3D Surface Simplification Based on Extended Shape Operator

Transcription

1 3D Surface Simlification Based on Extended Shae Oerator JUI-LIG TSEG, YU-HSUA LI Deartment of Comuter Science and Information Engineering, Deartment and Institute of Electrical Engineering Minghsin University of Science and Technology o.1, Hsin Hsin Road, Hsin Feng, Hsin Chu County TAIWA (corresonding author), Abstract: - In order to resent realistic scenes and models, comlex triangle meshes comrising large numbers of triangles are often used to describe 3D models. However, with an increase in the number of triangles, storage and comutation costs will raise. To reserve more geometric features of 3D models, Zhanhong and Shutian emloyed the curvature factor of collasing edge, Gaussian Curvature, to imrove the Quadric Error Metrics (QEM) simlification. Their method allows the QEM not only to measure distance error but also to reflect geometric variations of local surface. However, the method can only estimate the curvature factors of vertices in manifold surfaces due to Gaussian Curvature. To overcome this roblem, we roose a new simlification method, called Extended Shae Oerator. The Extended Shae Oerator estimates the local surface variation using three-rings shae oerator. The Extended Shae Oerator can be alied in the simlification of manifold and non-manifold surfaces. In our exeriment, we emloyed the error detection tool, Metro, to comare errors resulting from simlification. The results of the exeriment demonstrate that when the model has been simlified, the roosed method is suerior to the simlification method roosed by Zhanhong and Shutian. Key-Words: - surface simlification, extended shae oerator, curvature factor, quadric error metric 1 Introduction The increased revalence of information devices in recent years has increased the use of 3D modeling [14, 15, 16, 17]. Many 3D models comrise hundreds of thousands or even millions of triangles; for mobile devices, rocessing the large quantities of data in these models may affect the execution seed of rograms or games, and therefore, the most direct solution is to reduce the number of triangles used to describe the 3D model, the rocess of which is called surface simlification [3, 4]. The ultimate objective of surface simlification in 3D models is to reduce the number of triangles in the model without reducing the quality [5, 6, 12]. In addition, the characteristic contours of the model must also be taken into consideration during the rocess of simlification to avoid excessive damage to the resulting contours. Many simlification methods have been roosed [2, 13], including vertex decimation [10], vertex clustering [9], edge contraction [7], triangle contraction [8], and quadric error metrics simlification [1]. Among these aroaches, Quadric Error Metrics (QEM) simlification is considered the closest to otimal. The Curvature Factor [14] is based on the QEM simlification. This method used Gaussian Curvature to define the concet of curvature factors of collasing edge and embed it into the original QEM roosed by Garland. This method can revent more geometric features than original QEM. However, only the vertices of the full disc in the manifold meshes can this method estimate, shown in Fig.1. Fig. 1: Full disc in manifold mesh According to the definition of Chen et al.[18], a surface is a manifold if each oint has a neighborhood homeomorhic to a disc in the real lane. They take some examles of common mesh structures that are incomatible with the manifold definition are shown in Fig. 2. E-ISS: Issue 8, Volume 12, August 2013

2 Fig. 2: Some examles of non-manifold meshes [18] In fact, QEM simlification also suorts nonmanifold surface models besides manifold ones. Although the curvature factor [14] is based on the QEM simlification, it cannot be used to calculate the curvature factors of the vertices in non-manifold meshes, even the boundary vertices. To overcome this roblem, we roosed a new curvature estimation method, called Extended Shae Oerator (ESO). The ESO estimates the curvature factors using the shae oerators between the vertex and its neighbor vertices. Additionally, the ESO also extends the size of estimation from 1-ring neighborhood of the Curvature Factor to three rings. Exerimental results show the ESO can not only kee the geometric features but also decrease more errors caused by simlification. 2 Related Work 2.1 Quadric Error Metrics Simlification This method involves contracting vertex v 1 and vertex v 2 into a new vertex, v, the location of which is obtained using the distance values of a vertex and the conjoining triangles. The minimum distance value determines the location of the new vertex v. The estimated distance value also reresents the resulting error of simlification. In this manner, the QEM simlification moderately maintains the outer shae of the model while reducing the number of triangles within, thereby reducing the resulting errors. The estimated osition of v also strongly influences changes in the surrounding triangles (Fig. 3). v 1 v 2 Fig. 3: Quadric Error Metrics Simlification v In a three-dimensional sace, suose the equation of lane is ax+by+cz+d=0, where a 2 +b 2 +c 2 =1. The distance between this lane and any vertex v=[v x v y v z 1] T is thus as shown in Eq. (1): [1] 2 T 2 T T T ( D v ) ( v ) v ( ) v K v (1) Garland used quadric error metrics for the measurement of error in model simlification, defining error as: T Error( v) v K v (2) where K can be exressed as: a b K T a b c d c d (3) a 2 ab ac ad ab b 2 bc bd 2 ac bc c cd 2 ad bd cd d In the alication of K to measure surface error in QEM simlification, the error that each qualifying vertex air may otentially cause after simlification must be estimated; simlification is then alied to the vertex airs that will result in the least error. In this way, excessive damage can be revented and the contour of the object can be reserved. In addition, this action is reeatable for surface simlification. The main advantage of this simlification method is fast and kees low average error. However, this method considered only its distance metric, it is not suitable for simlifying models with shar angles. To solve this roblem, Zhanhong and Shutian roosed the Curvature Factor method to reserve quite a number of imortant shae features and reduce visual distortion effectively. [14] 2.2 Curvature Factor In order to reserve the shar features after simlifying, Zhanhong and Shutian defined an edge curvature factor. This factor is estimated by the Gaussian Curvature roosed by Meyer et al., the Gaussian Curvature formula as follow [19]: K 1 f v 2 # j j (4) i A( v ) i where A(v i ) resents the area sum of adjacent triangle of v i, θ j is the angle of the j-th face at the vertex v i, and #f denotes the number of faces around this vertex. E-ISS: Issue 8, Volume 12, August 2013

3 triangle. That is, the angle sum of all faces at the estimated vertex would be far less than 2π, even though the surface that the estimated vertex lies on is an even one. This case will take lowcurvature surfaces for high-curvature surfaces. Fig. 4: Estimation of the Gaussian Curvature of v i The edge curvature factor is defined as follow: # f v f v j j k k K (5) A( v1 ) A( v2) where v 1 and v 2 are the vertices of an edge, A(v 1 ) and A(v 2 ) resent the area sum of adjacent triangle of v 1 and v 2 resectively, θ j is the angle of the j-th face at the vertex v 1, θ k is the angle of the k-th face at the vertex v 2, and #f v1 and #f v2 denote the number of faces around v 1 and v 2 resectively. The edge curvature factor used the Gaussian Curvature to imrove the quadric error metrics and ket more shar features. The latter can simlify the manifold and non-manifold meshes, but the former is only suitable to be alied to the manifold ones. If the edge curvature factor is used to simlify the non-manifold meshes, the Eq.5 could not find out suitable curvature values. The examles that are unsuitable to use the Eq.5 are as follows: 1. An edge meeting the estimated vertex is shared by three triangles: Every edge in a manifold mesh should be shared by only two triangles. Therefore, if an edge meeting the estimated vertex is shared by three triangles, the angle sum of all faces at the estimated vertex would be more than 2π, as shown in Fig.5. That is, the estimation of Gaussian Curvature value would be minus if the surface the estimated vertex lies on is an even one. This value would resent the surface variation incorrectly. # Fig. 6: The estimated vertex o is a boundary vertex [18] 3. There are two incident cones on the estimated vertex: According to the definition of manifold surfaces, each vertex should have only one incident cone. Therefore, if there are two incident cones on the same estimated vertex, the angle sum of all faces at the estimated vertex would be far more than 2π, even u to 4π. This case will imact the determination of the edge curvature factor when collasing edge. Fig. 7: There are two incident cones on the estimated vertex o [18] To overcome these roblems, we roose a new simlification method, called Extended Shae Oerator. Our method can not only estimation surface variation on non-manifold meshes, but also decrease more errors caused by simlification than the one roosed by Zhanhong and Shutian. Fig. 5: An edge meeting the estimated vertex o is shared by three triangles [18] 2. Boundary vertex: When the estimated vertex is a boundary vertex, some edges will have only one incident 3 Shae Oerator Suose is a oint on surface M, and v is the tangent vector to M at. The shae oerator for tangent vector v at can thus be defined as S (v): S ( v) U (6) v where U is the normal vector field in the neighborhood of on M (Fig. 8); the shae oerator S (v) reresents the variation in normal vector U at on M in the direction of v (Fig. 9). E-ISS: Issue 8, Volume 12, August 2013

4 Fig. 8: ormal vector field on surface M [2] In ractice, this is based on a tangent lane. Suose there is a tangent lane TP at oint ; the rojections of 1, 2, 3,, and k are used to obtain the required tangent vectors t 1, t 2, t 3,, and t k, which are then emloyed to derive the shae oerators at in the direction of said vectors. Using the shae oerators enables the estimation of overall variations in the local surface area. Examles of imlementation results are resented in Fig. 11. high Fig. 9: Shae oerator estimation for tangent vector v at oint on surface M [2] The shae oerator refers to the variation in the normal vector field in the direction of vector v at oint on surface M. To calculate surface changes in the neighborhood of, Jong et al. [2, 13] used variations in the normal vectors of and its neighboring oints for estimation. Suose the k neighboring oints of are 1, 2, 3,, and k, and the tangent vectors of toward each neighboring oint are t 1, t 2, t 3,, and t k. Calculations from Eq. 6 rovide the shae oerators S (t 1 ), S (t 2 ), S (t 3 ),, and S (t k ) at in the directions of t 1, t 2, t 3,, and t k. Integrating all of the shae oerators enables the derivation of the following formula to estimate variations in the surface surrounding : S k i1 U ( ) U( ) k i1 i i (7) Fig. 11: Surface variation estimation; green reresents areas of minor variation, whereas red indicates areas of major variation The shae oerator is used to estimate surface variations by neighboring vertices on 3D models rior to vertex-air contraction in order to reduce the error caused by simlification. evertheless, for the simlification of models at lower resolutions, considering only the first ring of neighboring vertices is far from adequate. When simlification results in a smaller number of triangles, the simlified vertices cover a larger area. For this reason, the range under consideration should also be extended. This study rooses the Extended Shae Oerator to reduce error resulting from the simlification to the fewest ossible number of triangles. 4 Extended Shae Oerator low Fig. 10: Shae oerators of oint using the tangent vectors to each neighboring oint to estimate local surface variation 4.1 Analyzing the Imact Area of Simlified Vertices To analyze the imact areas of simlified vertices, we erform an exeriment to calculate the moving distance of each simlified vertex. This exeriment uses four models, including a cow, a dragon, a femur and an isis. The half edge collase [1] is adoted to record the moving ath of each simlified vertex in the simlification rocess because it will allow the vertex moving to the other one in the collasing edge. E-ISS: Issue 8, Volume 12, August 2013

5 In this exeriment, we simlify the above models into five different resolutions, include 40%, 30%, 20%, 10% and 5%. The exerimental results are shown in Tables 1 to 4. In these tables, the moving length resents the moving times of vertex when it is simlified. Taking the cow model (Table 1) as an examle, when the cow model is simlified into forty ercent of triangles, 1315 vertices are moved one time, 387 vertices are moved two times, 38 vertices are move three times, 2 vertices are moved more than 3 times, and 1162 vertices still stay at their original ositions. Table 1: Moving length analysis of simlified vertex (cow model; the number of vertices =2904) SP 5% 10% 20% 30% 40% ML The number of vertices > AML SP : simlification ercentage ML : moving length AML : average moving length Table 2: Moving length analysis of simlified vertex (dragon model; the number of vertices=25418) SP 5% 10% 20% 30% 40% ML The number of vertices > AML Table 3: Moving length analysis of simlified vertex (femur model; the number of vertices=76794) SP 5% 10% 20% 30% 40% ML The number of vertices > AML Table 4: Moving length analysis of simlified vertex (isis model; the number of vertices=100002) SP 5% 10% 20% 30% 40% ML The number of vertices > AML According to the exerimental results, the average moving lengths are less than three, even if the models are simlified into only five ercent of the number of triangles. Additionally, the ercent of the number of vertices that the moving lengths are less than or equal three are almost over ninety ercent, as shown in Table 5. Therefore, we extend the range from one ring to three rings for the estimation of surface variation. Table 5: The ercent of the number of vertices with moving length<=3 in different simlification ercentage SP 5% 10% 20% 30% 40% The ercent of the number of vertices Models with moving length<=3 (%) cow dragon femur isis SP: simlification ercentage 4.2 The Proosed Algorithm Shae oerators generally only consider the neighboring oints of, as shown in Fig. 12(a). To obtain a low-resolution model of higher quality after simlification, this study exanded the estimation range to rings of neighboring oints. As shown in Fig. 12(b), 1,1, 1,2, 1,3,, and 1,M1 reresent the first ring of oints neighboring ; 2,1, 2,2, 2,3,, and 2,M2 are the second ring of oints neighboring, and 3,1, 3,2, 3,3,, and 3,M3 comrise the third ring of oints neighboring. Likewise, we can define the th ring of oints neighboring as,1,,2,,3,, and,m. As simlification of the E-ISS: Issue 8, Volume 12, August 2013

6 model roceeds, the number of triangles in the mesh of the model dwindles. In other words, the imact range of each vertex exands and the data that each vertex encomasses increases as well. For examle, as shown in Fig. 13, oint is a vertex on a dragon model; during the first simlification, the area that influences lies within the first ring of neighboring oints. During the second simlification, the area that may influence extends to the second ring of neighboring oints, and so forth. After several iterations, the area influenced by each vertex exands. Therefore, we extended the estimation range of the shae oerator to the th ring. The formula to estimate the -rings shae oerator of the area where is located is: M j j i U( j i U 1 1, ) ( ) S (8) M j j1 i1 j, i where : the number of rings; j,i : the ith oint neighboring in the jth ring; U( j,i ) : the normal vector at oint j,i ; M j : the number of oints neighboring in the jth ring. The variations in the normal vectors of each ring are: M1 The first ring : U ( 1 1, ) ( ) i i U M 2 The second ring : U ( 1 2, ) ( ) i i U M3 The third ring : U ( ) U( : : : 1 3, ) i i i 1, i ) M The th ring : U( ) U( (a) the first ring (b) the first to third rings Fig. 12: rings centered at (=1 3) Basically, the Extended Shae Oerator simlification is an imroved algorithm for the QEM simlification. Prior to estimating the QEM, we first find out the -rings shae oerator S for each oint, calculate the surface variation within the range of rings neighboring, and then use S as the weight value to imrove the QEM. The 3,6 2,4 2,5 3,5 1,3 Extended Shae Oerator can reduce the error caused by simlification when the model is simlified to lower resolutions , ,4 1,1 1 3,2 1,2 2,1 3,1 1,5 1,4 2,2 3,3 3,M3 The third ring 3,3 3,2 3,1 3,M3 3,M3-1 2,2 2,3 The second ring 2,1 2,M2 2,M2-1 1,2 1,3 1,4 1,5 The first ring 1,1 1,7 1,6 Fig. 13: The -rings neighboring oints of oint E-ISS: Issue 8, Volume 12, August 2013

7 The rimary calculation stes of the roosed simlification aroach are as follows: 1. Calculate the shae oerator of each vertex {( 1,1, 1,2,, 1,M1 ), ( 2,1, 2,2,, 2,M2 ),, (,1,,2,,,M )}. Using Eq. (8), estimate the -rings shae oerator S of and change the QEM estimation formula to: Q ( ) S K (9) where f is the lane encomassing the triangles with as a vertex; K f reresents the 4 4 matrix of lane f. 2. Select each qualifying vertex air (v 1,v 2 ), and calculate the minimum error resulting from simlification. 3. Select the vertex air (v 1,v 2 ) with the minimum error for simlification. 4. Contract vertex air (v 1,v 2 ) into v, and calculate the QEM of v using Q=(Q 1 +Q 2 ), where Q 1 and Q 2 are the QEMs of v 1 and v 2, resectively. 5. Udate the information regarding the oints neighboring v 1 and v Reeat the stes above until the designated number of triangles is reached. f Curvature Factor method in reducing error. In the cow model, for examle, the number of triangles was reduced from 5804 to 2321, 1741, 1160, 580 and 290. The Curvature Factor resulted in errors of , , , , and , resectively, whereas the roosed method only caused errors of , , , , and This accounted for imrovements of 80.68%, 80.53%, 78.71%, 61.81%, and 62.46% comared to the Curvature Factor, resectively, as shown in Table 7, Fig.14 and Fig.21. Table 7: Hausdorff Distance Comarison (unit : 10 2 ) cow model SP Triangles CF ESO IR 40% % 30% % 20% % 10% % 5% % SP : Simlification Percentage IR : Imrovement Rate 5 Exerimental Results This study imlemented an exeriment with three rings to verify the Extended Shae Oerator (ESO) simlification method roosed in this study. We conducted comarisons with the Curvature Factor. Four models were emloyed for the simlification exeriment: a cow, a dragon, a femur, and an isis. The number of vertices and triangles are shown in Table 6. Table 6: Information of Exerimental Models Model Vertices Triangles Cow Dragon Femur Isis This study emloyed Hausdorff Distance imlemented in Metro [11] to measure the error caused by simlification and understand the variations in the shaes of the models before and after simlification. Errors occurring in the Curvature Factor (CF) [14] were also measured. The results of the four methods were then comared. The results of the exeriment show that the roosed method is more effective than the Fig. 14: Error measurement for cow model Besides taking the cow model for verification, the dragon, femur and isis models are taken for exerimental testing. In the dragon model, the original model contains vertices and triangles. Figure 22 shows the simlification results. Table 8 and Fig. 15 comare the error measurements with the Curvature Factor and the roosed method; the table shows that the roosed method can achieve an % error reduction. In the femur and isis models, the original models contain and triangles resectively. Figures 23 and 24 show the simlification results of the femur and isis models. Tables 9, 10 and Figs. 16, 17 comare the error measurements with the Curvature Factor and the E-ISS: Issue 8, Volume 12, August 2013

8 roosed method; the tables show that the roosed method can achieve % and % error reductions for the simlifications of the femur and isis models resectively. Table 8: Hausdorff Distance Comarison (unit : 10 3 ) dragon model SP Triangles CF ESO IR 40% % 30% % 20% % 10% % 5% % SP : Simlification Percentage IR : Imrovement Rate Table 10: Hausdorff Distance Comarison (unit : 10 3 ) isis model SP Triangles CF ESO IR 40% % 30% % 20% % 10% % 5% % Fig. 17: Error measurement for isis model Fig. 15: Error measurement for dragon model Table 9: Hausdorff Distance Comarison (unit : 10 3 ) femur model SP Triangles CF ESO IR 40% % 30% % 20% % 10% % 5% % In terms of feature reservation, Fig. 18 shows the comarison results of the QEM, the CF and the roosed method for simlifying the cow model into one with 580 triangles. Aarently, regarding the cow horn features, the CF can reserve the model features better than the QEM. However, the former also imact on the outline of cow neck. The roosed method can not only reserve the outline of cow horn, but also retain the shae of cow neck. In addition, Fig. 19 shows the comarison results of the CF and the roosed method for simlifying the dragon model into one with 5076 triangles. The figure shows that our method can retain more teeth and eye features of the dragon model than the CF. (a) original model Fig. 16: Error measurement for femur model (b) QEM (c) CF (d) ESO Fig. 18: Comarison of the features of cow horns and neck (580 triangles). E-ISS: Issue 8, Volume 12, August 2013

9 (a) original model (b) CF (c) ESO Fig. 19: Comarison of the features of dragon teeth (5076 triangles). Moreover, the Curvature Factor method will also destroy the features of hole easily due to the imroer angle estimation on non-manifold surfaces. Taking the femur model as an examle, although the boundary vertices by the hole are not on a highcurvature surface, the triangles meeting these vertices are still destroyed easily due to being taken for on a high-curvature surface. (a) original model boundary vertex (b) CF (c) ESO Fig. 20: Comarison of the hole features of femur (7666 triangles). 6 Conclusions The rimary objective of 3D model simlification is to maintain the characteristic contours of the model desite reducing the resolution. The Curvature Factor method emloys the Gaussian Curvature to reserve features of simlified models. However, it can only be used to simlify manifold models. To reserve the features of non-manifold model, this study rooses a novel method, called the Extended Shae Oerator. The roosed method uses threerings Shae Oerator to estimate the surface variation and reserves more features than the Curvature Factor. Exerimental results show that the Extended Shae Oerator can also reduce the error caused by simlification. References: [1] M. Garland and P. S. Heckbert, Surface simlification using quadric error metrics, SIGGRAPH 97 Conference Proceedings, 1997, [2] B. S. Jong, J. L. Tseng, W. H. Yang and T. W. Lin, Extracting Features and Simlifying Surfaces using Shae Oerator, International Conference on Information, Communications and Signal Processing, 2005, [3] J. L. Tseng, Shae-based simlification for 3d animation models using shae oerator sequences, 4th AU/WSEAS International Conference on Communications and Information Technology, Corfu Island, Greece, July 22-25, 2010, [4] P. F. Lee and C. P. Huang, The DSO Feature Based Point Cloud Simlification, 2011 Eighth International Conference on Comuter Grahics, Imaging and Visualization, 2011,.1-6. [5] Z. Y. Yang, S. W. Xu and W. Y. Wu, A Mesh Simlification Algorithm for Keeing Local Features with Mesh Segmentation, 2010 International Conference on Mechanic Automation and Control Engineering, 2010, [6] Y. Li, Q. Zhu, A ew Mesh Simlification Algorithm Based on Quadric Error Metrics, International Conference on Advanced Comuter Theory and Engineering, 2008, [7] A. Gu eziec, Surface simlification with variable tolerance, Second Annual Intl. Sym. on Medical Robotics and Comuter Assisted Surgery (MRCAS 95), 1995, [8] H. Bernd, A Data Reduction Scheme for Triangulated Surfaces, Comuter Aided Geometric Design, Vol.11, 1994, [9] J. Rossignac and P. Borrel, Multi- resolution 3D aroximations for rendering comlex scenes, Modeling in Comuter Grahics: Methods and Alications, 1993, [10] T. S. Gieng, B. Hamann, K. I. Joy, Gregory L. Schussman, and Issac J. Trotts, Constructing hierarchies for triangle meshes, IEEE Transactions on Visualization and Comuter Grahics vol.4, no.2, 1998, [11] P. Cignoni, C. Rocchini, R. Scoigno, Metro: measuring error on simlified surfaces, Comuter Grahics Forum, Vol.17, o.2, 1998, E-ISS: Issue 8, Volume 12, August 2013

10 [12] V. Ungvichian and P. Kanongchaiyos, Mesh Simlification Method Using Princial Curvatures and Directions, Comuter Modeling in Engineering & Sciences, Vol.77, o.4, 2011, [13] B. S. Jong, J. L. Tseng and W. H. Yang, An Efficient and Low-Error Mesh Simlification Method Based on Torsion Detection, The Visual Comuter, Vol.22, o.1, Jan. 2006, [14] T. Zhanhong and Y. Shutian, A Mesh Simlification Algorithm Based on Curvature Factor of Collasing Edge, 2nd IEEE International Worksho on Database Technology and Alications, 2010,.1-3. [15] M. H. Mousa and M. K. Hussein, Adative visualization of 3D meshes using localized triangular stris, WSEAS Transactions on Comuters, Vol.11, o.4, 2012, [16] H. U. Khan, Relica Technique for Geometric Modelling, WSEAS Transactions on Information Science and Alications, Vol.6, o.6, 2009, [17] C. D. Ruberto, M. Gaviano, A. Morgera, Shae Matching by Curve Modelling and Alignment, WSEAS Transactions on Information Science and Alications, Vol.6, o.4, 2009, [18] M. Chena, B. Tub, B. Lu, Triangulated manifold meshing method reserving molecular surface toology, Journal of Molecular Grahics and Modelling, Vol.38, 2012, [19] M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Discrete Differential-Geometry Oerators for Triangulated 2-Manifolds, Proceedings of VisMath, (a) 5804 triangles (original model) (b) 2321 triangles (40%) (c) 1741 triangles (30%) (d) 1160 triangles (20%) (e) 580 triangles (10%) (f) 290 triangles (5%) Fig. 21: Simlification results of the cow model using the ESO. (a) triangles (original model) (b) triangles (40%) (c) triangles (30%) E-ISS: Issue 8, Volume 12, August 2013

11 (d) triangles (20%) (e) 5076 triangles (10%) (f) 2538 triangles (5%) Fig. 22: Simlification results of the dragon model using the ESO. (a) (b) (c) (d) (e) (f) Fig. 23: Simlification results of the femur model using the ESO. (a) triangles (original model), (b) triangles, (c) triangles, (d) triangles, (e) triangles, (f) 7666 triangles. (a) (b) (c) (d) (e) (f) Fig. 24: Simlification results of the isis model using the ESO. (a) triangles (original model), (b) triangles, (c) triangles, (d) triangles, (e) triangles, (f) triangles. E-ISS: Issue 8, Volume 12, August 2013