COMPUTATIONS of a bijective mapping among different

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1 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 1 Efficient Optimiztion of Common Bse Domins for Cross-Prmeteriztion Tsz-Ho Kwok, Yunbo Zhng, nd Chrlie C.L. Wng, Member, IEEE Abstrct Given set of corresponding user-specified nchor points on pir of models hving similr fetures nd topologies, the cross-prmeteriztion technique cn estblish bijective mpping constrined by the nchor points. In this pper, we present n efficient lgorithm to optimize the complexes nd the shpe of common bse domins in cross-prmeteriztion for reducing the distortion of the bijective mpping. The optimiztion is lso constrined by the nchor points. We investigte new signture, Length-Preserved Bse Domin (LPBD), for mesuring the level of stretch between surfce ptches in crossprmeteriztion. This new signture well blnces the ccurcy of mesurement nd the computtionl speed. Bsed on LPBD, set of metrics re studied nd compred. The best ones re employed in our domin optimiztion lgorithm tht consists of two mjor opertors, boundry swpping nd ptch merging. Experimentl results show tht our optimiztion lgorithm cn reduce the distortion in cross-prmeteriztion efficiently. Index Terms complex domin, optimiztion, stretch, cross-prmeteriztion, surfce prmeteriztion. 1 INTRODUCTION COMPUTATIONS of bijective mpping mong different models re very useful for mny geometry processing pplictions, such s texture mpping, morphing, pir-wise model editing, shpe blending, detils trnsfer, model completion, shpe nlysis, nd model dtbse preprtion. A generl solution for constructing such mpping cn be computed through the globl prmeteriztion pproch (ref. [1], [2], [3]). However, for pplictions like morphing nd dtbse preprtion, prmeteriztion must be constrined by semntic fetures, which re correspondingly specified s nchor points on the surfce of input models. A method clled cross-prmeteriztion ws proposed in [4] to solve this problem by constructing consistent domins on pir of models linking the nchor points, nd Schreiner et l. [5] presented n pproch nmed s inter-surfce mpping for similr purpose. Without loss of generlity, models tht need to be crossprmeterized usully hve similr fetures, nd the correspondences between nchor points should respect such similrities (i.e., hnds mpped to hnds s mentioned in [4]). In ddition, the constructed cross-prmeteriztion should preserve the shpe of the models s much s possible. Similr to the 3D-to-2D surfce prmeteriztion problem (e.g., [6]), shpe preservtion is usully chieved by minimizing the distortion occurring in the bijective mpping. A relxtion bsed smoothing step ws introduced in [4] for this purpose. Nevertheless, our recent study shows tht the distortion is seriously ffected by the shpe dissimilrity of domins between the cross-prmeterized models, nd such distortion cn hrdly be reduced by the smoothing step in [4]. This motivtes our reserch. Given two surfce meshes, the source model M s nd the trget model M t, which hve the corresponding sets All uthors re with the Deprtment of Mechnicl nd Automtion Engineering, The Chinese University of Hong Kong, Shtin, NT, Hong Kong. Corresponding Author: Chrlie C.L. Wng (Tel: (852) ; Fx: (852) ; E-mil: cwng@me.cuhk.edu.hk). Mnuscript prepred - September 20, 2010; Accepted - June 8, of nchor points G s nd G t prescribed, linking the nchor points on both models in consistent wy cn construct two tringulr ptch lyouts P s nd P t hving the sme connectivity (e.g., the lgorithms in [4], [7]). Following [7], polygonl mesh M is represented by pir (V,K), where V is set of vertices nd K is n bstrct simplicil complex tht contins ll the topologicl (i.e., djcency) informtion. For convenience, we lso tret tringulr ptch lyout s specil mesh with nchor points being the vertices nd curved ptch boundries being the edges. Assume tht ech ptch P i in ptch lyout cn find corresponding plnr domin B i (e.g., by [6]), we cn obtin the 3D-to-2D mpping Γ s : P i s B i s nd Γ t : P i t B i t. After estblishing the mpping between 2D domins sγ st : B i s B i t (e.g., using men-vlue coordintes [8]), we hve estblished the cross-prmeteriztion Γ = Γ s Γ st Γ 1 t (1) to mpm s tom t. Bsiclly, such mpping is rrely stretchfree (i.e., isometric). However, pplictions like model synthesis nd shpe blending wish to minimize such distortion s it leds to unwnted shpe distortion on the results n exmple is shown in Fig.1. Mny prior reserchers hve dedicted on how to minimize distortions in 3D-to- 2D mppings s well s in 2D-to-2D mppings. Our recent study finds tht optimizing the topology of common bse domins in P s nd P t cn further reduce the stretch, nd the reserch presented in this pper works out n efficient lgorithm to optimize the topology of common bse domins for reducing the distortion in cross-prmeteriztion. The optimiztion is constrined by the nchor points. Specificlly, every nchor point should be prt of the simplicil complex of the ptch lyout nd the positions of nchor points cnnot be chnged during the optimiztion. After describing the overview of our domin optimiztion lgorithm in section 2, we investigte new signture, Length-Preserved Bse Domin (LPBD) (in section 3), which

2 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 2 Fig. 2: The flipping pth opertor proposed in [4] could mke the shpe of bse domins more irregulr when the nchor points re distributed very unevenly: (left) before pplying the opertor nd (right) fter flipping pths. Fig. 1: An exmple for illustrting the influence of n imperfect cross-prmeteriztion on the result of shpe blending. (Top row) Unwnted distortions (circled by red dsh lines) re generted on the hnd model synthesized from hlf of the hnd A nd hlf of the hnd B ccording to the crossprmeteriztion of [4]. (Bottom row) Such unrel distortions re vnished fter pplying our optimiztion lgorithm to the common bse domins generted by [4]. cn indicte the level of distortion between surfce ptches in cross-prmeteriztion efficiently nd effectively by mesuring the dissimilrity between bse domins. Bsed on this new signture, set of metrics re then studied (in section 4) nd the best ones re selected for our domin optimiztion lgorithm which is composed of two mjor opertors, boundry swpping nd ptch merging. Before presenting the technicl detils, we review the previous work nd brief the contributions below. 1.1 Previous Work As n essentil step for mny geometry processing pplictions, the construction of bijective mppings between models or domins hs been studied for mny yer. A couple of the previous pproches require bse domin in order to find such mppings. One of the most commonly used bse domins is the sphericl domin [9], which produces semless nd continuous prmeteriztions everywhere. However, it only works on genus-zero models. A more generl method is to use complexes bsed domins s [4], [5], [7], [10], [11], [12]. The whole surfce mesh is usully prtitioned into simplicil complexes, which re served s bse domins (or clled bse meshes in some pproches). Once the bse domins re obtined, the originl meshes cn be prmeterized to the bse domin, nd then through which the cross-prmeteriztion is estblished. Although there re mny excellent pproches in literture bout domin construction (e.g., [2], [3]) nd optimiztion (e.g., [12]) for single model prmeteriztion (see [13] for the comprehensive survey), it is not strightforwrd to extend them to multiple models when bijective mppings between different models re needed. The min difficulty is how to construct the consistent bse domins on input models hving similr fetures nd topology. Such common bse domins re the bsis for the rest of the crossprmeteriztion. Prun et l. [7] used the connectivity of predefined templte bse domin nd then trced the boundry of ptch lyouts on ech of the input meshes by linking the given feture points in consistent wy s tht in the templte bse domin. The method tht links feture points in consistent mnner is bsed on 1) finding the shortest pths between feture points nd 2) preventing the pths from intersecting, blocking nd being in wrong cyclicl order. Krevoy et l. [4] nd Schreiner et l. [5] further extended the ide of Prun et l. by utomting the genertion of the templte bse domin. Krevoy et l. [4] first compute the shortest pths between ll feture points, nd then select the best pir of corresponding pths from priority queue sorted by the sum of pth lengths on meshes. By tril-nd-error until ll the ptches hve been tringulted, they eventully obtin the ptch lyout for the bse domin. Nevertheless, the topology nd the shpe of the ptch lyouts re not optimized to hve less distortion in the cross-prmeteriztion. Specificlly, when the distribution of nchor points re very uneven, the distortion of mppings is still significnt fter pplying the flipping pth opertor nd the relxtion bsed smoothing opertor which is proposed in [4] for reducing the distortion in crossprmeteriztion. The min reson why their flipping pth opertor does not work is tht their metric for governing the opertor only considers the vlences of bse domin (i.e., the number of pths linking to n nchor point). For exmple, on the model shown in Fig.2, fter pplying the flipping pths opertor, the vlences t nchor points become closer to six (i.e., regulr); however, the shpe of domin ptches becomes more irregulr. The new signture, Length-Preserved Bse Domin (LPBD) proposed in this pper, cn solve this problem. The pproch of Schreiner et l. [5] constructs the bse domin in mnner similr to [4], nd tried to reduce the distortion in the corse-to-fine mpping optimiztion. However, they only chnge the connectivity of common bse domins by dding points to resolve the domin construction problem on high genus models. Wng et l. [15] extended the skin lgorithm [16] by simultneously growing two skins with identicl connec-

3 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 3 Fig. 3: An exmple for illustrting the overview of our domin optimiztion lgorithm. The left most column shows the two input hnd models with prescribed nchor points (in green). (First nd second rows) The topology nd shpe of common bse domins on M s nd M t re chnged during the optimiztion. (Third row) A uniform mesh of the source model, M s, is trnsferred onto the trget model, M t, using the estblished cross-prmeteriztion where the trnsferred mesh on the trget model should be smooth if the cross-prmeteriztion hs smll distortion. (Lst row) The distortion of cross-prmeteriztion cn lso be visulized by the distribution of L 2 -stretch [14] on the trnsferred mesh model. tivity over different skeleton models. This lgorithm hs less topologicl constrints on the models to be pproximted, nd the input models could be mesh, polygonl sop or point clouds. Another recent mesh fitting bsed pproch for generting comptible mesh surfces is [17], where the uthors pproximte the input geometry with linerized bihrmonic surfce. These templte fitting bsed pproches [15], [17], [18] in generl re slower thn the cross-prmeteriztion bsed remeshing pproches. Some reserches focus on mpping between models with different topologies [19], [20], [21]. The work of Zhng et l. [19] decomposes given model by brnches of its reeb grph, nd the pproch of Bennett et l. [20] is bsed on n initil lignment scheme tht llows users to identify topologicl chnges. Recently, Li et l. proposed pnts decomposition in [21] to prtition input models into pnts which re genus-zero with three boundries. After mtching ech corresponding pnt with two regulr hexgonl domins, the bijective mpping between two models is obtined. Their method cn hndle models with different genus numbers while ours focuses on solving distortion minimiztion problem by optimizing the common bse domins. 1.2 Contributions We develop domin optimiztion lgorithm for reducing the distortion in cross-prmeteriztion, which is constrined by the prescribed nchor points. The efficiency nd effectiveness of this optimiztion lgorithm is benefited by new signture, Length-Preserved Bse Domin (LPBD), investigted in this pper. Bsed on LPBD, set of metrics re studied nd the best ones re selected for the domin

4 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 4 Fig. 4: Illustrtion of bse domin construction in [4]. optimiztion lgorithm, which mnipultes the connectivity of common bse domins with two opertors, boundry swpping nd ptch merging. The mjor novelty comes from using the new signture, LPBD, to convey the distortion in cross-prmeteriztion to the shpe dissimilrity of bse domins therefore speed up the computtion. 2 DOMAIN OPTIMIZATION ALGORITHM Our domin optimiztion lgorithm strts from the common bse domins constructed by the method in [4] which is briefly introduced s Pre-step, nd the three mjor steps of our lgorithm re presented fter tht. An exmple is shown in Fig.3 to illustrte the functionlity of ech step. Pre-step: Initil bse domin construction The lgorithm presented in [4] is implemented in this step. Firstly, the shortest pths long polygonl edges between ll feture points re computed. Then, the best pirs of corresponding pths in terms of the sum of pth lengths re chosen s boundries to construct the common bse domins. Agin,the selection should prevent the pths from intersecting, blocking nd being in wrong cyclicl order (similr to [7]). Finlly, flipping pth nd smoothing opertors re pplied to generte the initil common bse domins (see Fig.4). Step 1: Boundry stretching This is step to reduce the differences between the ctul boundry shpe of surfce ptch in the ptch lyout nd the shpe of its corresponding plnr domin. The stretching opertor developed in [22] is pplied to the boundry curves one by one. The principle of this curve stretching opertor is to convert curve into geodesic curve loclly by using n edge opertor nd node opertor to mke the curve shorter dynmiclly detils cn be found in [22]. After severl itertions, the curve under stretching pproximtes geodesic curve linking its two endpoints on the given mesh surfce. Note tht the stretched curve does not necessrily pss long the edges of given mesh models ny more. The intersection between curve under stretching nd other sttic curves must be prevented by the intersection voidnce method presented in [7]. The resultnt ptch lyouts should hve very smooth boundries for ll ptches. However, it is not difficult to find from the exmple shown in Fig.3 tht the distortion in the cross-prmeteriztion is still very significnt lthough stretching the boundry of ptches cn reduce the distortion slightly. Step 2: Boundry swpping In this step of our lgorithm, the ptch lyouts on both models, M s nd M t, re djusted itertively nd loclly to reduce the distortion in cross-prmeteriztion. The boundry swpping opertor is conducted to optimize the topology of the ptch lyouts. Without loss of generlity, for two ptches shring common boundry curve, pplying the boundry swpping opertor on them will convert the ptches into two new ptches by replcing the boundry curve by new one linking its opposite nchor points in the tringulr ptches. The new boundry curve is constructed by first finding the shortest pth long the mesh edges linking the opposite nchor points nd then stretching the curve by the method in [22]. Agin, the intersection between the new curve nd other existing curves must be prevented. The priority list of pplying the boundry swpping opertor is built nd mintined bsed on the metrics of the shpe similrity of bse domins, nd serves s signture to indicte the level of distortion in cross-prmeteriztion. Detiled studies bout the new signture, Length-Preserved Bse Domin (LPBD), re presented in section 3. Step 3: Ptch merging A curved boundry shred by two ptches will be removed in this step if such removl helps reduce the distortion in the cross-prmeteriztion. Aprt from the reduction of distortion, other properties of the bse domin (like the convexity nd the flttenbility) re lso considered during ptch merging to generte vlid nd optiml result. Following these min steps, the given mesh surfces re trimmed by the boundry curves of the new ptch lyouts using the Constrined Deluny Tringultion (CDT) [23]. The newly constructed ptch lyouts led to crossprmeteriztion with less distortion (see Fig.3 for n exmple). 3 LENGTH-PRESERVED BASE DOMAIN This section investigtes new signture, Length-Preserved Bse Domin (LPBD), indicting the level of stretch (i.e., distortion) induced in cross-prmeteriztion. This signture cn be efficiently evluted on the ptch lyouts for the common bse domins of cross-prmeteriztion. 3.1 LPBD s Signture for Distortion From the problem description presented t the beginning of this pper, we know tht cross-prmeteriztion is composed of three mpping functions: Γ s, Γ st nd Γ t. Different from the domin optimiztion pproch for single model [12] tht lwys employs regulr tringles s the plnr bse domin in the mpping, we llow the shpe of the bse domin to vry in 2D. Specificlly, the plnr domin B i for 3D ptch P i on the given model is constructed by 1) preserving the length of boundry curves on P i (i.e., hving invrint length of boundries), 2) strightening ech of the 2D boundry curves, 3) trying to mimic the shpe of P i in 2D,

5 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 5 Fig. 5: A study of the robustness of using the dissimilrity on LPBDs to mesure the distortion in cross-prmeteriztion. For given source model M s shown in the upper-left corner of the figure, we test the distortions between it nd vrious of trget models M t, where the developbility of ptches keeps decresing from left to right for ptches in the sme row nd the boundries of ptches re chnged for the ptches in the sme column. The L 2 -stretches [14] in the cross-prmeteriztion, L 2 (Γ), nd the L 2 -stretches on the plnr bse domins, L 2 (Γ st), re lso listed in the figure. Note tht only the big tringles linking the nchor points used to evlute plnr domin distortion, L 2 (Γ st). The color mps re used to visulize the distribution of L 2 stretches generted by the cross-prmeteriztion. where the whole boundry of ptch is subdivided into curves by the nchor points. Obviously, the plnr bse domins generted in this wy, nmed s Length-Preserved Bse Domin (LPBD), hve different shpes nd sizes. The dissimilrity between the plnr bse domins Bs i nd Bt i is good signture to indicte the cross-prmeteriztion from Ps i to Pt i. By retining the length of boundry curves, the mpping from P i to B i is stretch-free long the boundries. If the boundry curves on P i re geodesic curves, flttening them into stright plnr line segments leds to stretchfree mpping round/cross the lines. If Ps i nd Pt i re developble surfces, the mppings Γ s nd Γ t from Ps i nd Pt i to the LPBD, Bs i nd Bt, i re stretch-free too. As the cross-prmeteriztion Γ from Ps i to Pt i is series-wound mppings,γ s Γ st Γ 1 t, the distortion ofγis only introduced by the 2D-to-2D mpping Γ st : Bs i Bt i in this scenrio. In other words, the dissimilrity between Bs i nd Bt i mesures the distortion of cross-prmeteriztion Γ. A more interesting study reltes to how significntly the distortion in Γ st contributes to the overll distortion in the cross-prmeteriztion Γ when the ptches Ps i nd Pt i re not developble. With given plnr shpe B i, we cn ssume tht the distortion in the 3D-to-2D mpping from P i to B i hs been minimized in our implementtion, the men-vlue coordintes bsed prmeteriztion [8] is used for this purpose. The mpping Γ st is gin estblished by men-vlue coordintes [8]. We now study how the developbility of 3D ptch ffects the robustness of using the signture, LPBD, to mesure the stretch in crossprmeteriztion. As shown in Fig.5, for n unchnged source model M s, when the trget model M t becomes more nd more nondevelopble while still keeping the sme boundry, the stretch error of the cross-prmeteriztion does not show significnt chnge. However, when keeping the geometric detils (i.e., the level of developbility) of ptches but chnging their boundries (i.e., the shpe of LPBD), the stretch errors of the cross-prmeteriztion chnge significntly (see the chnges in Fig.5 by columns). In summry, the dissimilrity between the LPBDs, B i s nd B i t, is robust signture to indicte the stretch in the overll cross-prmeteriztion. Different from the originl crossprmeteriztion pproch [4] tht constructs bse domins by linking nchor points with stright lines in 3D, mpping geodesic curves of the ptch boundries into plnr stright lines conveys more surfce dissimilrity to the shpe of 2D domins, B i s nd B i t. This enhnces the robustness of the LPBD signture. 3.2 LPBD Computing The Length-Preserved Bse Domin (LPBD), B i, of ptch P i R 3 on the given model cn be esily computed

6 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 6 by constrined optimiztion frmework similr to the computtion of Length-Preserved Free Boundry (LPFB) in [24]. Assume tht the boundry of P i is determined by linking n nchor points with pproximte geodesic curves, we compute the optiml plnr ngles α i on the i-th nchor point by solving constrined optimiztion problem s min n 1 (αi θi)2 i=1 2 s.t. nπ n αi 2π i=1 n i=1 licosφi 0 (2) n i=1 lisinφi 0 where θ i is the surfce ngle of the ptch t the i-th nchor point, nd l i denotes the length of the boundry curve between the i-th nd the (i+1)-th nchor points on the ptch. The computtion is tken in the ngle spce by setting the ngle vritions s objective function to generte plnr domin with shpe similr to the ptch s boundry in 3D. The first hrd constrint is derived from the closedpth theorem to prevent self-intersection, nd the ltter two hrd constrints re employed to ensure the genertion of closed boundry loop with φ i = iπ i b=1 αi. Once the 2D ngles on nchor points re determined, we cn esily locte the i-th nchor point of LPBD t ( i b=1 l bcosφ b, i b=1 l bsinφ b ). The LPBD cn be computed very efficiently becuse for ptch circled by n boundry curves, only n vribles plus three constrints re involved in the computtion (i.e., (n+ 3) (n+3) liner eqution system). This cn be esily solved by the Qusi-Newton s method. Notice tht we cn directly compute the plnr shpe of bse domin by the lengths of its boundries if the domin is tringulr. 4 BOUNDARY SWAPPING Edge swpping (or clled edge-flip) is widely used opertor to improve the qulity of tringulr meshes in mesh optimiztion (e.g., [25], [26], [27]); we employ similr concept to optimize the shpe similrity of common bse domins. Different from the edge swpping opertor, performing swpping on the curved boundries of bse domins is more difficult becuse of the following four resons: 1) The shpes of domins on both the source nd the trget models need to be considered. 2) The shpes of 2D domins fter swpping curved boundry cnnot be predicted by tht of 2D domins before swpping. 3) The shpe of 2D domin fter swpping could be degenerted, where the length of one boundry curve is longer thn the sum of the other two so tht the computed plnr tringle could be n invlid LPBD. 4) The computtion of mesuring the distortion in crossprmeteriztion bsed on dense meshes is timeconsuming. Our trget here is to exploit n effective metric tht cn be efficiently evluted. Before going to the study of finding good metric, let s ssume tht such metric for boundry swpping hs been well defined on every boundry curve ṽ v b, s Υ(ṽ v b ). For two ptches v v b v l nd v v b v r defined on four nchor points v, v b, v l nd v r nd shring the common Fig. 6: The boundry swpping opertor pplied on curved boundry edge ṽ v b shred by two ptches v v b v l nd v b v v r replces ṽ v b by new curve ṽ l v r nd therefore forms two new tringulr ptches v v r v l nd v b v l v r on the surfce. Algorithm 1 GreedyBndSwpping 1: Initilize n empty mximum hep H; 2: for every pir of ṽ svs b M s nd ṽ tvt b M t do 3: Evlute the metric Υ(ṽ v b ) on them; 4: Insert the curve ṽ v b into H when Υ(ṽ v b ) > 0; 5: end for 6: while H is NOT empty do 7: Remove curved boundry edge ṽ v b from H; 8: Apply the swpping opertor on ṽ v b ; 9: for ny of ṽ v l, ṽ v r, ṽ b v l nd ṽ b v r do 10: Evlute the metric Υ( ) on the boundry curve; 11: if the boundry curve is in H then 12: Updte its position in H; 13: else 14: Insert the curve into H when Υ( ) > 0; 15: end if 16: end for 17: end while boundry curve ṽ v b, pplying the boundry swpping opertor on them converts them into two new ptches v v r v l nd v b v l v r (see Fig.6). The metric Υ(ṽ v b ) returns vlue to indicte the level of distortion reduction in cross-prmeteriztion by swpping ṽ v b to ṽ l v r, the vlue of which is the greter the better. Bsed on this metric, greedy lgorithm cn be developed to conduct boundry swpping by using mximum hep keyed on Υ( ). Pseudo-code of the bsic lgorithm is given in the Algorithm GreedyBndSwpping. Notice tht, when considering the new boundry ṽ l v r, we need to prevent the pths from intersecting, blocking nd being in wrong cyclicl order. Moreover, we need to verify the vlidity of LPBD. If invlid LPBDs will be generted fter swpping, the opertion must be prevented. On the other hnd, if boundry swpping cn convert invlid LPBDs into vlid LPBD, it must hve the highest priority to perform. 4.1 Metrics Bsed on Dense Meshes A kind of obvious metrics for Υ(ṽ v b ) is the ones mesured by the deformtion between the dense meshes

7 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 7 of v s v b sv l s nd v s v r sv b s nd their trnsformed shpes Γ( v s v b sv l s) nd Γ( v s v r sv b s), which re defined by the cross-prmeteriztion Γ. The metrics bsed on ngle distortion or L 2 -stretch on these dense meshes re presented below. Angle Distortion Borrowing the ide of Sheffer et l. from [28], for ny tringle f i v s v b sv l s or v s v r sv b s, we define tht ngle distortion mesures the chnge of its three ngles i,1, i,2 nd i,3 on the source model compred with the corresponding ngles i,1, i,2 nd i,3 fter pplying the mpping Γ. If there re f tringles in these two ptches, the ngle distortion on the ptches cn be clculted by E ng(ṽ v b ) = 1 3 f f 3 i=1 j=1 ( i,j i,j i,j ) 2 (3) which hs lower bound zero on exmples without distortions. The boundry swpping metric bsed on the ngle distortion is then defined by the difference between the ngle distortions E ng(ṽ v b ) nd E ng(ṽ l v r ). Υ AD(ṽ v b ) = E ng(ṽ v b ) E ng(ṽ l v r ) (4) The ngle distortion for the whole model, E ng(m s), cn be evluted in wy similr to Eq.(3) by including ll tringles on M s. L 2 -Stretch Another widely used method for mesuring the distortion of surfce prmeteriztion is the L 2 -stretch introduced by Snder et l. in [14]. For tringle f s M s nd its corresponding shpe defined by the mpping Γ s Γ(f s), the L 2 -stretch is defined by computing the Eigen vlues of the Jcobin formed by prtil derivtives of unique ffine mpping between f s nd Γ(f s). To ese the computtion, for f s with three vertices (p 1,p 2,p 3) in R 3, we define locl plnr frme on Γ(f s) to obtin the plnr coordintes of its three vertices s (s k,t k ) (k = 1,2,3). Then, the L 2 -norm defined on the tringle f s bsed on the crossprmeteriztion is with L 2 (f s) = (τ s τ s +τ t τ t)/2 (5) τ s = ((t 2 t 3)p 1 +(t 3 t 1)p 2 +(t 1 t 2)p 3)/(2A(Γ(f s))) τ t = ((s 2 s 3)p 1 +(s 3 s 1)p 2 +(s 1 s 2)p 3)/(2A(Γ(f s))) nd A( ) defining the re of tringle. Therefore, the normlized L 2 -stretch of ll tringles in prticulr region Ω is E L 2(Ω) = f i Ω A(Γ(fi)) f i Ω ((L2 (f i)) 2 A(f i)) ( f i Ω A(fi))2, (6) which hs lower bound of 1.0. The boundry swpping metric bsed on the L 2 -stretch on dense meshes is then defined on ll tringles v s v b sv l s, v s v r sv b s, v v r v l nd v b v l v r s Υ L 2(ṽ v b ) = E L 2(ṽ v b ) E L 2(ṽ l v r ). (7) Similrly, the L 2 -stretch of the whole model, E L 2(M s), cn lso be evluted by including ll tringles on M s in the computtion of Eq.(6). Both of these two metrics, E ng(m s) nd E L 2(M s), reflect the distortion of cross-prmeteriztion in their own spects. An idelly optimized cross-prmeteriztion should reduce both of them. 4.2 Metrics Bsed on Shpe of Domins The min problem of directly pplying these metrics to the domin optimiztion for cross-prmeteriztion is tht the evlution of them is generlly time-consuming (especilly when the given models hve very dense tringulr meshes). Therefore, we hve studied some other metrics which re defined directly on the LPBD signture nd wish to find one tht well blnces the qulity of results nd the speed of computtion. As the metrics, Υ AD(ṽ v b ) nd Υ L 2(ṽ v b ), re tightly coupled with the qulity of cross-prmeteriztion, the results of pplying them in the Algorithm GreedyBndSwpping re served s benchmrks for the selection of good metric below. Are Similrity The first tested metric is stimulted by the nlysis given in [24] tht surfce prmeteriztion with smll distortion usully hs smll re vritions (e.g., flttening developble mesh surfce). Therefore, n re similrity metric is defined s where Υ re(ṽ v b ) = E re(ṽ l v r ) E re(ṽ v b ) (8) E re(ṽ c v d ) = (ψ(p l s) ψ(p l t)) 2 +(ψ(p r s) ψ(p r t )) 2 with P l nd P r denoting the surfce ptches on the left nd the right of the curved boundry ṽ c v d in the ptch lyout. The function ψ(p i ) returns vlue bsed on the re difference between surfce ptch, P i, nd its LPBD, B i. { A(P ψ(p i i )/A(B i ) (A(P i ) A(B i )) ) = A(B i )/A(P i ) (A(P i ) < A(B i )) A(B i ) is the re of bse domin, which cn be directly clculted by Heron s formul using the lengths of boundry curves on P i. However, it should be noted tht the Heron s formul which is bse on lengths my fil if the length of boundry curve is lrger thn the sum of other two boundry curves lengths i.e., degenerted cse hppens. For degenerted cse, we simply define the re ofb i with very smll vlue (e.g., 10 8 ). Domin Angle Distortion In order to mesure the similrity of tringulr domins, one of the methods is to clculte the difference of corresponding ngles in the plnr domins, Bs i nd Bt. i Therefore, using the nottion illustrted in Fig.6 s the ngles in LPBDs, we cn define the following metric to govern the swpping opertor. 6 6 Υ DAD(ṽ v b ) = k=1 ( αk s α k t α k s ) 2 k=1 ( βk s β k t β k s ) 2 (9) Agin, curved boundry edge ṽ v b is inserted into the priority list only when Υ DAD(ṽ v b ) > 0. Domin L 2 -Stretch Another method tht mesures the similrity between the tringulr domins is through the L 2 -stretch which

8 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 8 swpping e.g., when ṽl sv r s < ṽ sv b s but ṽl tv r t > 2 ṽ tv b t. Similrly if the curves re elongted on both M s nd M t, the swpping is lso prevented since the vlue of Υ SDL is negtive. When the ptches P i s nd P i t for the common bsed domins on both models pproches regulr tringles, the shpes of their corresponding LPBDs, B i s nd B i t, should be similr to ech other s well (i.e., the distortion in the mpping Γ st is reduced). Gretest Angle Reduction In mesh optimiztion, nother widely used rule which detects the possibility of pplying the edge swpping opertor is bsed on checking whether the gretest ngle in two tringles djcent to the edge is reduced. Assuming tht α k nd β k (k = 1,...,6) re the six ngles in the two LPBD tringles djcent to the edge ṽ v b nd the swpped one ṽ l v r respectively see Fig.6 for n illustrtion, we define the gretest ngle reduction metric s Fig. 7: An exmple to illustrte the fct tht highly irregulr shpes in LPBDs will lso led to high distortion in crossprmeteriztion (top row). The color mp shows the distribution of L 2 -stretch. The distortion cn be reduced by using domins with more regulr shpes (bottom row). hs been used in section 4.1. Differently, the L 2 -stretch here mesures the difference between plnr tringulr domin of the source model nd its corresponding domin of the trget model. For exmple, L 2 ( p i p j p k ) gives the L 2 -stretch between p i sp j sp k s nd p i tp j tp k t. For tringulr domins djcent to curved boundry edge (s illustrted in Fig.6), the metric of domin L 2 -Stretch is defined s with Υ DL 2(ṽ v b ) = E L 2(ṽ v b ) E L 2(ṽ l v r ) (10) E L 2(ṽ v b ) = L 2 ( p p b p l )+L 2 ( p b p p r ), (11) E L 2(ṽ l v r ) = L 2 ( p p l p r )+L 2 ( p b p r p l ). (12) Shorter Digonl Length During our experimentl tests, we find tht the distortion in cross-prmeteriztion will still be high if the shpes of LPBDs of two models re similr (not exctly sme) to ech other but re fr from regulr tringles. See Fig.7 for n extreme exmple to demonstrte this observtion. Therefore, the fourth metric tries to mke the common bse domins on both the source nd the trget models become regulr tringles. A rule employed in the mesh optimiztion to generte regulr tringles is tht digonl edge will be swpped if the swpped one is shorter. The metric Υ SDL below is introduced for the sme purpose. Υ SDL(ṽ v b ) = 2 ( ṽl svs r + ṽl tvt r ), (13) ṽ svs b ṽ tvt b where ṽ v b denotes the length of the curve linking the nchor points v nd v b, while ṽ l v r denotes the length of the curve linking the nchor points v l nd v r. In this metric, the length chnges of the boundry curves on both the source nd the trget models re considered. Swpping is prohibited if either side hs become much worse fter Υ GAR(ṽ v b ) = mx k {αs,α k t} mx k {βs,β k t}. k (14) The swpping is pplied only when Υ GAR(ṽ v b ) > 0. Similr to using Υ SDL to govern boundry swpping, the metric Υ GAR lso tends to mke the shpes of bsed domins on both models pproch regulr tringles. Therefore, it cn reduce the distortion in cross-prmeteriztion. It should be remrked tht, ll the ngles of LPBDs in 2D re clculted by the length of the boundry curves directly using the Lw of Cosine. For the degenerted cses (e.g., the length of one side is longer thn the sum of the other two), the computed vlue ς for rccos by the Lw of Cosine does not fll in the rnge [ 1,1]. For these cses, we simply ssign the ngle s ςπ ( ς < 1) or (1 ς)π ( ς > 1). Anlysis nd discussion of ll metrics In order to select the best metric for the GreedyBndSwpping lgorithm whose purpose is to reduce the distortion in cross-prmeteriztion through optimizing the shpe of common bse domins, we pply the lgorithm to vriety of models by using ll bove metrics. The distortion in crossprmeteriztion bsed on the resultnt bse domins is mesured by wrping source model M s to the shpe of the trget model M t nd clculting the ngle distortion E ng(m s) nd the L 2 -stretch E L 2(M s). Figure 8 shows the br chrts for the sttistics of E ng(m s) nd E L 2(M s) on different models. When nlyzing the results, severl interesting phenomen cn be observed. Firstly, the evlution results obtined from using E ng(m s) nd E L 2(M s) to mesure the distortion in cross-prmeteriztion re not consistent with ech other. Tke Dvid-ege models s n exmple, fter using the ngle distortion metric Υ AD bsed on dense meshes to optimize the common bse domins, the L 2 - stretch, E L 2(M s), becomes even worse thn the crossprmeteriztion before domin optimiztion. Another exmple is the models of men. Using the L 2 -stretch metric Υ L 2 to govern the boundry swpping turns out hve enlrged the globl ngle distortion, E ng(m s). Secondly, the results obtined from using the L 2 -stretch metric Υ L 2 re better thn tht obtined from using the metric Υ ng in terms of the L 2 -stretch error, E L 2(M s). However, using Υ ng does not lwys give the result k

9 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 9 Fig. 9: The ptch merging opertor pplied on curved boundry edge ṽ v b (the one shown in dsh line) djcent to two tringulr ptches. Fig. 8: The sttistics of using different metrics in the greedy lgorithm when pplying the boundry swpping opertor to improve the shpe similrity of common bse domins nd thus reduce the distortion in cross-prmeteriztion. The top br chrt shows the ngle distortion E ng(m s), nd the bottom one gives the results of E L 2(M s). tht smller vlue is on E ng(m s) thn the results obtined by using Υ L 2 in the optimiztion. Thirdly, when using the metrics defined on the dissimilrity of LPBDs to govern optimiztion, Υ DL 2 results in cross-prmeteriztions with less distortion thn Υ re nd Υ DAD bsed on the evlutions of both E L 2(M s) nd E ng(m s). Lstly, fter considering the regulrity on the shpe of bse domins, the optimiztions using metrics Υ SDL nd Υ GAR lwys give better results thn one using the metrics, Υ re, Υ DAD nd Υ DL 2, tht consider only the dissimilrity. Among them, using Υ SDL genertes better results in most cses. Bsed on these observtions, we cn conclude tht the metric Υ SDL is the best choice (bsed on the results of our experimentl tests) to reduce the distortion in crossprmeteriztion under the greedy boundry swpping lgorithm. Furthermore, the metric Υ DL 2 gives better results in mesuring the similrity of bse domins thn Υ DAD. 4.3 Blnced Algorithm Using the metrics defined on the shpe of LPBDs insted of dense meshes cn gin some speedup in the optimiztion procedure; however, we find tht more significnt speedup cn be obtined if some of the swpping opertions on boundry curves re skipped. Specificlly, we skip the opertions on those similr-enough common bse domins. In detil, the blnced lgorithm for boundry curve swpping consists of three mjor steps. Step 1) The domin L 2 -stretch, E DL 2(ṽ v b ), defined on Algorithm 2 GreedyPtchMerging 1: Initilize n empty mximum hep H m; 2: for every pir of ṽ sv b s M s nd ṽ tv b t M t do 3: Evlute the metric Υ merge(ṽ v b ) on them; 4: Insert ṽ v b into H m when Υ merge(ṽ v b ) > 0; 5: end for 6: while H is NOT empty do 7: Remove curved edge ṽ v b from H m; 8: if Υ merge(ṽ v b ) > 0 then 9: Apply the ptch merging opertor on ṽ v b ; 10: Remove the edges ṽ v l, ṽ v r, ṽ b v l nd ṽ b v r from H m if ny of them hs lredy been in H m; 11: end if 12: end while every curved boundry edge ṽ v b is first evluted by Eq.(11). Step 2) Among ll curved boundries, only those hving E DL 2 > 2.0+(1 ϕ)mx{e DL 2 2.0} re defined s ctive boundry curves. The golden rtio, i.e., 0.618, is selected for ϕ. Step 3) Applying the greedy boundry swpping lgorithm only on the ctive boundry curves nd the degenerted cses to optimize the common bse domins by using the metric Υ SDL. This blnced lgorithm gives good trde-off between the qulity nd the speed. Around 5 to 15 times speedup cn be gined more discussion on the results cn be found in section 6. 5 PATCH MERGING The ptch merging opertor cn be pplied to further reduce the distortion in cross-prmeteriztion. Without loss of generlity, when pplying the ptch merging opertor to curved boundry edge ṽ v b djcent to two tringulr ptches v v b v l nd v b v v r, the tringulr ptches re merged into qudrilterl ptch v v r v b v l. Although merging ptches into n-sided ptches (n > 4) could possibly further reduce the distortion in cross-prmeteriztion, polygon with n > 4 is esier to hve concve corners using which s the 2D domin my violte the correctness

10 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 10 Fig. 11: The bse domins of horse-cmel models before (top row) vs. fter (middle row) optimiztion, nd their corresponding L 2 -stretches in the color mp. The tringles belonging to different domins re displyed in different colors. Note tht, the smoothing technique presented in [4] hs lredy been pplied to reduce the distortion. Bottom row shows the shpe interpoltion results before (left) nd fter (right) domin optimiztion. Fig. 10: The shpe of blended model of dinosur nd mn cn be improved by the optimized cross-prmeteriztion. (Top row) The initil common bse domins generted by [4], nd (Middle row) the optimized bse domins the L 2 -stretches re shown by the color mp. (Bottom row) The blending results obtined by the cross-prmeteriztions before vs. fter domin optimiztion, where the red dsh lines circle the unrel distortions cused by the lrge stretch in cross-prmeteriztion. of cross-prmeteriztion (see the discussion in detil below). Thus, we choose not to do tht in order to ese the implementtion nd shorten the computing time. The metric for governing the order of ptch merging is defined bsed on the shpe similrity of the LPBDs mesured by ngle distortions. Υ merge(ṽ v b ) = α k s α k t 1 4 k=1 4 γs k γt, k (15) k=1 where α k nd γ k re ngles on the LPBDs before nd fter merging respectively (s illustrted in Fig.9). More thn tht, the metric is set to be or removed from hep to prevent ptch merging opertion when ny of the following cses occurs. Any of the ptches djcent to the curved boundry ṽ v b is qudrilterl ptch. There is one edge of the to-be-merged qudrilterl ptch longer thn the sum of the other three this will led to degenerted result when computing the LPBD of the ptch. γ k > π mong k = 1,...,4. This cse is prevented since ny ngle greter thn π will mke LPBD concve polygon, which will generte self-overlps on the 3D-to-2D prmeteriztion such tht the bijective mpping for the cross-prmeteriztion cnnot be estblished. Detiled discussion bout self-overlpping in mesh prmeteriztions cn be found in [29]. Agin, we use the greedy strtegy with the metric Υ merge(ṽ v b ) to perform ptch merging. See the pseudocode in Algorithm GreedyPtchMerging for more informtion. 6 RESULTS AND DISCUSSION We hve implemented the proposed lgorithm in C++ plus OpenGL. All the experimentl tests shown in this pper re

11 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 11 Fig. 12: An exmple of hed models Dvid nd Ege. (Top) The cross-prmeteriztion by [4]. (Bottom) The result fter optimizing the common bse domins by our pproch. Fig. 13: Morphing between dilo model nd mnnequin cn be estblished by the cross-prmeteriztion with optimized common bse domins. run on stndrd PC with Intel Core 2 Duo CPU E6750 t 2.67GHz plus 2GB RAM. The first exmple is pir of hnd models. From the blending result shown in Fig.1, we cn see tht the unwnted distortion in the blending cn be eliminted fter optimizing the common bse domins by our method. More exmples, dinosur-mn pir, horse-cmel pir, pir of hed models Dvid nd Ege, nd dilo-mn pir, re shown in Figs The results re quite encourging. The distortions in cross-prmeteriztion (mesured by the L 2 -stretch [14] nd displyed by the color mp) on ll pirs of models re significntly reduced fter pplying the proposed domin optimiztion pproch. Note tht, ll results shown in Figs re generted by the blnced boundry swpping lgorithm in optimiztion. The initil bse domins re generted by our implementtion of pproch in [4]. The globl ngle distortion E ng(m s) nd the L 2 -stretch E L 2(M s) on the resultnt models re listed in Tble 1 nd compred between the originl cross-prmeteriztion results of [4] nd the optimized ones by the metrics defined on the dense meshes, Υ ng nd Υ L 2. The blnced boundry swpping lgorithm followed by the GreedyPtchMerging lgorithm genertes results tht re similr to tht of the optimiztion which is bsed on the rel prmeteriztion (i.e., the metrics evluted on the dense meshes). However, the speed of the blnced lgorithm is bout 5 to 15 times fster (see the sttistics shown in Tble 1). In short, the LPBD signture newly introduced in this pper nd the proposed Fig. 14: The sttistics of results obtined by using different metrics in the GreedyBndSwpping lgorithm on the hnd exmple shown in Fig.3 but hving different sets of feture points. The top br chrt shows the ngle distortion E ng(m s), nd the bottom one gives the results of E L 2(M s). blnced lgorithm cn efficiently improve the qulity of cross-prmeteriztion results by optimizing the shpe of common bse domins, which re constrined by the nchor points. In the boundry swpping lgorithm, we do not explicit prevent letting the boundry curves pss through the highly stretched re; however, s it will enlrge the dissimilrity on the LPBD signtures, such cses will be voided utomticlly by the metrics ccording to LPBDs. Another interesting study is bout how the positions of feture points (constrints) ffect the distortion in crossprmeteriztion. See the sttistics in Fig.14 on the sme pir of hnd models shown in Fig.3 before, the level of distortion could chnge significntly when different sets of feture points re specified. Such vrition occurs even if the sme number of feture points re used. To compre with the results generted by [4], we pply our pproch on pir of hed models provided by uthors of [4] on their webpge (see Fig.15). We tried severl sets of feture points on this pir of models, nd our method cn lwys further reduce the overll L 2 -stretch, E L 2(M S), from to 1.043, nd respectively. The color mps for illustrting the vlue of L 2 -stretch re lso shown in Fig.15. In these tests, s the input models re not provided by the uthors of [4], we cn only work on the remeshed models (i.e., using the output of [4] s our input). Although this is not direct comprison to the originl implementtion in [4], it still somewht proves the performnce of our pproch. In order to further verify the performnce of our optimiztion lgorithm, we chnge the wy to construct initil common bse domins in [4], where the query is bsed

12 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 12 TABLE 1: Comprisons between different optimiztion methods. Models (M s/m t) Hnd1/2 Dilo/Mn Dinosur/Mn Horse/Cmel Dvid/Ege Mle1/2 Femle1/2 Sizes (#v) 14k / 1,515 27k / 11k 56k / 11k 20k / 9,770 50k / 8,268 3,856 / 8,950 11k/2230 Feture points # Originl [4] E L 2(M s) E ng(m s) By Υ AD E ng(m s) E L 2(M s) Time (sec.) E L 2(M s) By Υ L 2 E ng(m s) Time (sec.) E L 2(M s) Blnced E ng(m s) Time (sec.) E L 2(M s) Merging E ng(m s) Time (sec.) * [4] is the originl cross-prmeteriztion before re-meshing. The reported time is the processing time of domin optimiztion. Fig. 15: Applying our pproch on the result of cross-prmeteriztion generted by [4], the L 2 -stretch cn still be further reduced: () the result of cross-prmeteriztion downloded from the webpge of uthors [4], (b) our result generted on the set of feture points shown in the left - the L 2 -stretch is E L 2(M S ) = 1.043, (c) the result on nother set of feture points with E L 2(M S ) = 1.037, nd (d) the result with being the vlue of L 2 -stretch. on computing the shortest pth between feture points by the Dijkstr s lgorithm. Here, the priority query is keyed ccording to the sum of Geodesic distnces between feture points (see Fig.16 for n exmple), nd we compute the Geodesic distnces pproximtely by the method in [22]. It is found tht using Geodesic distnces cn somewht improve the shpe of initil common bse domins thus lso the cross-prmeteriztion. However, our lgorithm cn still further reduce the distortion by optimizing the topology of bse domins. Tble 2 gives the relted sttistics. Lstly, we test the results of cross-prmeteriztion constructed by our method in two pplictions. The first ppliction shown in Fig.17 is to estblish humn body dtbse from the scnned 3D models. After reconstructing mesh surfce from the scnned point cloud by [30], we wish to generte consistent connectivity for ll humn models stored in the dtbse. More thn tht, the semntic fetures predefined on the templte model must be ble to be esily trnsfered to ll the models by the consistent connectivity. As shown in Fig.17, if lrge distortions re embedded in the cross-prmeteriztion, the semntic feture curves will be highly distorted, which is intolerble for downstrem pplictions (e.g., mnnequin fbriction). The second ppliction is to pply the optimized crossprmeteriztion in design utomtion of pprel products. After constructing correspondences between the tringles on the source model M s nd the trget model M t, we cn use some deformtion trnsformtion [18] techniques or sptil wrping techniques [31], [32] to deform n pprel product designed for the modelm s to new shpe fit for the new model M t. However, if lrge distortions re embedded in the cross-prmeteriztion, the resultnt model of the pprel product will hve highly uneven distortions i.e., the shpe of the new product is uncceptble. From Fig.18, we

13 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 13 Fig. 16: The initil common bse domins constructed ccording to Geodesic distnces insted of the lengths of shortest pths (s [4]) cn still be improved by our optimiztion pproch, which cn be seen from the color mp for L 2 -stretches in crossprmeteriztion. The sttistic dt hs been listed in Tble 2. TABLE 2: Sttistics of distortions on the initil bse domins constructed by Geodesic distnces nd the optimiztion results. Models (M s/m t) Hnd1/2 Horse/Cmel Feture points # E L 2(M s) Geodesic distnces E ng(m s) Time (sec.) By Υ AD E ng(m s) E L 2(M s) Time (sec.) E L 2(M s) By Υ L 2 E ng(m s) Time (sec.) E L 2(M s) Blnced E ng(m s) Time (sec.) E L 2(M s) Merging E ng(m s) Time (sec.) Construction ccording to the Geodesic distnces tkes very long time to obtin the initil bse domins. cn find tht the shpe of n utomticlly designed dress for new client (i.e., M t) hs been improved to n cceptble level by using the optimized cross-prmeteriztion. 6.1 Limittion The min limittion of the pproch proposed in this pper is tht, like other greedy lgorithm bsed mesh optimiztion techniques, the result of this pproch my be only locl optimum s it depends hevily on how good the given common bse domins (generted by [4]) re. Some topologicl obstcles my prevent the lgorithm from generting further optimized common bse domins. We pln to develop new common bse domin construction method to generte common bse domins which hve better shpe similrities from the beginning of domin construction. This will be our ner future work. Fig. 17: An ppliction of dt preprtion for estblishing humn body dtbse. From left to right, the given source model M s served s templte model, the input mesh surfce s trget model M t, the result bsed on the originl crossprmeteriztion [4] (with unwnted distortion on the semntic feture curves), nd the result bsed on optimized domins. 7 CONCLUSION We present new signture, Length-Preserved Bse Domin (LPBD), in this pper for mesuring the level of stretch between surfce ptches in cross-prmeteriztion. This new signture well blnces the ccurcy of mesurement nd the computtionl speed. A set of metrics hve been studied nd compred to find good criterion to govern the optimiztion of common bse domins. A greedy optimiztion lgorithm is dopted to reduce the distortion in cross-prmeteriztion by repetedly pplying two opertors, boundry swpping nd ptch merging, to the input bse domins. A vriety of models hve been tested to demonstrte the functionlity of our pproch. Experimentl tests prove tht our method cn efficiently nd effectively improve the qulity of cross-prmeteriztion through optimizing the connectivity of the common bse domins, which re constrined by the nchor points. ACKNOWLEDGMENTS The uthors would like to thnk the vluble comments given by the reviewers. The work presented in this pper is

14 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 14 Fig. 18: An ppliction of the design utomtion for pprel products. (Left) The source model M s wering well designed dress. (Middle) A new model M t nd the dress wrped bsed on the cross-prmeteriztion between M s nd M t uneven distortions re generted on the wrped dress which re cused by the lrge distortion in the cross-prmeteriztion. (Right) The distortion becomes smooth on the deformed dress bsed on the optimized cross-prmeteriztion. prtilly supported by the HKSAR Reserch Grnts Council (RGC) Generl Reserch Fund (GRF) CUHK/ nd CUHK/ Thnks Chen et l. [33], AIM@SHAPE Shpe Repository, nd Imge-bsed 3D Models Archive for shring some 3D models used in this pper. REFERENCES [1] X. Li, Y. Bo, X. Guo, M. Jin, X. Gu, nd H. Qin, Globlly optiml surfce mpping for surfces with rbitrry topology, IEEE Trns. on Visuliztion nd Computer Grphics, vol. 14, no. 4, pp , [2] N. Ry, W.C. Li, B. Lévy, A. Sheffer, nd P. 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15 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED 15 Tsz-Ho Kwok received his B.Eng. degree in Automtion nd Computer-Aided Engineering from the Chinese University of Hong Kong (2009). He is currently PhD student in Deprtment of Mechnicl nd Automtion Engineering, the Chinese University of Hong Kong. His reserch interests include imge processing, geometric nd solid modeling, nd visul Computing. Yunbo Zhng is Ph.D Student of Deprtment of Mechnicl nd Automtion Engineering t The Chinese University of Hong Kong. His reserch interests include CAD&CAM, physicl bsed geometric modeling, nd computer grphics. Zhng received his Mphil in digitl design nd mnufcture from Ntionl Engineering Reserch Center for CAD, Huzhong University of Science nd Technology. Chrlie C.L. Wng is currently n Associte Professor t the Deprtment of Mechnicl nd Automtion Engineering, the Chinese University of Hong Kong, where he begn his cdemic creer in He gined his B.Eng. (1998) in Mechtronics Engineering from Huzhong University of Science nd Technology, M.Phil. (2000) nd Ph.D. (2002) in Mechnicl Engineering from the Hong Kong University of Science nd Technology. He is member of IEEE nd ASME, nd n executive committee member of Technicl Committee on Computer- Aided Product nd Process Development (CAPPD) of ASME. Dr. Wng hs received few wrds including the ASME CIE Young Engineer Awrd (2009), the CUHK Young Resercher Awrd (2009), the CUHK Vice-Chncellor s Exemplry Teching Awrd (2008), nd the Best Pper Awrds of ASME CIE Conferences (in 2008 nd 2001). His current reserch interests include geometric modeling in computer-ided design nd mnufcturing, biomedicl engineering nd computer grphics, s well s computtionl physics in virtul relity.