Mesh Simplification Using Vertex Clustering Based on Principal Curvature

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1 Intenational Jounal of Multimedia and Ubiquitous Engineeing, pp Mesh Simplification Using Vetex Clusteing Based on Pincipal Cuvatue Zhen Hua, Zilong Huang and Jinjiang Li Shandong Institute of Business and echnology, Yantai, China Shandong Nomal Univesity, Jinan, China Shandong Institute of Business and echnology, Yantai, China Abstact Accoding to the poblem that how to incease the fidelity of simplified model, this pape put fowad a method which using vetex clusteing based on pincipal cuvatue. Ou measue taking the gadual simplify thinking and simplifying the mesh model successive by quadic eo metic and pincipal cuvatue based on integal invaiants. We impove the defect of eo accumulation which consisted in the pevious algoithms. Not only ensuing the display effect fo the model, but also aising the efficiency of algoithm. eywods: 3D mesh simplification, vetex clusteing, tiangle mesh, quadic eo metic; integal invaiants, pincipal cuvatue 1. Intoduction With the development of compute technology and lase scanning equipments constantly, the 3D models become complex and detailed which applied to compute gaphics, compute aided geometic design, vitual eality, evese engineeing and othe field. In geneal, the 3D models ae epesented by tiangle mesh fom econfiguation (polygon mesh can be conveted into tiangle mesh though tiangulated, so, the eseach fo tiangle mesh models have geneal meaning. he models has become moe and moe detailed, it means that the numbe of tiangle patches which constitute the mesh model has up to millions o even billions as usual. Due to the positive coelation between the numbe of tiangle patches and the cost of pocessing them, the stoage, eal-time endeing and delivey fo mass infomation of models has bought an unpecedented challenge fo compute hadwae and netwok tansmission. Evey categoy of mesh model simplification methods has its own chaacteistic. Accoding to the diffeence of executing state fo simplification, we can divide the simplification algoithms into two kinds, static and dynamic. And also can fom the viewpoint whethe the topology stuctue has changed in the simplification pocess, divided into topology peseving and topology change. Howeve, these divese classifications ae had to contain all the simplification algoithms that coss each othe. he essence of model simplification is actually using appopiate methods to ISSN: IJMUE Copyight c 015 SERSC

2 Intenational Jounal of Multimedia and Ubiquitous Engineeing minimize the numbe of vetices, edges and patches, while maintaining the geometical chaacteistics popety of oiginal model. In othe wods, that is ensuing cetain accuacy, at same time, to educe the scale of data as much as possible. In this pocess, how to impove the efficiency and contol the simplification eo have attacted a lot of attention fom domestic and foeign scholas [1]. his pape pesents a new vetex clusteing simplification method based on pincipal cuvatue. Ou method choose the pincipal cuvatue of vetex as chaacteistic, using the chaacteistic points as less as possible to epesent the detailed infomation of oiginal model as much as possible and keep the loss of local visual chaacteistic to a minimum. And it also eflect the two geneal pinciples of model simplification, fist, the least numbe of vetices pinciple, second, the minimal diffeence pinciple which between oiginal model and simplified model. he pecondition of all those two pinciples is eo in the contollable ange. Fistly, accoding to the diffeent opeational objects in the pocess of model simplification, this pape summaizes the cuent situation and eseach achievements of 3D mesh model simplification; secondly, making a detailed intoduction of basic concepts, elated pinciples and the main steps which poposed in ou pape; finally, veifying the feasibility of this method, analyzing the supeioity that compaed with the simila algoithms. o summaize the content which needed to supplement and impove.. Related Wok In ecent yeas, thee ae a lot of fuitful eseaches in the mesh model simplification, while most of them ae woking based on the geometical elements that emoved in the pocess of simplification, such as vetex deletion, edge collapse and patch combination. As a widely used mesh model simplification algoithm which based on vetex clusteing, it is easiest to achieve and has the highest efficiency, but the simplification esult can t be guaantee is one of the main dawbacks []. Rossignac poposed a simplification method which is faste and easy to opeate based on vetex clusteing, nevetheless, this method has some limitations: on one hand, due to it didn t take the vetex distibution into consideation, the algoithm esulted in wasting time and space, on the othe hand, thee is no method given to contol the eos when we detemine the new vetices [3]. Zhou et al., made a self-adaptable patition on bounding box of mesh model by using the Octee, and put fowad an eo contol method based on the distance fom point to suface. Compaed with Rossignac s algoithm, applied extensively and obvious effects ae both the advantages [4]. uk pesented a mesh e-tiling simplification algoithm based on positioning vetices, the method distibutes a cetain numbe of vetices on the oiginal mesh model by using the point epulsion. And then, using the new vetices and old vetices to stuctue a middle mesh, finally, deleting the old vetices and e-tiangulated to get the simplified model. he limitations of this method concentate on the distibution of initial vetices. Only using the point epulsion as a efeence is not enough to make a bette esult on the models which has the ough 100 Copyight c 015 SERSC

3 Intenational Jounal of Multimedia and Ubiquitous Engineeing suface, and also inceasing the eo because of its huge calculation [5]. Afte taking a sufficient analysis fo the defects of point epulsion, Zhou and Ma applied the cuvatue and aea of tiangle as essential factos to eflect the chaacteistic infomation of mesh model, made a coss-conside on impotance degee of these two factos and designed a new positioning vetices algoithm, got fine esults [6]. Hoppe fist poposed the edge collapse simplification algoithm that applied to the two-dimensional manifold of tiangle mesh models. o measue the oiginal mesh with an enegy function, they established the complex equation fo optimizing global enegy. he non-lineaization of optimization pocedue fo mesh model has led to huge calculation and low aithmetic speed [7]. In ode to ovecome these shotcomings, ao and Pan intoduced the enegy value to evaluate the meta opeation and adjust the matching elation between oiginal mesh and simplified mesh. hei algoithm not only be applied to abitay topological shape, but also can change itself accoding to diffeent equiements [8]. In ode to contol the eo effectively and incease the accuacy of simplification, Zhou et al. aimed at the suface econstuction of evese engineeing, poposed a new algoithm that has a high degee of automation and no eo accumulation by combining the contolling based on examining ball with edge deletion [9]. In the pocess of edge collapse, ibbelt et al. egaded one of the endpoints on the edge as the epesentative vetex instead of adding a new vetex. his method impoves the efficiency of endeing while using less memoy [10]. he scientists also concentate on the deletion and combination of tiangle patch in the mesh model simplification algoithms. Hamann calculated the eo fo simplification based on the cuvatue and equal angle of the tiangle. He egaded the mean of cuvatues fo vetices as weight, emoved the tiangles which long and naow. Although it is easy to opeate, it has the huge calculation [11]. Gieng et al. poposed a simila tiangle collapse algoithm. he diffeence is that the latte egaded the poduct of aea and cuvatue as the cost of collapse. his change can maintain the shape chaacteistics pefeably, but it costs the algoithmic complexity [1]. Accoding to the method pesented by Hamann, Ma et al. impoved its tiangle emoval citeion to decease the whole calculation and peseve the featue availably [13]. Zhou et al. descibed the squae volume eo, shape facto and nomal constaint facto as a quadatic objective function, aftewads, put fowad a new tiangle collapse simplification algoithm which can keep the bounday featue effectively [14]. Anothe impotant banch of mesh simplification is dynamic defomation mesh simplification. Shami pesented a global multi-esolution stuctue based on time-dependant diected acyclic gaph. It is the fist simplification algoithm fo distoted suface, but the data stuctue is too complex to contol [15, 16]. iche et al. egaded the successive edge collapse as e-clusteing. o obtain the simplified model fo cuent fame, they used the vetices exchange opeations of pevious fame. he multi-esolution method which has a dynamic connectivity, not only aims at distoted suface but also with high efficiency [17]. In ode to get the pefect Copyight c 015 SERSC 101

4 Intenational Jounal of Multimedia and Ubiquitous Engineeing effect, Huang applied the vetex tee to change the dynamic connectivity, this opeation inceases the quality of epoduction, but it also bings huge calculation and complex pocess [18]. hee ae many scholas consideed simplifying the models fom multiple angles and obtained some bette esults. Lv, Zhang and Sun collapsed the edge pefeentially which has less eo afte collapsed and got a coase mesh with good topological connectivity. hen, simplify the coase mesh though intoducing new vetices with good valance. he method geatly inceases the quality of simplified mesh [19]. Cohen-Steine et al. divided the oiginal mesh into seveal clustes of simila sufaces and optimized them via the algoithm of Lloyd. hey gained the simplified mesh by using polygons to epesent evey cluste. Although the algoithm has pefeable stability, howeve, due to the time of iteation is unknown, it is difficult to impove its speed [0]. In the field of vitual eality, the oiginal model contains not only the geometic popeties, but also a lot of textue, colo and othe infomation. How to integate the chaacteistic infomation into the pocess of simplification is the popula eseach in ecent yeas. In ode to solve the discontinuities of appeaance attibutes in the simplification, Lu, Zeng and Pan poposed a new fomula fo the cost of edge collapse by analyzing the elationship between the discete appeaance seam and vetices, and then, they wee applying the half-edge collapse to impove the EM. hei algoithm has a good pefomance in featue-peseving [1]. Zhang et al. used the symmetic Hausdoff distance in n R space to contol the ode of eo and accuacy of simplification. he algoithm can peseve the additional infomation of colo and textue []. Liu, Xie and Jin aimed at the mesh models which have textue attibutes, then maked the bounday edges and textue edges of the model in advance, weighted them accoding to the impotance degee. Futhemoe, they combined the half-edge collapse with weight in the simplification to confim the ode of collapse and get the simplified model [3]. Accoding to the calculation fo geometical eo based on EM, Feng and Zhou added the textue eo calculation in the fome. he algoithm etains the textue of suface fo the model successfully [4]. 3. Desciption of the Algoithm his pape take a hieachical and gadual simplify mode. Fist, we classify and choose the vetices of mesh model peliminaily, delete the edundant vetices which have simila chaacteistic. And then, accoding to the pincipal cuvatue, we get the final epesentative vetices though conducting the cluste on the emaining vetices which have moe impotant degee fo the simplified model Model Pepocessing Due to the stability of tiangle patch, thee is an ieplaceable advantage of tiangle mesh fo indicating the fee-fom suface and dawing the 3D models. And it 10 Copyight c 015 SERSC

5 Intenational Jounal of Multimedia and Ubiquitous Engineeing also becomes the mainly method fo epesenting the 3D models in the field of inteactive compute gaphics. Unde odinay cicumstances, 3D model mostly epesented by polygon mesh, and the polygon mesh can divide into seveal tiangles by connecting the vetices. With this opeation, we can tiangulate evey suface of the pending polyhedal model to get the tiangle mesh. We numbeed all those tiangles and ecoded them in tiangle table. Figue 1 shows an example of dividing a polygon into seveal tiangles. 3.. he Classical EM Algoithm Figue 1. he Way to Divide Polygons Galand fist pesented the concept of EM and made it as the standad fo measuing the cost of edge collapse. he algoithm defined the squae of distance fom point to suface as the eo to contol the simplified pecision of models. It has high efficiency and minimal occupancy of compute memoy [5]. In the ange of Euclidean geomety, define a vetex: Make [ p p p 1 x y z (1 p ] o indicate the tiangle patch: [ a b c d ( f ] ax a by b cz c d 1 0 (3 he nomal vecto is: n ( a, b, c (4 hen the quadatic fom of eo fo P is: p p p ([ p p p 1] ( p f x y z (5 F is a set of tiangles which both take the p as thei own vetex. By futhe p F Copyight c 015 SERSC 103

6 Intenational Jounal of Multimedia and Ubiquitous Engineeing ewiting, we can get anothe epesentation like: ( p f ( f p p ( ff p p ( p p p F p F p F f (6 p is used to detemine the squae of distance fom a cetain vetex to f, a ab ac ad ab b bc bd ff p (7 ac bc c cd ad bd cd d Make the sum of p fo all neighbohood tiangles which use the p as thei own vetex. In that way, we can get the value of eo Peliminay Classify and Simplify fo Vetices Based on uadic Eo Metic Step 1: Apply the bounding box to suound the oiginal mesh model, subdivide the bounding box into N cells aveagely based on its edges. When some vetices into the same cell, they can fom a cetain class. We numbeed all those classes and ecoded them in vetex index table N. Step : When one tiangle s thee vetices all belong to the same class which in the tiangle table, these thee vetices degeneate into a new point, the point is located in the cente of gavity. Step 3: When a tiangle s two vetices belong to the same class, we apply the EM to calculate the value of eo fo the two vetices, and then, emain the vetex which has less value than the othe one. Figue shows that how to combine vetices when two vetices belong to the same class. Figue. he Eo Value of p is Less than that of q, so Delete the q Step 4: When the class of a vetex is diffeent fom othe two vetices in one tiangle, keep this vetex intact. hough the above seveal steps of vetex clusteing and delete, we have aleady eliminated the edundant vetices which can epesent the same chaacteistic 104 Copyight c 015 SERSC

7 Intenational Jounal of Multimedia and Ubiquitous Engineeing infomation in the mesh model. hese opeations decease the space occupancy and impove the efficiency of whole algoithm; they also educe the time complexity and povide advantage fo the futhe simplification he Selection Algoithm of Repesentative Vetices Based on Pincipal Cuvatue Although doing a seies of peliminay simplification, thee ae still many unteated edundant vetices in the mesh model due to the simple classification. We need impoving the classification to futhe simplify the vetices. Owing to the pinciple cuvatue of a vetex on the suface can eflect the geometical chaacteistics moe than othe elements. And also the cuvatue is insensitive to the noise intefeence so that it has bette obustness. his is the eason why we apply the pinciple cuvatue to obtain the epesentative vetices. Fistly, we apply the integal invaiants to estimate the pincipal cuvatue fo evey cell among the bounding box that used by peliminay simplification. Assuming a point set is a small set nea the point p which adius is. We calculate the volume of the point set [6, 7]: V dx Secondly, calculating the gavity cente of its: (8 S V 1 xdx (9 then, obtaining its covaiance matix: J xx dx V S S (10 Whee XX is a 3x3 matix has the ank is 1 (We use the column vecto. If is the ball neighbohood, accoding to the fomulae above, we can get the eigenvalues of the covaiance matix fo : M b M b 5 6 (3 k k O (, ( k 3 k O (, When gets close to 0, we can calculate the two pincipal cuvatue values by fomula (13 and fomula (14: (11 (1 6 ( M 3 M b, b,1 b, (13 6 ( M 3 M b,1 b, b, (14 If is the spheical neighbohood, similaly, gaining the fomula (15 and Copyight c 015 SERSC 105

8 Intenational Jounal of Multimedia and Ubiquitous Engineeing fomula (19: M s M s he two pincipal cuvatue values: 4 5 (3k k O(, ( k 3 k O (, (15 (16 ( M 3 M s, s,1 s, (17 ( M 3 M s,1 s, s, (18 Secondly, compaing the pincipal cuvatue of evey cell with the pinciple cuvatue of vetices in it, and then choose the vetices which pinciple cuvatue is equal with its own cells espectively, egad them as epesentative vetices. If thee is no vetex to confom to this condition, go on subdividing the bounding box until finding the epesentative vetices by iteative algoithm. Finally, afte all the epesentative vetices ae found, we can get the simplified mesh model by tiangulated the new vetices. 4. Expeiments When we take the expeiments, we choose the epesentative vetices based on geneal cuvatue and pinciple cuvatue espectively fo contast. Figue3 povide the selected esults of epesentative vetices and thei coesponding simplified models. he fist column is based on geneal cuvatue; the second is based on pinciple cuvatue, the last two columns show the simplified models coespond to the fist and second. Accoding to the compaison between a and b of the face models, the numbe of epesentative vetices chose by the geneal cuvatue is moe than using the pinciple cuvatue on the same pats which have emakable chaacteistics. Howeve, in ea positions, because of the suface undulation is less pominent, so that the numbe of epesentative vetices has not much diffeence. It is obvious that the featue extaction based on pinciple cuvatue is bette than the geneal cuvatue in the visual effect fom compaing c with d. Similaly, the geneal cuvatue method needs a lot of epesentative vetices to epesent the local positions that have ich lines like the legs of hose. In othe wods, the pinciple cuvatue method not only educes the computational scale, but also the simplified efficiency exceeds the fome. 106 Copyight c 015 SERSC

9 Intenational Jounal of Multimedia and Ubiquitous Engineeing (a (b (c (d (a (b (c (d Figue 3. Selected Results and thei Coesponding Simplified Models Based on diffeent Algoithms We apply ou algoithm, Rossignac s clusteing algoithm and Galand s EM algoithm to simplify the thee models in Figue4 espectively. he esults show by Figue 4. he fist column is oiginal models, the second and thid columns ae the esults of using the clusteing algoithm and EM, the last column is based on ou algoithm. Like this pape, Rossignac s algoithm also subdivided the model by bounding box, so its simplified degee of each pat is the same and visual effect looks evenly. Fom the expeimental esults we may see that the algoithm can guaantee the pofile of oiginal models, but it also lost many pominent local details, especially in some impotant featue positions of face. Fo example, both the eyes position of the fist model and the uppe half of the second model have insufficient visual pefomances. Due to apply the EM to contol the simplified eo, Galand s algoithm impoved the details pefomance. Howeve, when the simple eo contol stategy simplifies the model moe in depth, it may cause the excess simplification and distoted effect, especially concentated in the positions which have a lot of chaacteistics. Such as the hais and mouth cack of the fist model. hee ae geat Copyight c 015 SERSC 107

10 Intenational Jounal of Multimedia and Ubiquitous Engineeing diffeence between the oiginal model and simplified model. On one side, this pape daws on the expeience of advantages which in the two algoithms above, and also combines with the pinciple cuvatue, on the othe side, we let the bounding box intesect with EM afte taking thei defects into consideation. It not only makes the model simplified evenly, but also peseves its chaacteistics and contols the simplification degee effectively. he expeimental esults show that, ou method is much bette than the othe two in handling details and holding the whole algoithm. he simplified model is the most close to oiginal model based on ou algoithm. (a (b (c (d (a (b (c (d 108 Copyight c 015 SERSC

11 Intenational Jounal of Multimedia and Ubiquitous Engineeing 5. Conclusion (a (b (c (d Figue 4. Seveal Expeimental Results his pape intoduces a simplified algoithm using vetex clusteing based on pincipal cuvatue, and combining the EM with clusteing. he algoithm only opeates the vetices and tiangles, don t involve the topological elation of too much vetices, line and sufaces. his tait povides the conditions fo meging the tiangles which geomety adjacency but topology-independent. he expeiments pove: the algoithm emoves vetices and tiangles gadually. It also maintains the chaacteistics of oiginal model at the geatest extent, especially fo obvious chaacteistics but iegula model. he simplified model has high quality of epoduction. he algoithm should pay attention to the following seveal aspects which need to impove in the futue: we can adopt the inhomogeneous manne to subdivide the bounding box, so that the method may give consideation to featue infomation in diffeent positions of the model, and make it without loss of geneality. Moeove, although using the gadual simplified thinking, the eo contol method should still be pefected to incease the pecision of whole algoithm. Acknowledgements Poject suppoted by the National Natue Science Foundation of China (No , , 61745, 61477; the Povincial Natual Science Foundation of Shandong unde Gant No. ZR013FM015. Refeences [1] He H. G., ian J., Zhang X. P., Zhao M. C. and Li G. M., A Suvey of Mesh Simplification, vol. 1, no. 13, (00. [] Luebke D., A Develope s Suvey of Polygonal Simplification Algoithms, IEEE Compute Gaphics & Applications, vol. 3, no. 1, (001. [3] Rossignac J. and Boel P., Multi-Resolution 3D Appoximation fo Rendeing Complex Scenes, In: Falcidieno, B. unii,. L, eds. Poceedings of the nd Confeence Modeling in Compute Gaphics: Copyight c 015 SERSC 109

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