Evolutionary Multi-Objective Optimization for Mesh Simplification of 3D Open Models

Transcription

1 Evolutonary Mult-Objectve Optmzaton for Mesh Smplfcaton of 3D Open Models B. Rosaro Campomanes-Álvarez a,*, Oscar Cordón abc and Sergo Damas a a European Centre for Soft Computng, Gonzalo Gutérrez Qurós s/n, 33600, Meres, Asturas, Span b Department of Computer Scence and Artfcal Intellgence, Unversty of Granada, Perodsta Danel Saucedo Aranda s/n, 18014, Granada, Span c Research Center on Informaton and Communcaton Technologes (CITIC-UGR), Unversty of Granada, Perodsta Rafael Gómez 2, 18014, Granada, Span Abstract. Polygonal surface models are typcally used n three dmensonal (3D) vsualzatons and smulatons. They are obtaned by laser scanners, computer vson systems or medcal magng devces to model hghly detaled object surfaces. Surface mesh smplfcaton ams to reduce the number of faces used n a 3D model whle keepng the overall shape, boundares and volume. In ths work, we propose to deal wth the 3D open model mesh smplfcaton problem from an evolutonary multobjectve vewpont. The qualty of a soluton s defned by two conflctng objectves: the accuracy and the smplcty of the model. We adapted the Non-Domnated Sortng Genetc Algorthm II (NSGA-II) and the Mult-Objectve Evolutonary Algorthm Based on Decomposton (MOEA/D) to tackle the problem. We compare ther performance wth two classc approaches and two sngle-objectve mplementatons. The comparson has been carred out usng sx dfferent datasets from sx correspondng real-world objects. Expermental results have demonstrated that NSGA-II and MOEA/D performs smlarly and obtan the best solutons for the studed problem. Keywords: 3D modelng, mesh smplfcaton, evolutonary mult-objectve optmzaton, NSGA-II, MOEA/D 1. Introducton Polygonal surface models are the representaton of 3D vsualzatons and smulatons. They are obtaned by laser scanners, computer vson systems or medcal magng devces to model hghly detaled object surfaces. These surface models are used n many dfferent areas such as computer vson, computer-aded desgn, medcne, and topography [34, 5, 55, 31, 2]. Typcally, a surface model conssts of thousands of polygons. The sze of the correspondng fle usually causes long processng tmes [36]. Obtanng a reduced model wth a smaller polygonal surface and a smlar accuracy s a challenge n the area. Surface mesh smplfcaton s the process that ams to reduce the number of polygons used n a surface whle preservng the overall shape, volume, and boundares as much as possble [12]. There are several technques n order to smplfy a mesh. Decmaton [50], clusterng [42], and energy functon optmzaton [30] are the most popular and classc methods. Recently, some other proposals have been made consderng alternatve ways to smplfy 3D surfaces based on advanced optmzaton technques as evolutonary algorthms [18, 32, 62]. Regardless the approach followed, all the latter methods consder the mesh smplfcaton problem as a sngle-objectve optmzaton task, ether n a drect or ndrect way. Nevertheless, two man crtera can be dentfed measurng the qualty of the reduced 3D model, namely ts accuracy to approxmate the orgnal mesh and ts assocated sze, measured n terms of the number of polygons composng t. These two goals are clearly conflctng n nature as the more complex a mesh s, the more accurate t wll be, and vce versa. * Correspondng author. E-mal: rosaro.campomanes@softcomputng.es 1

2 Hence, formulatng mesh smplfcaton as a multobjectve optmzaton problem (MOP) [11, 13, 1] can arse as a promsng and very novel alternatve to tackle ths complex 3D modelng task. In partcular, mult-objectve evolutonary algorthms (MOEAs) [13, 57] have largely demonstrated ther capablty to deal wth many dfferent knds of MOPs n a very effcent way [11, 13, 15, 69]. Ths famly of methods shows the mportant advantage of beng able to provde the user wth a Pareto set of non-domnated solutons wth dfferent trade-offs on the satsfacton of the optmzed objectves n a sngle run. In the mesh smplfcaton framework, ths would mean obtanng several and dverse alternatve reduced 3D models wth dfferent compromses between ther accuracy and ther complexty (number of trangles), thus allowng the user to select the most approprate for hs/her specfc condtons. Thus, the formulaton of mesh smplfcaton as a MOP and the comparson wth classc methods s an nterestng research lne whch has been studed n ths work. Up to our knowledge, ths s the frst study where mesh smplfcaton s tackled from a multobjectve vewpont n the lterature. Our methodology s thus based on the smplfcaton of a 3D open model by an evolutonary multobjectve algorthm. An open model refers to a surface wth open ends. The problem s based on the locaton of a certan number of ponts n order to approxmate a mesh as accurately as possble to the ntal surface. It wll consder two conflctng objectves, the accuracy and the smplcty of a smplfed mesh. In a prevous research [8], we proposed the use of the computatonally fast and extended nondomnated sortng genetc algorthm (NSGA-II) [15] MOEA to tackle the mesh smplfcaton problem as a proof of concept. In the current contrbuton, we am to demonstrate the good performance of our methodology by: ) showng how any other MOEA can be consdered, and ) developng a deeper expermentaton to valdate t wth respect to both some classcal and sngle-objectve evolutonary mesh smplfcaton methods. To do so, we have used the mult-objectve evolutonary algorthm based on decomposton (MOEA/D) proposed n [66], whch decomposes a MOP nto a number of scalar optmzaton sub-problems and optmzes them smultaneously. Moreover, we also analyzed the method developed by Huang et al. n [32], that uses a sngleobjectve algorthm to smplfy 3D facal meshes, as well as two classc approaches, edge collapse decmaton wth a quadrc error metrc [27, 50, 20, 59], and vertex clusterng wth topology preservng [42]. Ths work s structured as follows. Secton 2 ntroduces a short survey regardng classc technques for mesh smplfcaton and some others that consder evolutonary algorthms for that task. Secton 3 descrbes all the components of the proposed approach. Secton 4 presents the performed experments and the results obtaned. Secton 5 concludes the whole work. 2. State of the Art n Mesh Smplfcaton Many dfferent mesh smplfcaton approaches have been proposed n the specalzed lterature [12, 27]. They can be ether local or global. The former methods smplfy a mesh by the teratve use of some local operator. The latter are appled to the nput mesh as a whole. The followng two subsectons revew local and global famles of methods respectvely. Fnally, subsecton 2.3 presents the exstng evolutonary approaches to the problem Incremental Methods Based on Local Updates The methods n ths group run the smplfcaton process as a sequence of local updates. Each update reduces the mesh sze and decreases the surface approxmaton accuracy. The decmaton method [50, 29, 23] can be classfed nto three dfferent approaches accordng to the dfference of the selected objects: removal of an edge, removal of a trangle, and removal of a vertex. The latter approach can be guded by a quadrc error metrc algorthm [20, 59] based on the teratve contracton of vertex pars. Another example of an teratve method s the energy functon optmzaton approach [30, 46]. The mesh reducton s teratvely obtaned by performng legal moves on mesh edges: collapsng, swappng, or splttng. The progressve meshes method [29] s an enhanced verson of the splttng technque. Other methods based on surface sgnal [46] are used to smplfy meshes by manly usng textures and colors Non Incremental Global Methods The non ncremental global methods smplfy the mesh as a whole. Among them, the coplanar method [28] uses a named detectng plane to determne whether a vertex s near enough. 2

3 The re-tlng method [60] starts wth a polygonal surface and creates a trangulaton of t wth a userspecfed number of vertces. The number of polygons shared by any gven edge s the man restrcton. The clusterng technque [42] s based on geometrc closeness. It uses the cube or octree neghborhood structured to group nearby vertces nto a cluster. For each cluster, the method generates a new representatve vertex [22]. Ths method preserves the topology of the mesh. Besdes, the algorthms based on wavelets [21, 26] and the smplfcaton usng envelopes [14] provde tght error bounds on arbtrary trangulated meshes whle allowng topologcal changes durng the smplfcaton Mesh Smplfcaton Approaches Based on Evolutonary Algorthms Computatonal ntellgence technques have dealt wth a wde varety of problems rangng from operatonal cost optmzaton [51, 45, 4], engneerng desgn [35, 40, 52, 10], to copyrght protecton and data authentcaton [58]. Evolutonary algorthms have been largely and successfully appled to many dfferent computer graphcs, computer vson, and mage processng tasks [9, 7, 64, 65, 53, 43, 44, 67, 47, 25]. In partcular, there are a few studes based on applyng evolutonary computaton to deal wth the mesh smplfcaton problem [18, 32, 62]. In [18], Fujwara and Sawa tackled the problem of approxmatng a human facal surface by constructng a trangular mesh wth a lmted number of sample ponts. They developed a sngle-objectve genetc algorthm that selects a gven number of ponts from the whole dataset and consders a Delaunay trangulaton to buld the smplfed 3D model. Huang and Ho [32] proposed an evolutonary algorthm as an extenson of Fujwara and Sawa s proposal. The mprovement s based on usng the orthogonal array crossover (OAX) (see Secton 3.3). Fnally, Xandong et al. proposed a method of trangular mesh reducton based on a new concept called super-face and the use of a genetc algorthm n [62]. 3. Evolutonary Mesh Smplfcaton of 3D Open Models In ths secton, we formulate the 3D open model mesh smplfcaton problem. We frst detal a proposal based on Huang and Ho s method [32] and then descrbe our mult-objectve approaches to tackle ths complex problem Problem Formulaton Let M be a scanned 3D open model (Fgure 1). It s possble to reduce ths polygonal surface to a two dmensonal problem. A 3D surface of ths knd can be represented by the functon: 2 f : ( x, y) R z R (1) We have defned the mesh smplfcaton problem n the 2D space due to effcency purposes. The algorthm works wth 2D models, whch store fewer ponts than a 3D surface 1. Fgure 1. Surface representaton n the standard Cartesan coordnate system. Therefore, we need to locate n 2D ponts wth n beng less than N (the number of ponts of the orgnal mesh). The number of ponts to be located can be ether fxed a pror or automatcally chosen by the algorthm. Our experments are based on the latter opton,.e., the algorthm attempts to solve the problem wth few data. After locatng the n ponts, a trangulaton s performed usng the new number of ponts n order to generate an approxmate polygonal (trangular) mesh surface. In partcular, methods descrbed n Secton 2.3 consder the Delaunay trangulaton [3]. It s commonly used n problems such as the mesh generaton process [63]. Gven a set of ponts P n the plane, a Delaunay trangulaton s a trangulaton D(P) such that no pont n P s on the crcumscrbed 1 Note that, n some concave surfaces or wth hgh topologcal homotopy class, the 3D-2D mappng s restrcted to some regons of the whole set [24]. However, ths lmtaton s not present n our approach. 3

4 crcle of any trangle of D (P). Delaunay trangulatons maxmze the mnmum angle of all angles of the trangle. They tend to avod sknny trangles. 2 Let Pn be an ntal set of ponts n R, ts correspondng Delaunay trangulaton s denoted by D P ). In our problem, a chromosome encodes a ( n confguraton P n of n ponts n the 2D space. The genotype space conssts of those 2D confguratons whle the phenotype space ncludes the correspondng 3D models obtaned usng D ( P n ). Let us have a populaton of N pop ndvduals, each ndvdual representng a certan mesh confguraton. An ndvdual s a smplfed mesh,.e., a mesh wth fewer ponts than the reference model. The members of the populaton share a global vector wth the orgnal mesh coordnates (orgnal model). Each poston of ths structure stores the x, y and z coordnates. Every ndvdual s defned by a bnary chromosome wth n genes (the number of ponts whch wll be located on the new smplfed mesh). A one value n the th poston of the chromosome vector means that the th vertex of the orgnal model remans n the smplfed mesh represented by such chromosome. On the contrary, a zero means that there s not a pont on the grd plane for ths poston. A graphcal representaton of the problem structures s shown n Fgure 2. number of ponts. Then, t carres out the smplfcaton and obtans a new 2D mesh wth n ponts ( P n ), where n < N. It apples Delaunay trangulaton to the new 2D mesh gettng D P ). Ths 2D trangulated fnal ( n mesh D( P n ) s converted nto 3D and thus a fnal 3D open model wth n ponts and ts correspondng number of trangles s obtaned. The algorthm selects the model that best approxmates the orgnal one,.e. the lowest error mesh. Fgure 3 presents the scheme to obtan a smplfed mesh from an orgnal 3D open model by means of a bnary-coded genetc algorthm. Fgure 3. Bref scheme of the mplemented method for mesh smplfcaton Objectves to Be Optmzed Fgure 2. Problem structures scheme. The four ponts of the mesh corners are ncluded n every chromosome. Ths avods an abnormal boundary shape thanks to the mantenance of the rectangular shape as a whole [18, 32]. The algorthm performs the followng smplfcaton process. Frst, t converts the orgnal 3D model of N ponts P ) nto a 2D model wth the same ( N We have consdered two objectves to be jontly maxmzed, accuracy and smplcty. For a better formulaton, the former s guded by the mnmzaton of an error metrc. The latter s gven by the mnmzaton of the number of trangles that compose the mesh (the lower the number of trangles s, the hgher the model smplcty s). Therefore, we am to mnmze two objectves, the error and the number of trangles. We have followed the same procedure presented by Huang et al. n [32] to calculate the approxmaton error,.e., the error between the orgnal and the approxmated meshes. Each Delaunay trangle T D( P n ) contans a certan number of mesh ponts (x, y). The dstance at each sample pont p s defned as follows: d p p p d p = z ~ z, (2) 4

5 where z p s the heght value of the surface at the pont p and z~ p s the lnearly nterpolated value of heght at p determned by the trplet of heghts for the three vertces of trangle T. The error e for T s the sum of these dstances over all the sample ponts p nsde T where: e = (3) d p p T The total approxmaton error e, defned by Eq. (4), s the sum of the errors e of all trangles that form the smplfed mesh. e = (4) e T D( P n ) 3.3. Recombnaton Recombnaton s the process n whch a new ndvdual soluton s created from the nformaton contaned wthn two (or more) parent solutons. It s one of the most mportant operators n genetc algorthms. Recombnaton s randomly appled accordng to a crossover ranges n rate p c, whch typcally ranges n [0.5, 1]. To apply recombnaton, two parents are selected and a random value u s generated unformly. If u s lower than p c, two offsprng are created va recombnaton of the two parents. Otherwse, they are created by drectly copyng the parents [17]. We have used two varants for the crossover n our MOEAs: the classc unform crossover [56] and the OAX recombnaton [41]. The unform crossover [56] s based on dvdng the parents nto a number of sectons of contguous genes and reassemblng them to produce offsprng. It works by treatng each gene ndependently and makng a random choce as to whch parent t should be nherted from. Ths s mplemented by generatng a strng of L random varables from a unform dstrbuton over [0, 1]. In each poston, f the value s below a parameter p (usually 0.5), the gene s nherted from the frst parent; otherwse from the second. The second offsprng s created usng the nverse mappng. The OAX recombnaton technque performed n [32] has also been consdered n order to mprove the performance of crossover. An effcent way to study the effect of several factors smultaneously s to use an orthogonal expermental desgn (OED) wth orthogonal arrays (OAs) and factor analyss (FA). Ths knd of desgn s consdered to provde the treatment settngs at whch one conducts the all-factors-atone statstcal experments [41]. The orthogonal desgn defnes some combnatons for each experment by usng factor levels. An OA s a matrx of numbers arranged n rows and columns where each row represents the levels of factors n each run and each column represents a specfc factor. In the context of expermental matrces, orthogonal means statstcally ndependent. FA can evaluate the effects of factors and determne the best level for each factor such that the evaluaton s optmzed. Usng OAX the offsprng chromosomes are formed from an ntellgent combnaton of the good genes from ther parents rather than the conventonal random combnaton. The best of the two genes n the two parents s chosen by evaluatng the contrbuton of ndvdual genes to the ftness functon based on OED. Let there be α factors wth two levels for each factor. The total number of experments s 2 α for the popular one-factor-at-a-tme study. The columns of two factors are orthogonal when the four pars (1,1), (1,2), (2,1) and (2,2) occur equally frequently over all experments. Generally, levels 1 and 2 of a factor represent selected genes from parents 1 and 2, respectvely. To establsh an OA of α factors wth two levels, we obtan an nteger β = [ log 2( α + 1) ] 2, buld an β 1 orthogonal array L β ( 2 ) wth ß rows and ß-1 columns, use the frst α columns, and gnore the other ß- α-1columns. The algorthm for constructng OAs can be found n [39]. An OED can reduce the number of experments for FA. The number of OA experments requred to analyze all ndvdual factors s only ß where α+1 ß 2α. After proper tabulaton of expermental results, the summarzed data are analyzed usng FA to determne the relatve effects of levels of varous factors. Let fe t denote a ftness value, n our case fe t s the total approxmaton error determned by Eq. (4), of the combnaton correspondng to the experment t, where t= 1,,ß. It defnes the man effect of factor j wth level k as S jk where j=1,, α and k=1, 2: β S = fe F, (5) jk t= 1 t t where F t = 1 f the level of factor j of experment t s k; otherwse, F t =0. In a mnmzaton problem, the level 1 of factor j makes a better contrbuton to the ftness functon than the level 2 of factor j does when S j1 < S j2 (the opposte stuaton occurs n maxmza- 5

6 ton problems). The most effectve factor j has the largest man effect dfferent (MED) S j1 - S j NSGA-II NSGA-II [15] s one of the most effcent and effectve MOEAs followng the eltst approach. Its partcular ftness assgnment scheme conssts of sortng the populaton n dfferent fronts wth a nondomnaton order relaton. Then, the algorthm combnes the current populaton and ts offsprng generated wth the standard bmodal crossover and polynomal operators to form the next generaton. Fnally, the best ndvduals accordng to non-domnance and dversty are chosen. Ths new verson of NSGA [54] s characterzed by a low computatonal complexty: O(NlogN), where N s the populaton sze. The pseudo-code of the NSGA-II method [15] s outlned n the Appendx C of the supplementary nformaton avalable at [73] MOEA/D A MOP can be stated as follows: T mnmze F ( x) = ( f ( x),..., f ( x)) 1 m, subject to x Ω, (6) m where Ω s the decson space, F : Ω R conssts of m real-valued objectve functons, and R s m called the objectve space. MOEA/D [66] s a recent proposal of a MOEA based on explctly decomposng the MOP showed n Eq.(6) nto N scalar optmzaton sub-problems. The algorthm solves these sub-problems smultaneously by evolvng a populaton of solutons. At each generaton, the populaton s comprsed by the best soluton found so far (.e. snce the start of the run of the algorthm) for each sub-problem. The neghborhood relatons among these sub-problems are defned based on the dstances between ther aggregaton coeffcent vectors. The optmal solutons to two neghborng sub-problems should be very smlar. Each sub-problem (.e. scalar aggregaton functon) s optmzed n MOEA/D by usng nformaton only from ts neghborng sub-problems. There are several approaches for convertng the problem of approxmaton of a Pareto front nto a number of scalar optmzaton problems. In the followng, we ntroduce the Tchebycheff approach, whch has been used n our expermental study due to the good results n terms of feasblty and effcency obtaned n [66]. Let λ = ( λ,..., 1 λ ) W m be a weght vector, and m λ be the number of sub-problems,.e., 0 for all =1,,m and = λ = 1 m 1.The Tchebycheff approach consders a scalar optmzaton problem n the form: te * * mnmze g ( x /, z ) mn{ f ( x) z } λ Ω z ( z,..., ) = 1 m subject to x, (7) * * W where * = s the reference pont. It s 1 z m ntalzed as the lowest value of the objectve functon f found n the ntal populaton. * For each Pareto optmal pont x there exsts a * weght vector λ such that x s the optmal soluton of Eq. (7) and each optmal soluton of Eq. (7) s a Pareto optmal soluton of the objectve functon. Therefore, the desgner s able to obtan dfferent Pareto optmal solutons by alterng the weght vector Desgn of a Sngle-Objectve Genetc Algorthm for Mult-Objectve 3D Open Model Mesh Smplfcaton As a frst approxmaton to the mult-objectve problem, we wll extend Huang and Ho s proposal [32] n order to compare t wth our Pareto-based evolutonary approach. Ths algorthm conssts of several components such as populaton ntalzaton, selecton scheme, genetc operatons, and termnaton crteron. As already ntroduced, each chromosome encodes a selecton of 2D ponts P, havng a phenotype gven by ts n Delaunay trangulaton D ( P n ). The ftness functon to be mnmzed has been adapted n order to consder the two objectves ntroduced n Secton 3.2: F( m) = we( m) + (1 w) T ( m), (8) where m s the smplfed mesh encoded n the chromosome, D(P n ); w [0, 1] s a weght, E(m) s the error determned by Eq. (4) to be mnmzed, and T(m) s the total number of trangles of m. The extenson of the orgnal method to tackle the ftness functon n Eq. (8) s based on an evaluaton functon combnng several objectves usng a weghted sum [11, 13]. Ths method generates a set of Pareto optmal solutons by gvng dfferent weghts to the functon and runnng repeatedly the algorthm. The pseu- λ 6

7 do-code of the extended sngle-objectve method s explaned as follows: Populaton ntalzaton: The populaton s ntalzed by randomly locatng n ponts to each ndvdual. The four ponts of the 2D mesh corners are fxed to mantan the boundares. Selecton scheme: It uses an eltst selecton model n such a way that the ndvdual wth better ftness s always kept n the new populaton. It ranks N ndvduals accordng to ths ftness gven by Eq. (4). Then, t apples tournament selecton [6]. Mutaton: Each located pont, coded at a gene n each ndvdual, s randomly moved to one of the nearest neghbors on the vector wth a probablty pm called the mutaton rate. It makes mutaton n each gene wth a mutaton rate equal to 1/chromosome_length. Crossover: From two randomly chosen parents, we generate two offsprng usng two possble crossover operators, OAX [41] or unform crossover [56] (see Secton 3.3). Termnaton crteron: To reach a maxmum number of generatons The NSGA-II Proposal As sad, two conflctng objectves are consdered: accuracy and smplcty. Therefore, the two objectves to mnmze are the error and the number of trangles of the mesh [8]. Gven a mesh m for the mult-objectve method, the ftness functon s as follows: 1 mn FM ( m) = E( m) (9) 2 mn FM ( m) = T( m) The method scheme s detaled below: Populaton ntalzaton: Ths procedure s the same than n the sngle-objectve algorthm populaton ntalzaton. Selecton scheme: The algorthm combnes the current populaton wth the obtaned offsprng usng recombnaton n order to generate the next generaton. The best ndvduals accordng to nondomnance and dversty are selected to be reproduced regardng the non-domnated fronts of NSGA- II (see Secton 3.4). Mutaton: Each located pont, coded at a gene n each ndvdual, s randomly moved to one of the nearest neghbors on the vector wth a probablty p called the mutaton rate. It makes mutaton n m each gene wth a mutaton rate equal to 1/chromosome_length. Crossover: From two randomly chosen parents, we generate two offsprng usng two possble crossover operators, OAX [41] or unform crossover [56]. Termnaton crteron: To reach a maxmum number of generatons The MOEA/D Proposal We use the ftness functon defned by Eq. (7). The pseudo-code of the method [66] s outlned as follows: Intalzaton: Randomly generate W weght vectors λ. Compute the Eucldean dstances between any two weght vectors and then work out the W closest weght vectors to each weght vector. For each = 1,2,..., N set B ) = {,..., }, where λ,...,λ W ( 1 W 1 are the W closest weght vectors to Generate an ntal populaton FV = F x. Set ( ) Intalze 1 N x,..., x λ. randomly. W = ( z 1,..., z m by a problem-specfc z ) method. In our case, we have m=2 objectves to mnmze, so m=2 sub-problems are to be mnmzed. Update: For = 1,..., N do Reproducton: Randomly select two ndexes k, l from ), and then generate a new soluton y from B ( k x and l x by usng genetc operators: The mutaton s the same than n the NSGA-II proposal. The crossover conssts of generatng two offsprng from two randomly chosen parents usng unform crossover [46]. j = Update of z: For each 1,..., m, f f j (y') z j = f j (y'. Update of Neghborng Solutons: For each ndex z j <, then set ) te j te j j j B(), f g ( y' λ, z) g ( x λ, z), then set x j = y' and F(y' ). Termnaton crteron: Stop f a maxmum number of generatons have been performed. Otherwse, go to Update. FV j = 7

8 4. Experments Sx datasets have been consdered to accomplsh all the mesh smplfcaton experments. Three of those fles correspond to synthetc meshes, Laurana.ply, Cheff.ply and Ramses.ply 2. These models are provded courtesy of the Shape Repostory 3. The rest of the datasets correspond to real-world models we are dealng wth n some research projects wthn the forensc scences area [49, 33]: two human skulls meshes (Skull1.ply and Skull2.ply), gven by the Physcal Anthropology Laboratory of the Unversty of Granada, Span; and the mesh of the scanned Spansh hstorcal monument face, the lady of Elche (FaceLadyElche.ply), kndly provded by the ITMA Materals Technology Centre n Asturas, Span. All of them are 3D open models (see Fgure 4). Fgure 4. Problem meshes. (a) M1 dataset (Laurana.ply) 922 vertces, 1667 trangles. (b) M2 dataset (Cheff.ply) 2622 vertces, 4864 trangles. (c) M3 dataset (Ramses.ply) 1420 vertces, 2734 trangles. (d) M4 dataset (Skull.ply) 1055 vertces, 2004 trangles. (e) M5 dataset (Skull2.ply) 5196 vertces, trangles. (f) M6 dataset (FaceLadyElche.ply) 1777 vertces, 3254 trangles. 2 The ply format s the polygon fle format. It descrbes an object as a collecton of vertces, faces and other elements, along wth propertes such as color and normal drecton that can be attached to these elements Expermental Setup The sngle-objectve algorthm, NSGA-II, and MOEA/D have been mplemented n C/C++ and all the experments have been performed on an Intel Core 2 Quad CPU Q GHz, wth 4 GB RAM, runnng Wndows 7 Professonal. We have dstngushed two varants per dataset for the sngle-objectve algorthm and NSGA-II. Varant 1 (v1) uses unform crossover because t tends to produce more dversty than 2-pont crossover, and varant 2 (v2) uses OAX wth L4 matrx (based on Taguch matrces proposed n [41]). The man reason why we selected OAX wth a L4 matrx and not L8 or L12 s effcency, as L4 took around 25 mnutes per tral, so L8 and L12 are not tractable n terms of run tmes n practce. Respect to MOEA/D we have drectly chosen the unform crossover because t has a smple desgn and obtaned better results than the OAX. The used parameter values for the sngle-objectve algorthms, NSGA-II and MOEA/D, have been the followng: the populaton s set to be 100, 50 generatons, crossover probablty of 0.8, and mutaton probablty of 1/chromosome_length. NSGA-II and MOEA/D have been run 10 tmes wth dfferent seeds. The ntal populatons are generated by unformly randomly samplng from the feasble search space. z n MOEA/D s ntalzed as the lowest value of f found n the ntal populaton and the settng of weght vectors λ s the same as n [66]. W s set to be 20. The prevous parameters were selected after analyzng the performance of the algorthm proposals [15, 66], and some detaled senstvty studes of genetc algorthm parameters [13, 16]. Both algorthms present a robust desgn that mples results less senstve to parameter varaton. Hence, ths parameter settng could be used wth other 3D nputs. Regardng the sngle-objectve algorthm, we consdered dfferent weght vectors. The weght of the frst objectve functon, the total approxmaton error (Eq.(4)), ranges from 1 to 0 (step 0.1), and the smplcty (number of trangles) weghts from 0 to 1 n the same step sze. The algorthm has been run for each of the 11 weghted vectors so obtaned. 8

9 4.2. The Classc Methods We have also performed the mesh smplfcaton process usng two classc algorthms of dfferent famles of methods (ncremental and non-ncremental) n order to compare ther results wth our evolutonary proposals. The two chosen methods are edge collapse decmaton based on quadrc error metrc [27, 50, 20, 59] and vertex clusterng wth topology preservng [42]. They are representatve technques among varous surface based algorthms [68]. The edge collapse decmaton has been selected because t leads to hgher defned meshes and provdes reasonable effcency, whle the vertex clusterng s usually very robust and can be fast [27]. We developed the experments by usng the Meshlab software [72], whch provdes some tools and flters to apply both smplfcaton technques. We have carred out the followng procedure n order to run the classc methods: let us have an orgnal mesh to be smplfed usng the quadrc edge collapse decmaton and the vertex clusterng technques. Ten dfferent smplfcatons are performed changng the reducton percentage,.e., from 5% to 50% of reducton wth a step sze of 5%. A specfc reducton percentage value means that the algorthm reduces the mesh nto a certan number of faces or trangles. After obtanng the smplfed mesh wth a specfed number of trangles, we calculated the error between the new mesh and the orgnal one n the way descrbed n Secton 3.2. Therefore, we wll have ten solutons per algorthm that wll be compared wth those obtaned by the sngle-objectve algorthm, NSGA-II, and MOEA/D Performance Comparson: Pareto Fronts The true Pareto-optmal front s usually consdered to compare the performance among mult-objectve algorthms. However, ths front cannot be calculated n reasonable tme n many real-world problems. That s the case of ths study. Hence, we have consdered an approxmaton to the true Pareto front approxmaton (called pseudooptmal Pareto front), whch s obtaned from the aggregaton of the set of solutons P produced by every method n all the runs performed. We calculated the Pareto front approxmaton of the sngle-objectve algorthm as follows: we frst merged the 11 solutons obtaned by each weghted vector. Repeated solutons are removed. We fnally produced the Pareto front approxmaton by performng a later domnaton check accordng to the Pareto domnance defnton (the nterested reader s referred to a bref revew on MOP n Appendx A of the supplementary nformaton avalable at [73]). The Pareto front approxmatons for NSGA-II and MOEA/D have been calculated by mergng the solutons obtaned n the ten runs. Then, the repeated solutons are removed and the Pareto domnance s appled n order to get the fnal Pareto front approxmaton. Fnally, n the case of the two classc methods, the Pareto front approxmaton s calculated jonng the ten calculated solutons n the way shown n Secton Metrcs of Performance: Qualty Indcators Mult-objectve qualty ndcators represent a means to measure qualty dfferences between Pareto front approxmatons on the bass of addtonal preference nformaton. It s possble to check whether an algorthm provdes sgnfcantly better approxmaton sets than another wth respect to the preferences represented by the consdered ndcator [70, 37, 38]. We have used three of the most extended multobjectve performance ndcators: the unary hypervolume ndcator (HVR) [70], the bnary epslon ndcator (Iε) [71], and the bnary coverage metrc (C) [70] (see Appendx B of the supplementary nformaton at [73] for ther detaled formulaton). In addton, the cardnalty of the Pareto set approxmatons obtaned,.e., the number of solutons composng them, wll also be reported and analyzed Analyss of the Results We have performed experments usng the sx mentoned datasets. Fgures 5 and 6 show the aggregated Pareto front approxmatons (see Secton 4.3) for two of those datasets, M1 and M2, for llustraton purposes (the Pareto front approxmatons of the rest of the datasets are avalable at [73]). The Pareto front approxmatons of the classc approaches are not shown because they obtan hgher errors than the rest of the methods. Ther Pareto front approxmatons are far from the other fve algorthms n the graphcal representaton. The sad fgures show that the mult-objectve methods have a better convergence than the sngleobjectve ones. The NSGA-II proposals obtan solutons coverng the whole space and acheve a greater 9

10 dversty. In general, MOEA/D and NSGA-II obtan solutons that domnate those acheved by the remanng methods n every dataset. The sngle-objectve technques just tend to fnd solutons n a specfc regon of the search space. That s a typcal behavor for weghted combnatonbased algorthms where specfc weght vectors bas the search drectons n dfferent runs avodng the obtanng of well spread Pareto fronts [13]. Table 1 presents the mean values for the C ndcator of the two best algorthms, NSGA-II v1 and MOEA/D. Ths table reveals that n terms of the coverage metrc, the fnal solutons acheved by NSGA- II v1 are better than MOEA/D for the M1, M3, and M5 datasets. Otherwse, MOEA/D outperforms NSGA-II for the M2, M4, and M6 nstances. Table 2 compares NSGA-II wth MOEA/D by usng the mean and the standard devaton values for the epslon metrc. In ths case, NSGA-II v1 s better than MOEA/D for all the datasets but M2, although slght dfferences are always found. Ths shows the smlar performance of both methods. Fgure 7 shows the mean values for the HVR metrc of all the consdered algorthms n the studed datasets. The fgure reveals that NSGA-II v1 and MOEA/D outperform the rest of the methods n all datasets but M1. Fgure 8 presents the evoluton of the cardnalty of all the algorthms n the problem nstances. NSGA-II acheves more dversty than MOEA/D n ths study, beng useful to fnd good compromses or trade-offs wthn the search space of our problem. The sngle-objectve algorthms can only detect one optmal soluton (n our case, pseudo-optmal soluton) n a sngle run whle the mult-objectve algorthms obtan a whole set of optmal (pseudooptmal) solutons. So, multple sngle-objectve method runs are needed to acheve the same level of nformaton that can be obtaned from a sngle multobjectve method run, thus showng the capabltes of our proposal for the 3D open model mesh smplfcaton problem solvng. The classc approaches obtan the hghest errors n ths study. The edge collapse decmaton strategy acheves a good approxmaton and preserves the topology of the orgnal mesh but at the cost of usng a hgh number of trangles. The effect of the decmaton s small and hghly localzed, so t tends to smplfy the mesh by regons. A smlar behavor corresponds to vertex clusterng wth topology preservng. We would need many smplfcatons of the orgnal mesh to acheve a good relatonshp between accuracy and complexty n comparson wth evolutonary algorthms. Hence, the hgher the number of smplfcatons n the orgnal mesh wll be, the hgher the total approxmaton error wll be. From the above results, we can conclude that both NSGA-II v1 and MOEA/D outperform the rest of the methods n the studed datasets,.e., they acheve a better trade-off between the complexty and the accuracy of the resultng meshes. Regardng the run tme, all methods show a smlar behavor but MOEA/D. The latter method needs less CPU tme than the others. It spends between 6 and 8 mnutes per run. The rest of the evolutonary algorthms take around mnutes per run for varant 1, a smlar run tme than the classc methods, and 20 mnutes per run n varant 2. The nterested reader s referred to a bref revew on computatonal costs between NSGA-II and MOEA/D at [66]. Fgure 5. Pareto front approxmatons of the evolutonary algorthms for the M1 dataset. Fgure 6. Pareto front approxmatons of the evolutonary algorthms for the M2 dataset. 10

11 Table 1 Mean values for the bnary C ndcator (Sgnfcant bold values treated as best result) Dataset C(NSGA-II v1,moea/d) C(MOEA/D,NSGA-II v1) M M M M M M Table 2 Mean and standard devaton for the bnary Epslon ndcator (Sgnfcant bold values treated as best result) Dataset NSGA-II v1 MOEA/D M (0.0714) (0.0108) M (0.0459) (0.0010) M (0.0122) (0.0298) M (0.0032) (0.0376) M (0.0098) (0.0218) M (0.0126) (0.,0105) Fgure 7. Evoluton of the mean values of the HVR ndcator n all the algorthms for all the datasets The Wlcoxon Test We have performed a Wlcoxon sgned-rank test [61] to analyze the sgnfcance of the results n the comparson of the qualty of the Pareto front approxmatons obtaned by the sngle-, mult-objectve, and classc algorthms by means of the prevously explaned unary and bnary ndcators. Ths s done n order to avod the fact that one exceptonally good result n any of the compared algorthms produces a wrong analyss. Unlke the commonly used t-test, the Wlcoxon test does not assume normalty of the samples and t has already demonstrated to be helpful analyzng the behavor of evolutonary algorthms [19]. Nevertheless, we should remark the fact that there s not any reference methodology to apply a statstcal test to a bnary ndcator n mult-objectve optmzaton. Thus, we have decded to follow the procedure descrbed n [48]. The sgnfcance level consdered n the performed test s p=0.05 for the I ε ndcator. For the C metrc we chose a threshold of In vew of ths statstcal study, avalable at [73], we can draw the concluson that the mult-objectve algorthms are sgnfcantly better n behavor than the rest of the methods. On the contrary, the two classc approaches perform sgnfcantly worse than the evolutonary ones wth the appled sgnfcance level. Specfcally, the analyss reveals that NSGA-II v1 and MOEA/D are sgnfcantly better, n terms of accuracy and complexty, than the rest of the technques for the C and epslon ndcators. Meanwhle, the results reveal that there are no dfferences among the solutons obtaned by NSGA-II v1 and MOEA/D wth the consdered sgnfcance level. Regardng the sngle-objectve methods, the solutons acheved by varant 2 have a better tradeoff than the results obtaned by varant 1 n the two metrcs. Fnally, the test does not obtan sgnfcant dfferences between the performances of the two classc algorthms Analyss and Comparson of Some Selected Solutons Fgure 8. Evoluton of the cardnalty n all the algorthms for all the datasets. We have selected three dfferent solutons from each Pareto set approxmaton n order to evaluate the qualty of the solutons obtaned. Namely, the one wth the best value n the frst objectve (mnmum error soluton), the one wth the best value n the second objectve (mnmum number of trangles solu- 11

12 ton), and a compromse soluton wth the best tradeoff value (best trade-off soluton). The trade-off soluton s selected as follows. We compute 1000 random weghts w [0, 1] and take the average value of the aggregaton functon of both objectves Obj1 (error) and Obj2 (number of trangles): 1000 w 1( ) + (1 ) 2( ) ( ) = = j jobj s w j Obj s F s (9) 1000 Due to the fact that the objectves Obj1 and Obj2 are not normalzed we need to apply a factor n order to scale them: PAa Obj2( s ) Obj1( s ) α =, (10) PA a a where PA s the cardnalty of the Pareto front approxmaton and s s a soluton of ths Pareto front. The fnal aggregaton formula to compute the average value s the followng: F( s ) 1000 αw Obj1( s ) + (1 w = = j j 1000 j ) Obj2( s ) (11) The soluton wth the lowest aggregated value s selected. For each of the three chosen solutons, we present the values of the two objectves, error and number of trangles. Fgure 9 contans the representatons of the three best solutons obtaned by the evolutonary methods for the proposed models. We have excluded the classc approaches because ther representatons are far from the rest of the algorthms n the graphcs. Comparng the solutons obtaned by each algorthm n the M1 dataset, MOEA/D, NSGA-II v2, and NSGA-II v1 acheve the best results. MOEA/D obtans the best mnmum error soluton followed by NSGA-II v2. MOEA/D and NSGA-II v1 perform the best tradeoff soluton, and the best mnmum number of trangles soluton corresponds to NSGA-II v2. Regardng the M2 dataset, NSGA-II v1 acheves the best results, although MOEA/D also obtans a good set of solutons. Related to the M3 dataset, NSGA-II v1 performs the best mnmum error soluton. The latter algorthm and MOEA/D acheve smlar results for the mnmum number of trangles and the best trade-off solutons. NSGA-II v1 and MOEA/D lead the three best solutons n the M4 and M6 datasets. We emphasze that NSGA-II v1, v2, and MOEA/D have a better performance than the other technques for the M5 dataset. For llustraton purposes, fgure 10 presents the best smplfed 3D models acheved by NSGA-II v2 and MOEA/D tacklng the M1 dataset. NSGA-II v2 obtans very good results n ts mnmum error soluton (Fgure 10b, 10g). Ths 3D mesh rghtly approxmates the contour of the face and other dffcult regons such as nose, mouth, lps, forehead and eyes, obtanng a hgh qualty approach comparng to the orgnal model (Fgure 10a, 10f) and (Fgure 10b, 10g). However, the mnmum number of trangles soluton draws a worse shape of the face wth respect to the orgnal mesh, as expected (Fgure 10a, 10c). Areas lke the chn and rght cheek have not been properly approxmated (Fgure 10c). The approxmaton of the forehead and the mouth s worse than the prevously obtaned soluton as can be seen n Fgure 10c, 10h. Regardng the results of MOEA/D, the algorthm obtans a good contour of the face n the rght sde for both solutons (Fgure 10d, 10e). The mnmum error soluton rghtly approxmates the mouth, nose, and eyes (Fgure 10d, 10). The accuracy of the smplfed mesh showng the mnmum number of trangles s good, but t approxmates the nose worse than the prevous soluton (Fgure 10e, 10j). In general, NSGA-II and MOEA/D acheve some smlar results (Fgure 10b-e, 10g-j). They manly dffer n the approxmaton of the face contour (Fgure 10b, 10d). We have also analyzed the best solutons of the NSGA-II v1 and the clusterng approach tacklng the M6 dataset n [73] as supplementary nformaton. 12

13 Fgure 9. The best three selected solutons for each evolutonary algorthm n the analyzed datasets. Fgure 10. Frst row. M1 Dataset orgnal model and solutons for NSGA-II v2 (b and c) and MOEA/D (d and e) algorthms, frontal vews. (a) Orgnal model, 1667 trangles. (b) Mn. error soluton, error= , trangles=387. (c) Mn. number of trangles soluton, error= , trangles=276. (d) Mn. error MOEA/D soluton, error=3808, trangles=401. (e) Mn. number of trangles MOEA/D soluton, error=4033, trangles=297. Second row. M1 Dataset orgnal model and solutons for NSGA-II v2 (g and h) and MOEA/D ( and j) algorthms, lateral vews. (f) Orgnal model, 1667 trangles. (g) Mn. error soluton, error= , trangles=387. (h) Mn. number of trangles soluton, error= , trangles=276. () Mn. error MOEA/D soluton, error=3808, trangles=401. (j) Mn. number of trangles MOEA/D soluton, error=4033, trangles=

14 5. Concludng Remarks We have proposed an evolutonary mult-objectve framework to solve the 3D open model mesh smplfcaton problem. The mult-objectve approach has been mplemented by usng two specfc MOEAs, NSGA-II and MOEA/D. They both have allowed us to fnd a set of solutons wth dfferent trade-offs between the accuracy and the smplcty of the smplfed 3D open models. In order to compare the performance of our proposals, we have consdered three benchmarkng methods, a sngle-objectve evolutonary algorthm and two classc technques. The experments developed have been based on three publcly avalable datasets and three real-world 3D open models, one of them beng a mesh of a scanned Spansh hstorcal monument and the other two beng human skull models from a forensc scence research project. A Wlcoxon rank sum test has also been accomplshed to analyze the sgnfcance of the obtaned results. From the analyss of those results, we can conclude that both the frst NSGA-II varant proposed and MOEA/D obtan solutons that domnate those acheved by the rest of the methods. Ther resultng meshes are more accurate, have a fewer number of trangles, and present a better trade-off between the two tackled objectves. On the opposte, the classc approaches have obtaned the hghest modelng errors. They showed a partcular behavor n the studed datasets. Many smplfcatons of the orgnal mesh were requred to acheve a good relatonshp between accuracy and complexty, n contrast to the consdered evolutonary algorthms. Besdes, the classc methods may excessvely smplfy a certan regon, leavng other parts, whch are expected to be smplfed, untouched. Hence, the hgher the number of smplfcatons n the orgnal mesh s, the hgher the total approxmaton error wll be. Despte achevng a good precson n other studes [20, 42, 59, 68], ths has not been our case. The reason could be that we consdered meshes wth open boundares, where t s possble that large errors are generated n correspondence wth the mesh boundary [12]. Nevertheless, both classc algorthms present reasonable processng tmes (smlar to NSGA-II v1 and the sngle-objectve v1, although larger than MOEA/D ones), an easer mplementaton desgn, and a good stablty and robustness. Besdes the performance mprovement, we should agan remark that the mult-objectve proposal allowed us to obtan a set of trade-off solutons n a sngle run of the algorthm. In ths way, the decson maker can choose the most sutable soluton (e.g. the most accurate, the smplest, or the one wth the desred trade-off between both objectves) dependng on the context the smplfed 3D models wll be used. Ths s a clear advantage over sngle-objectve methods whch try to fnd the best soluton n a sngle run of the algorthm. Hence, more runs of the algorthm are needed to obtan a smlar level of nformaton to that acheved by the mult-objectve algorthm, whch mples a sgnfcantly hgher computatonal effort. Fnally, as future works, t could be nterestng to extend the MOEAs framework n order to tackle complete 3D surface approxmatons. Acknowledgements Ths work has been supported by the Spansh Mnstero de Educacón y Cenca under project TIN ncludng European Development Regonal Funds (EDRF) and the project SOCOVIFI2 (references TIN C02-01/TIN C02-02). The authors would lke to thank the team of the Physcal Anthropology Laboratory of the Unversty of Granada and the ITMA Materals Technology Centre for provdng us wth real-world cases for our analyss. References [1] H. Adel and S.L. Hung. Machne Learnng - Neural Networks, Genetc Algorthms, and Fuzzy Sets, John Wley and Sons, New York, [2] H. Adel and S. Kumar. Dstrbuted Computer-Aded Engneerng for Analyss, Desgn, and Vsualzaton, CRC Press, Boca Raton, Florda, [3] F. Aurenhammer. Vorono Dagrams. A Survey of a Fundamental Geometrc Data Structure, ACM Computng Surveys, 23(3), pp , [4] P. Barald, R. Canes, E. Zo, R. Seraou, and R. Chevaler. Genetc algorthm-based wrapper approach for groupng condton montorng sgnals of nuclear power plant components, Integrated Computer-Aded Engneerng, 18(3), pp , [5] F. Bernardn and H. Rushmeer. The 3D model acquston ppelne, Computer Graphcs Forum, 21(2), pp , [6] T. Bckle. Tournament selecton. Handbook of Evolutonary Computaton. T. Back, D.B. Fogel, Z. Mchalewcz edtors. IOP Publshng Ltd and Oxford Unversty Press, pp. C2.3:1-4,

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