parameerizaion for simplificaion purposes ha is independen of he parameerizaion provided by he aris for exuring. The main drawback of his echnique is he use of a common bijecive parameerizaion for all he levels of deail Our conribuions In his paper, we presen a mehod ha generaes a single and accurae, low-resoluion approximaion of a muli-char exured model ha preserves geomeric fideliy while avoiding exure disorions and arefacs. The main conribuions of his work are as follows: Bijecive mappings: For each edge collapse, he mesh is reparameerized by local bijecive mappings o avoid disorions o he appearance of he simplified mesh produced by geomeric changes. These bijecive mappings are GPU-friendly: fully suppored by he exure mapping hardware, he render-o-exure feaure and he fragmen shading. To avoid blurring arefacs due o an excessive number of resampling operaions, he mappings are applied o an index exure. Each exel of his index exure sores he exure coordinaes referring o he original exure. The echnique suppors any arbirary muli-char parameerizaion wih he only requiremen being ha he index exure has o be empy near concave pars of is char boundaries. In Secion 7, we will see ha his is no a srong requiremen. Subsequenly, our approach allows he char boundary edges o be simplified, guaraneeing geomeric fideliy and avoiding arefacs. Weighed QEMs: We have modified he QSlim [GH97] simplificaion mehod by weighing he quadrics wih a local area disorion measure o preserve highly curved

3 1844 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models regions and consequenly beer apply he exure of he original model. 2. Overview of our Proposal When simplifying a exured surface, S, he riangles creaed for each edge collapse have o be parameerized, ha is exure coordinaes have o be assigned o he verices of he new riangles. However, his is no sufficien o preven exure disorions due o he lineariy of he exure mapping. Therefore, we propose no only o modify he exure coordinaes, bu also o modify he exure conen in order o preserve he appearance of he model more accuraely. Firs he model is parameerized by an index exure, I, whose exels sore exure coordinaes referring o he original exure, T. This index exure, I, is modified in he course of edge collapsing. We denoe by S i and I i he simplified surface and he index exure before he collapse of he edge, e i. The edge-collapse operaor modifies I i and we obain I i+1 from he riangles involved in S i, heir corresponding exure riangles in I i and he riangles involved in S i+1. A firs, i seems reasonable o assign o a poin on S i+1 he exure aribue of is closes poin on S i.however,his crierion does no produce a bijecion beween S i+1 and S i and, consequenly, exure aribues may be los. In [CC06], his crierion is applied only for a reduced number of poins in he following way: given wo edges of he involved riangles, one of S i and one of S i+1, whose corresponding exure edges inersec, heir closes poins are compued and he exure aribue of he poin on S i is assigned o he poin on S i+1. These assignmens creae wo cell pariions, one in I i and one in I i+1, and a cell correspondence beween hem. The exure modificaion is done by mapping a cell in I i o is corresponding cell in I i+1. The main drawbacks of his echnique are ha cell overlapping may be produced and ha a large number of poin assignmens and cells are needed for compuaion when full-edge collapses are used. In conras, our mehod akes advanage of he orhogonal projecions of he surface riangles involved in S i and S i+1 ono he angen plane, π, of he poin of collapse, v i.only when hese wo projecions are bijecive is he edge-collapse allowed. The new riangles in I i+1 are filled wih a fragmen shader by firs mapping he riangles in I i o he projeced riangles in plane π, and hen mapping hese laer riangles o he new riangles in I i+1. In fac, hese wo mappings are linear bijecions and can be merged ino jus one by using heir composiion. In his way, S i and S i+1 have he same appearance from a viewpoin placed on he normal direcion of v i. Observe ha, in conras o poin-o-poin correspondence sraegies, in our approach, every exure aribue will appear on he simplified surface. Moreover, he exure deviaion has o be minimized, ha is he disance from he poin on S i of an aribue in exure I i and he poin on S i+1 of he same aribue in exure I i+1 mus be minimal. Consequenly, he more similar a surface riangle is o is projeced riangle ono he angen plane and o is corresponding exure riangle, he more minimal he exure deviaion is. Because he posiion of a poin in a polygon can be derived from he areas of he riangles deermined by he poin and he polygon edges, we use he area as a measure of similariy beween riangles. Consequenly, our simplificaion process incorporaes a measure of disorion beween he area of he surface riangles and he area of heir projecions ono he angen plane, and he new riangles in I i+1 are deermined by he areas of he new involved surface riangles in S i+1. Therefore, he fundamenals of our mehod, which will be described in more deail in he following secions, are as follows: Using an index exure ha avoids blurring (Secion 3). A simplificaion process guided by a local area disorion measure (Secion 4). An edge managemen ha (Secion 5). decides if a candidae edge can be collapsed by aking ino accoun is adjacen surface riangles and heir corresponding exure riangles and deermines he new surface riangles and he new exure riangles. GPU-friendly bijecive mappings ha fill he new exure riangles (Secion 6). 3. Index Texure A he beginning of he process, an index exure, I, encoded wih wo floa channels o ge enough precision, is creaed wih he same size as T, and each one of is exels sores is respecive coordinaes. Texure I can be creaed wih he same parameerizaion as he original model or wih anoher one. When using a parameerizaion mehod ha minimizes he disorion beween surface riangles and exure riangles, such as [LPRM02, SCOGL02, SWG 03] or [ZSGS04], more accurae resuls are obained. The whole process is applied o I, producing he successive modificaions, I i. In his way, he original exure, T, is no under-sampled and, consequenly, blurring arefacs produced by he ieraive resampling do no appear during he simplificaion process. Once he simplified level is obained, we generae he exure of he oupu model by ransferring he conens of he original exure via he I exure coordinaes. 4. Weighed Quadric Error Merics The QSlim [GH97] algorihm is an incremenal mehod ha simplifies a mesh by ieraively collapsing edges ordered by increasing errors provided from he QEM.

4 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models 1845 weighed he QEM of each verex by is saliency. They define he mesh saliency using a cenre-surround operaor on Gaussian-weighed mean curvaures o capure wha mos would classify as visually ineresing regions on a mesh. The main drawback of his mehod is he empirical and heurisic naure of he saliency measure, which needs user parameers. Figure 1: Curved regions (horns and ears) no preserved by he QSlim approach. In conras, hey are well preserved by our approach. Boh simplified versions have he same number of faces. Oher merics o guide he simplificaion process have been proposed. Among hem, Park e al. [PSC06] proposed an area-based meric. The conracion cos of every edge is compued by he absolue difference beween he areas of he mesh before and afer he conracion. In conras, we wan o minimize he disorion beween he area of surface riangles and he area of heir projecions ono he angen plane. For his, we propose o preserve he curved feaures by weighing he quadrics by a local area disorion measure. In he following secions, we will define his measure and how i is used in our simplificaion process. Le F be he se of faces inciden a verex v. Each face f F is conained in a plane π f defined by n u + d = 0, where n is is uni normal. The squared disance of any poin u o his plane is (n u + d) 2 = u T (n n T )u + 2(dn) T u + d 2. The fundamenal quadric Q f of π f is defined by Q f = (A, b, c) = (n n T, dn, d 2 ), and he squared disance can be compued as Q f (u) = u T Au + 2b T u + c. The quadric of he verex, v, is defined as he weighed sum of hese fundamenal quadrics, Q = f F Q f w f,wherew f is he area of he riangle, f. Given a quadric Q, poin u = A 1 b minimizes he quadraic error Q(u). Afer compuing quadrics for all verices, he conracion cos or he error of every edge, e = (v i,v j ), is compued by (Q i + Q j )(u ij ), where poin u ij minimizes he quadraic error (Q i + Q j )(u), where Q i and Q j are he quadrics of verices v i and v j. The algorihm has many advanages, which are jusified heoreically in [HG99]. The mos imporan quadric properies are he preservaion of he Gaussian curvaure of he model and he good aspec raio of he simplified model riangles. The auhors prove ha, in he limi, he quadric error is minimized by riangulaions wih opimal aspec raio. Nowihsanding all hese advanages, imporan feaures of he objecs may no be well preserved due o heir low geomeric error wih respec o he whole model. To illusrae his, we show an example in Figure 1. In he image, we can see ha he ears and he horns of he simplified cow are no well preserved. Moreover, his behaviour produces a poor visual appearance of a simplified exured model due o high exure disorions near hese feaures. Kho and Garland [KG03] inroduced a user-guided simplificaion mehod by weighing he QEM of each verex. The order of he edge conracions is manipulaed by muliplying he quadric of each verex by some scalar facor a he iniializaion sep. By selecing differen weighs he user can conrol he relaive imporance of differen surface regions o preserve he desired feaures. Lee e al. [LVJ05] 4.1. Local area disorion measure Firs, we need o recall some heoreical aspecs of he QEM ([HG99]) ha are used o define and jusify he proposed local area disorion measure.lenbe he surface normal a verex v, lek 1,k 2 be is principal curvaures and e 1,e 2 heir corresponding principal direcions. In he orhogonal coordinae frame wih origin a v and axes e 1,e 2,n, he second-order local approximaion of he surface is a pach of he form S(x,y) = (x,y, k ) 1x 2 + k 2 y 2, 2 (x,y) [ ε 1,ε 1 ] [ ε 2,ε 2 ]. By considering only he lower erms of he Taylor series approximaion, we have A is a diagonal marix wih enries a 11 = 4 3 ε3 1 ε 2k 2 1,a 22 = 4 3 ε 1ε 3 2 k2 2, a 33 = 4ε 1 ε 2 a 11 + a The area of he surface pach is ( a 11 + a 22 + a 33 = 4ε 1 ε ε2 1 k2 1 + ) ε2 2 k Thus, he disorion facor beween he area 4ε 1 ε 2 of he domain and he area of he surface pach is a 11 + a 22 + a 33 = 1 + ε2 1 k2 1 + ε2 2 k2 2. 4ε 1 ε 2 6 Observe ha he area disorion depends on he erm ε2 1 k2 1 +ε2 2 k2 2 6, which we call he local area disorion measure and can be compued by

5 1846 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models a 11 + a 22 = race(a) a 33. a 11 + a a 33 race(a) + a 33 Due o he invariance of he eigenvalues of a marix under bijecive linear ransformaions, he local area disorion measure can be compued in he canonical coordinae frame by race(a) λ race(a) + λ, where λ is he A eigenvalue corresponding o he eigenvecor closes o he surface normal a v Simplificaion process We guide he order of he edge collapses using he weigh derived from he local area disorion measure o guaranee a balance beween he minimizaion of he geomerical error and he minimizaion of he exure deviaion. This weigh can be compued for he iniial quadrics and for he newly creaed quadrics. Inuiively, recompuing he weighs during he process seems a beer soluion. Experimenal resuls have confirmed his and we have modified he simplificaion process in his way. A he iniial sep each verex, v i,has assigned o i he quadric Q i = (A i,b i,c i ). Then, we have o compue he conracing cos of each edge e = (v i,v j ) o creae a keyed heap. We use he quadric w i Q i + w j Q j, where he weighs w i are compued by w i = race(a i) λ i. race(a i ) + λ i Consequenly, he conracion cos of every edge e = (v i,v j ) is compued by (w i Q i + w j Q j )(u ij ), where poin u ij minimizes he quadraic error (w i Q i + w j Q j )(u). Moreover, he quadric assigned o he new verex u ij has o be w i Q i + w j Q j w i + w j in order no o overesimae is weigh since i will be recompued. This is a simple and efficien soluion, which allows us o preserve he curved feaures of he models. Consequenly, alhough he global error increases slighly, he exure will be beer mapped o he simplified model. Figure 1 shows an example of he resuls obained by our approach using he cow model. There are eviden differences beween our resuls and hose obained by he QSlim approach. Wih our approach, he horns and he ears are preserved wihou losing hecorrecshape.figure2showsawholeviewandaclose-up of he Armadillo model simplified by he QSlim approach and ours. I can be seen ha, in conras o QSlim, our approach Figure 2: Simplificaion of he Armadillo model wih QSlim and our approach. Comparison of he whole model and a close-up deail of is hand. preserves all he fingers even when he approximaion has only 0.125% faces of he original one. 5. Edge Managemen The edges have o be collapsed while boh he surface and he parameerizaion opology are preserved. Thus, necessary consisency checks mus be carried ou ([DEGN99]). In he following a riangle on S i is denoed by, is corresponding riangle in I i by, a riangle on S i+1 by and is corresponding riangle in I i+1 by. The same noaion is applied for edges and verices. Le e be a candidae edge o be collapsed and v be is poin of collapse wih angen plane π. Le{ 0,..., k } and { 0,..., k 2 } be he riangles involved in he collapse on Si and on S i+1, respecively. Edge e is no allowed o be collapsed if a leas one of he following consrains holds: e has only one endpoin on a char boundary; e is a non-char boundary edge wih boh endpoins on a char boundary; e has an endpoin lying on more han wo chars; he orhogonal projecion ono π of riangles { 0,..., k } or riangles { 0,..., k 2 } overlap. These consrains enforce char compliance and ensure a bijecion beween S i and S i+1. When an edge e does no fulfil any of he previous consrains, we need o deermine he new riangles { 0,..., k 2 }

6 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models 1847 Figure 3: New exure riangles of a char boundary edge. in I i+1. There are wo possible cases depending on wheher e lies on a char boundary or no. Char inerior edge. LeU be he union of exure riangles { 0,..., k }. For each boundary edge b j of U, le a j (x) be he relaive area of he riangle deermined by b j and a poin x U wih respec o he area of U, and le a j be he relaive area of he riangle j wih respec o he oal area of he riangles { 0,..., k 2 }. Because we wan o minimize he area disorion measure beween riangles in S i+1 and riangles in I i+1,weakeashe corresponding poin of v in I i+1 he poin v U ha minimizes he quadraic funcion ( ) aj (x) 2 1. a j j Taking ino accoun he poin v and he boundary edges b j, we obain he riangles { 0,..., k 2 }. If hese new riangles overlap, he edge e is also no allowed o be collapsed. Char boundary edge (Figure 3). There are wo exure edges e 1, e 2 corresponding o surface edge e. Each one is a boundary edge of is own char. For each subse of riangles of he same char, we apply he same process as in he firs case, resricing he opimal poin o lie on he boundary edge, e 1 or e 2. Observe ha, when a char is no convex, he new riangles inersec he exerior of concave pars of he char. For his reason, we need he exure o be empy near concave pars of he char boundaries. Therefore, in he wors case, he convex hull of each char should be empy Edge flip To improve he fiing of simplified models o original models and o creae well-shaped riangles in fla regions, we use an edge flip operaor. I akes wo adjacen riangles as parameers and swaps he shared edge wih is opposie diagonal. This operaor has o be guided by an accurae measure o decide which of he wo edges, he acual or he opposie one, mus be kep in order o beer approximae he mesh o he iniial one. Figure 4: Mappings for filling he new exure riangles. For each edge collapse, non-char boundary edges of riangles { 0,..., k 2 } are flip-esed. The crierion used o decide he flip operaion is he one proposed by Jiao e al. [JCNH06], which uses he resricion of he QEM ono he angen plane of each verex as a local meric, and applies he modified Delaunay crierion proposed by Bossen and Heckber [BH96]. The non-overlapping checks are carried ou on each angen plane of he four verices for he surface riangles and on he exure map for he exure riangles. 6. Bijecive Mappings Once an edge collapse has been carried ou, he exels in riangles { 0,..., k 2 } mus be filled wih he conen of exels in riangles { 0,..., k }. As we briefly described in Secion 2, our mehod akes advanage of he orhogonal projecions of he involved surface riangles in S i and S i+1 ono he angen plane π of he poin of collapse v. Thus, our basic idea consiss of ransferring he conen of a exel p o a exel p having he same projecion ono π of heir corresponding surface poins. Figure 4 illusraes he siuaion and we will now describe how o implemen i. On plane π, we consider an orhogonal frame cenred a v. For any riangle or on he surface, we have wo linear ransformaions, M and P. The ransformaion of M maps o is corresponding riangle in he exure, while he ransformaion of P maps o he orhogonal projecion of ono π. LePM be he linear ransformaion P M 1. For each exel p of a riangle, we have he poin p on saisfying M (p ) = p and he poin p conained in a riangle of S i saisfying P (p) = p = P (p ), where p π. Le

7 1848 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models Figure 5: Blurring arefacs disappear when using an index exure during he simplificaion process. p be he poin in deermined by p = M (p). Thus we have PM (p) = (P M 1 )(p) = (P M 1 )(p ) = PM (p ). Our approach consiss of filling exel p wih he conen of exel p wihou compuing i explicily. Due o he bijecive naure of he defined linear ransformaions for a fixed riangle we have If j l, (PM 1 PM j )( j ) (PM 1 PM l )( l ) =. j (PM 1 PM j )( j ) =. Consequenly, by using sandard hardware capabiliies, we can fill he exels in a riangle by mapping ono he riangles (PM 1 PM j )( j ) wihou explicily compuing heir common inersecion. However, because we are dealing wih a muli-char alas, he ransformaion ha, for each I i exel, maps he T exure coordinaes sored in i, is no coninuous. Thus, he sandard hardware bilinear inerpolaion of T exure coordinaes migh cause invalid T exure coordinaes. To solve his problem, we map he riangles (PM 1 PM j )( j ) wih a fragmen shader. For each fragmen wihin wih I i coordinaes (u, v), we compue he four closes exels o (u, v), wih a being he closes. Then, we compue he index of he fragmen by a bilinear inerpolaion of he I i indices sored a hese four exels, aking ino accoun only hose he indices ha lie in he char of he index sored a a. Figure 5 shows he good performance in reducing blurring arefacs resuling from using an index exure. 7. Resuls Our approach allows us o simplify large models while preserving heir mos imporan deails wihou losing fideliy. We obain single approximaions as close as possible o he original models, which may be useful in various fields of applicaion where he frame rae is more imporan han insignifican deails. Several ess have been done on differen models o sudy he accuracy of our mehod. Figure 6: Simplificaion of a exured Cow model. Comparison beween our approach and hree differen exure preserving mehods. The empy space of he alas has been filled in wih red. Figure 6 compares our exured cow approximaion resul wih he resuls obained using he open-source ool Mesh- Lab [CCR08, CCC 08], he commercial package Polygon Cruncher [Moo] and Hoppe s approach [Hop99]. The firs wo ools also allow us o preserve he boundary edges by simplifying char ineriors only. Acivaing his opion generaes a poor qualiy, low-resoluion model full of seams is generaed ha canno achieve he desired simplificaion level. To es how well our approach behaves, we have filled in he empy space of he alas wih red. Observe how our approach is he only one able o generae simplified models wihou exure disorions or arefacs near he char boundaries by using only one addiional exure during he simplificaion process. Anoher comparison is presened in Figure 7. As you can see, he bunny model generaed wih [SSGH01] presens

8 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models 1849 Figure 7: Simplificaion of a exured Bunny model. Comparison beween [SSGH01] and our approach. Figure 9: Two differen parameerizaions for he Vase model. a high-qualiy mesh and correcly preserved exure wihou disorions. Figure 8 shows he Buffle model parameerized wih a high number of small sized chars. A close-up view of he model shows ha neiher exure disorions nor arefacs appear in he simplified version, despie he presence of char boundaries. Figure 8: Buffle model simplificaion. The alas, he mesh wih char boundaries depiced in red, he exured model and a close-up view are shown for original and simplified models. percepible exure disorions because of he reparameerizaion, wih convex and sraigh char boundaries, ha is carried ou o be used on a progressive mesh. In conras, wih our approach we obain an accurae approximaion wih Parameerizaions wih a lo of chars reduce he level of simplificaion because he chars mus all be preserved. Figure 9 shows a scanned Vase model parameerized wih LSCM [LPRM02] and Iso-chars [ZSGS04]. Noice ha he firs one has a huge number of chars, whereas he second one only has 24 chars. In Figure 10, we can see he resuls of our simplificaion of hese wo parameerizaions of he Vase model a he same level (90% simplificaion). Observe ha parameerizaions wih less chars allow us o obain beer geomeric qualiy and also o subsanially increase he simplificaion level (98% simplificaion). The flexibiliy ha resuls from using parameerizaions ha admi non-convex chars is an imporan advanage wih respec o [SSGH01].

9 1850 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models Figure 10: Vase model simplificaion resuls according o he parameerizaions in Figure 9. imporan geomeric feaures are los a low resoluions wih any simplificaion mehod. To capure all hese deails, heir normal maps were generaed before applying our mehod. Faihful and realisic approximaions are obained by applying bijecive mappings on colour and normal maps even a a simplificaion level of 95% for he Imperia and of 99% for he Lion. 8. Conclusions and Fuure Work Figure 11: Close-up view of he Vase model exure before and afer applying our bijecive mapping approach. Figure 11 shows a close-up view of he Vase model parameerized wih Iso-chars. Char boundaries have been properly simplified o give accurae approximaions wihou seams beween chars. Observe ha, even hough chars have inerchanged par of heir conens, heir global shapes are preserved due o he low sreching of he parameerizaion used. For his reason, if we use a good parameerizaion mehod o generae he index exure, empy exure near concave pars of char boundaries is no a srong requiremen. Figures 12 and 13 are represenaive examples of large meshes obained from real objecs. For each of hese, we would like o have a simple version ha shares he same special characerisics. No all he deails of hese models are represened in heir exure maps, since small bu We have developed a mehod for he auomaic simplificaion of a highly deailed polygonal surface model ino a single faihful approximaion conaining fewer polygons. The mehod consiss, on he one hand, of weighing he QEM by a local area disorion measure and, on he oher, of bijecive mappings ha properly modify an index exure for each edge collapse o avoid disorions on he appearance of he simplified mesh. In his way he char boundary edges are no penalized. The main benefi of his approach is ha realisic and accurae approximaions of complex exured models can be generaed. Exending our approach o obain a progressive mesh [Hop96, SSGH01] of he model wih all he inermediae meshes sharing a common exure parameerizaion could be significan. Because he bijecive mappings can be invered, our echnique could be applied on he fly while rendering a change in he level of deail. However, we believe ha his sraegy can only be applied for simple models. Consequenly, dealing wih progressive meshes will be an imporan par of our fuure work. Acknowledgemens The auhors hank he anonymous reviewers for proofreading various versions of his paper and for providing us wih

10 N. Coll & T. Paradinas / Accurae Simplificaion of Muli-Char Texured Models 1851 Figure 12: Imperia model simplified by our approach wih exure and normal maps. Observe ha, in boh global and close-up views, he visual appearance is preserved from he original one o he simplified one. Char boundaries (in red) have been simplified wihou producing arefacs. Figure 13: Lion model simplificaion resuls wih and wihou normal mapping. consrucive commens and criicism. This projec was funded by gran TIN C02-02 from he Spanish Miniserio de Educación y Ciencia. References [BH96] BOSSEN F. J., HECKBERT P. S.: A plian mehod for anisoropic mesh generaion. In Proceedings of he 5h Inernaional Meshing Roundable (Oc. 1996), pp [CC06] CHEN C.-C., CHUANG J.-H.: Texure adapaion for progressive meshes. Compuer Graphics Forum 25, 3 (Sepember 2006), [CCC 08] CIGNONI P., CALLIERI M., CORSINI M., DELLEPIANE M., GANOVELLI F., RANZUGLIA G.: Meshlab: An

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