Simplification of Meshes into Curved PN Triangles

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1 Simplification of Meshes into Curved PN Triangles Xiaoqun Wu, Jianmin Zheng, Xunnian Yang, Yiyu Cai Nanyang Technological University, Singapore Abstract Complex and highly detailed models are very common in computer graphics due to the rapid development of data acquisition technology. However, such huge data in surface representation may cause inconvenience in applications such as data storage, transmission or real time rendering, etc. Therefore, simplification of complex surfaces into meshes with fewer data plays important roles in many applications. Different from many existing surface simplification methods which construct new base meshes with smaller number of flat triangles, we propose a new simplification method in this paper, simplifying an original mesh into a new base mesh with curved point normal (PN) triangles and constructing detailed mesh automatically from the base mesh by tessellation. To obtain the base mesh, a new error metric for curved PN triangles is given so that the simplified mesh with curved PN triangles can approximate the original mesh with lower errors than surface simplification with flat triangles. We also develop a new feature sensitive metric to simplify meshes with better feature preservation. I. INTRODUCTION Due to the development of 3D model scanning technologies, geometric models could be very complex and highly detailed to maintain realism in 3D computer graphics and video games. However, surfaces represented by huge data meshes are notoriously difficult for hardware and software to store, render and transmit. Especially in interactive computer graphics, it is a big challenge to render or manipulate such complex models at interactive rates. Representing a surface mesh with small number of base triangles or representing a complex geometric surface with levels of details can help to solve above mentioned problems. When a surface have been decomposed into a simplified base model M 0 together with a sequence of refinements or details M i, one can then load surfaces with various levels of details (LOD) for high speed rendering or local close view of a scene. The decomposition of an original mesh into base mesh and details is closely related to model simplification. The simplification techniques construct coarser models which approximate the original surfaces as close as possible from high detailed models with less vertices, edges and faces. Sometimes, important features within the original surfaces should also be preserved along with surface simplification. Considerable number of algorithms have already been developed on simplifying those high resolution models [1-6] or representing meshes with level of details [7-13]. Most of existing simplification algorithms in this area focused on the simplification of polygonal models, such as triangular meshes or quad meshes, into new polygonal meshes. Reference [14,15] extended the simplification idea to subdivision surfaces. Subdivision surface simplification try to approximate an original dense triangular mesh or a subdivision surface with dense control mesh by another subdivision surface with simpler control mesh. But subdivision surfaces need information of the neighborhood of each vertex. It is not explicit to operate on the control mesh or to preserve surface features by modifying the control mesh of a subdivision surface, either. In this paper, we replace every flat triangle of a triangular mesh by a curved PN triangle [16] and simplify the original surface mesh into a new surface consisting of curved PN triangles. To construct the base mesh with curved PN triangles, we introduce a modified quadric error metric (QEM) [2] that measures distances between curved PN triangles surface patches. We also develop a feature sensitive metric which can be used to detect surface features. By a feature sensitive error metric, important surface features can be preserved well during surface simplification. When the base mesh of a simplified surface is obtained, curved PN triangles interpolating base triangles can be constructed easily. If a surface mesh with smooth silhouette or appearance is needed, the curved PN triangles can be tessellated into small flat triangles efficiently. Compared with the techniques of approximating dense meshes by subdivision surfaces or representing of an original mesh by a progressive mesh, curved PN triangles can be constructed very easily and be used to represent surfaces with smooth regions with salient features as well. In section 2, we simply introduce the concept of curved PN triangles and QEM-based mesh simplification algorithm. In Section 3, a modified quadric error metric for curved PN triangles and a new feature sensitive error metric will be developed. Section 4 gives some experimental results and in Section 5 we conclude the paper and discuss some future works as well. II. CURVED PN TRIANGLES AND QEM-BASED MESH A. Curved PN Triangles SIMPLIFICATION Curved PN triangles were firstly introduced to improve the visual quality of polygonal surfaces by smoothing out silhouette edges and providing more sample points for vertex shading operation [16]. A curved PN triangle is derived from a flat triangle equipped with unit normals at three vertices. The geometry and normals of a curved PN triangle are defined separatively from the given vertices and normal vectors. The geometry of a curved PN triangle is defined by a cubic Bézier patch b(u, v) that interpolates end vertices and end normals of APSIPA. All rights reserved. Proceedings of the Second APSIPA Annual Summit and Conference, pages , Biopolis, Singapore, December 2010.

2 the given triangle. b : R 2 R 3,u+v +w = 1,u,v,w 0 3! b(u,v) = b kij k!i!j! wk u i v j i+j+k=3 = b 300 w 3 +b 030 u 3 +b 003 v 3 +b 210 3w 2 u+b 120 3wu 2 +b 201 3w 2 v +b 021 3u 2 v +b 102 3wv 2 +b 012 3uv 2 +b 111 6wuv. where the coefficients n kij are given as follows: n 200 = n 1, n 020 = n 2, n 002 = n 3, w ij = 2 (v j v i ) (n i +n j ) (v j v i ) (v j v i ) R, n 110 = n 011 = n 101 = h 110 h 110,h 110 = n 1 +n 2 w 12 (v 2 v 1 ), h 011 h 011,h 011 = n 2 +n 3 w 23 (v 3 v 2 ), h 101 h 101,h 101 = n 3 +n 1 w 31 (v 1 v 3 ). Fig. 1. The control net of a triangular Bézier patch Coefficients b kij are called the control points and connected to form a control net (see Fig. 1). Given three vertices v 1,v 2,v 3 and the corresponding unit normal vectorsn 1,n 2,n 3 of a triangle, the coefficients are defined as follows: b 300 = v 1, b 030 = v 2, b 003 = v 3, w ij = (v j v i ) n i R, b 210 = (2v 1 +v 2 w 12 n 1 )/3, b 120 = (2v 2 +v 1 w 21 n 2 )/3, b 021 = (2v 2 +v 3 w 23 n 2 )/3, b 012 = (2v 3 +v 2 w 32 n 3 )/3, b 102 = (2v 3 +v 1 w 31 n 3 )/3, b 201 = (2v 1 +v 3 w 13 n 1 )/3, E = (b 210 +b 120 +b 012 +b 021 +b 102 +b 201 )/6 V = (v 1 +v 2 +v 3 )/3, b 111 = E (E V)/2. The normal component of a curved PN triangle is defined by a quadratic function of position and normal data. The quadratic function n is defined as follows [16] (see Fig. 2): n : R 2 R 3,u+v +w = 1,u,v,w 0 n(u,v) = n kij w k u i v j, i+j+k=2 = n 200 w 2 +n 020 u 2 +n 002 v 2 +n 110 wu+n 011 uv +n 101 wv, Fig. 2. The coefficients of the normal component of a curved PN triangle Fig. 3. Tessellation: (a) level 0, (b) level 1, (c) level 2, (d) level 4, (e) level 5, (f) level 6 Curved PN triangles smooth out contours with cubic Bézier patches and the normal components. It is shown in reference [16] that each cubic boundary curve stays close to its corresponding edge due to the special setting of Bézier coefficients. Therefore the PN triangle does not deviate too much from the original triangle. In order to generate smooth visual effects, the curved PN triangle needs to be tessellated into smaller triangles (see Fig. 3). The vertices and normals of the tessellated triangles can be computed from the interpolating surface or normal function by de Casteljau algorithm [17]. 599

3 Note that curved PN triangles are usually generated and tessellated in the rendering period between the vertex-andprimitive-assembly stage and the vertex-shading stage [16]. The input and output of the original triangular meshes and the curved PN triangles are of the same format and thus the change on the design on adjacent pipeline stages is minimal. B. QEM-based Mesh Simplification The QEM-based mesh simplification contracts a mesh edge into a point by minimizing the sum of distance squares from the point to the planes through either vertices of the edge [18]. Two triangles, three edges plus one vertex will be removed after one time of edge contraction. The square distance of a vertex v = [x y z 1] T to a plane P can be represented in the quadratic form d(v) = v T Qv, where the plane is defined by equation ax + by + cz + d = 0, a 2 + b 2 + c 2 = 1 and the matrix Q is obtained as Q = ab ac ad ab b 2 bc bd ac bc c 2 cd ad bd cd d 2 a 2 The matrixqis called fundamental error quadric. So the error from a vertex v to a set of adjacent planes which are denoted by adj(v) is obtained as d(v) = v T Q t v t adj(v) = v T t adj(v) = v T Q v v. Q t v The matrix Q v = t adj(v) Q t is then used as the fundamental error quadric at vertex v. When the error quadric for every vertex has been obtained, the error quadric for arbitrary edge e = (v 1,v 2 ) is defined as Q e = Q v1 + Q v2. In order to perform the contraction (v 1, v 2 ) v, one can choose the new position v by the minimizing the squared distance d( v) = v T Q e v. Since v can be determined by solving a standard least square problem, it only needs to solve the following linear equation system: d( v) x = d( v) = d( v) = 0. y z The system can be reformulated as a 2 ab ac ad ab b 2 bc bd ac bc c 2 cd v = and the coordinates of v are obtained as x 1 ȳ = a2 ab ac ab b 2 bc ad bd z ac bc c 2 cd The quadric error metric d( v) is the cost for contracting the edge (v 1,v 2 ) into point v. Arrange the costs in an increasing Fig. 4. Sub-triangulate a PN patch once or decreasing order after the costs for all valid edges of a mesh have been computed. The edge with minimum cost is chosen and replaced by a point. After contraction of an edge into a new vertex, the costs for all updated edges are recomputed and the costs are rearranged for next round edge contraction. This process can be implemented iteratively until a prescribed number of vertices or triangles have been left. III. SIMPLIFYING MESHES INTO CURVED PN TRIANGLES In this section we propose a new error metric that measures distances between surface meshes consisting of curved PN triangles, and a new mesh simplification algorithm based on this new error metric will be proposed. We also present a feature sensitive metric which can be used to preserve surface features well during mesh simplification. A. Preprocess for Curved PN Triangles Simplification The original QEM in [18] is defined on surface meshes consisting of flat triangles and a given triangular mesh is simplified into another mesh consisting of fewer triangles. When we simplify a surface mesh consisting of curved PN triangles, the original distance metric should be modified too. The new distance metric does not represent the distances to the original triangular mesh, but to the piecewise curved PN triangles that interpolate vertices and normals of the original triangular mesh. In order to estimate the distances to any curved PN triangle efficiently, we tessellate the geometry of a curved PN triangle into smaller flat triangles (see Fig. 4). The boundary curves of a curved PN patch are cubic Bézier curves. The midpoints of boundary curves can be evaluated by de Casteljau algorithm [17] as follows v 12 = 1 8 (v 1 +v 2 )+ 3 8 (b 210 +b 120 ) v 13 = 1 8 (v 1 +v 3 )+ 3 8 (b 201 +b 102 ) v 23 = 1 8 (v 2 +v 3 )+ 3 8 (b 012 +b 021 ) When the midpoints of boundary curves are obtained, a curved PN triangle will be tessellated into four sub-triangles. The distances to the curved PN triangle will be estimated as the distances to the sub-triangles approximately. As a curved PN 600

4 parallel to (n v1 + n v2 ). Then we can represent ū in matrix form: ū i = M i v, where M i = 1 2 M i1 1 8 M i M i3. Fig. 5. Edge collapse of curved PN triangles triangle always approximates its base triangle compactly, the approximate distances are accurate enough. Similar to QEM, the initial error quadric Q vi t for vertex v i to the curved PN triangle is defined as a quadric that measures the square distance from vertex v i to the corner triangle A i (see Fig. 4). The error quadric from vertex v i to a set of joining curved PN triangles is obtained as Q vi = t adj{v i} Qvi t. B. Error Metric Based on Curved PN Triangles Assume e = (v 1,v 2 ) is an arbitrary edge of the original mesh, it is collapsed into the new vertex v. Let u ij s be the midpoints of boundary curves of original curved PN triangles, ū i s be the midpoints of boundary curves of simplified curved PN triangles (see Fig. 5). The error between the simplified curved PN patches and original ones consists of three parts: The square distances from the new vertex v to the planes of old tessellated triangles adjacent to v 1 or v 2. The square distances from midpoints ū i s to the old subtriangles adjacent to u ij. The square distances from new vertex v to v 1 and v 2. We let Qv1, Qv2 be the quadrics of vertices v 1 and v 2 computed in last subsection. Similar to QEM, the square distance from v to the tessellated old triangles is computed by Q v = Q v1 + Q v2 (1) d v = v T Qv v (2) Just like mid-edge vertices u ij s which have close relationship with vertices v i s, the position of the new vertex v also influences the positions of mid-edge vertices ū i of new neighboring curved PN triangles. By the definition of PN triangle, we can calculate these mid-edge vertices by de Casteljau algorithm. u ij = 1 2 (p i +v j ) 1 8 [((p i v j ) n vj )n vj +((v j p i ) n pi )n pi ] ū i = 1 2 (p i +v) 1 8 [((p i v) n v )n v +((v p i ) n pi )n pi ] Let the homogeneous coordinates of the mentioned vertices and normals be v = (x y z 1) T, p i = (x i y i z i 1) T,n pi = (a 1 a 2 a 3 1) T,n p = (b 1 b 2 b 3 1) T. Note that n p is also Let d 1 = (n pi p i ), d 2 = (n p p i ), the matrices M i1, M i2 and M i3 are given as follows x i M i1 = y i z i M i2 = M i3 = a 1 2 a 1 a 2 a 1 a 3 a 1 d 1 a 1 a 2 a 2 2 a 2 a 3 a 2 d 1 a 1 a 3 a 2 a 3 a 3 a 3 a 3 d b 2 1 b 1 b 2 b 1 b 3 b 1 d 2 b 1 b 2 b 2 2 b 2 b 3 b 2 d 2 b 1 b 3 b 2 b 3 b 2 3 b 3 d Now, the distances from ū i to the old subtriangles adjacent to u ij are defined by D i = ū T i Q t ū i = = v T ( t adj(u ij) t adj(u ij) = v T Q w i v t adj(u ij) v T M T i Q tm i v M T i Q tm i )v Here, Q t means the fundamental error quadric of subtriangle t adj(u ij ). Let Qw = i Qw i, the total square distances from mid-edge points ū i s relating to new vertex v to old subtriangles are obtained as d w = v T Qw v. (3) The position of new vertex v is also constrained by the distances to edge ends d e = v v v v 2 2. (4) Equation (4) can also be easily represented into a quadratic form d e = v T Qe v. (5) Finally, the error cost of new vertex v of the simplified curved PN triangles is defined as d = λ 1 dv +λ 2 dw +λ 3 de, λ 1 +λ 2 +λ 3 = 1. Here,λ 1,λ 2,λ 3 are a set of positive weights. The error quadric of the edge (v 1,v 2 ) is Q = λ 1 Qv +λ 2 Qw +λ 3 Qe (6) and the error cost can also be represented as d = v T Qv. In our experiments, we choose λ 1 = 0.5, λ 2 = λ 3 = 0.25, which can give good results in practice. 601

5 Fig. 6. Feature metrics computed on a surface mesh C. Feature Sensitive Error Metric Surface features are important geometric entities for shape representation, so it is desirable to preserve surface features as much as possible when an original surface mesh is simplified. We propose a metric measuring features by a function in terms of mean curvature normals. The mean curvature of each vertex of a mesh is computed by the method in [19]. The feature metric for a vertex v i is defined as: 1 F(v i ) = k i n vi k j n vj 2 (7) adj(v i ) j adj(v i) Here k i,k j are the mean curvatures of vertices v i and v j, respectively, n vi, n vj are the unit normals of vertices v i, v j, and adj(v i ) is the neighborhood of vertex v i. In Fig. 6 we plot the feature metric for a surface mesh, where vertices with red color have high feature metric values while blue color means lower feature metric values. From this figure we can also see that important surface features have been properly identified by our proposed feature metric. When feature metrics for all vertices are obtained, we assign a feature weight to each vertex for feature preservation in mesh simplification. Assume α be the number greater than feature metrics F(v i )s of 70% vertices of a mesh, we define w vi = { lf(vi) α if F(v i ) > α 1 else Here, we assume l > 1. In all our examples, we set l = 7. An arbitrary edge e = (v 1, v 2 ) of the original mesh, the weighted error that results from collapsing the edge into a new vertex v is defined as e(v) = (w v1 +w v2 )v T Qv. D. The Simplification Algorithm Based on the proposed weighted error metric for surface meshes of curved PN triangles, any triangular mesh can be simplified in the following main steps: 1) Subdivide the input mesh once, construct curved PN triangles for every triangle of the input mesh. 2) Compute error quadric Q vi for every vertex v i of the mesh. 3) Compute error quadric Q for every edge by Equation (6). (8) Fig. 7. The base meshes generated by our feature preserving metric (b), (c) and by QEM (e), (f) 4) Compute mean curvature normal for every vertex and feature sensitive error metric for each edge. 5) Compute candidate collapsed vertex for every edge and built the priority queue by the weighted edge error metrics e(v)s. 6) Update the priority of all the edges related to v 1 and v 2 after the collapse of an edge on the top of the queue. We note that when an edge is contracted to a vertex, vertex number will be reduced by one and triangle number will be reduced by two. The edges connecting to either ends of original edge are reconnected to the new vertex. Then weighted error quadric relating to all refined vertices and edges and the priority queue of edge error metrics will be updated. The edge collapse operation can be implemented recursively, until a fixed number of vertices or triangles are reached. As an optional postprocess step, an obtained simplified mesh with curved PN triangles can be tessellated into meshes with smaller flat triangles. The tessellation is necessary only when smooth silhouettes or smooth appearance of a simplified mesh is desired. The tessellation of curved PN triangles can be implemented by sampling surface points and vertex normals from the interpolating surface or quadratically interpolating normal function directly. IV. EXPERIMENTAL RESULTS We have tested our proposed algorithm with many data sets and we present a few interesting examples here. Comparisons 602

6 (a) Model 66128f (b) QEM 3000f (c) Ours 3000f Fig. 8. Error Comparison with the simplified results by traditional QEM based mesh simplification method will also be given. In Fig. 7 we illustrate the simplified surfaces by our feature sensitive error metric for curved PN triangles. Fig. 7 (a) is the original model with triangles. Fig. 7 (b) and (c) are the coarser models with 5000 faces and 1000 faces, respectively, by our proposed simplification algorithm. Fig. 7 (e) and (f) are the coarser models with respective 5000 faces and 1000 faces by QEM based mesh simplification method. Fig. 7 (d) shows the zoomed details originally in the red boxes of Fig. 7 (c) and (e). From this figure we can see that our proposed method can preserve important surface features like the eyes and mouth of the model much better than the QEM based mesh simplification method. We have also refined the simplified meshes in Fig. 7 (c) and (f) by tessellating the curved PN triangles that interpolate all simplified base triangles. After that, we measure the error between the refined meshes with the original given dense mesh. The error between triangular meshes are computed by the method given in [20]. From Fig. 8 we can see clearly that the simplified mesh with curved PN triangles by our proposed new algorithm has lower error than by the QEM based mesh simplification method. In Fig. 9 we present some examples for mesh simplification by our proposed algorithm and the traditional QEM based mesh simplification method. As seen in Fig. 9 (a), (d), (g) are the original models. Fig. 9 (b), (e), (h) are the simplified meshes by the QEM-based mesh simplification algorithm. Fig. 9 (c), (f), (h) are the simplified base meshes by our proposed surface simplification method. This figure demonstrates that salient surface features like mouth and nose on the original surface can be preserved better by our new proposed method. When a base mesh M 0 which preserves features has been obtained, a sequence of refinements M i can be generated by tessellating the PN triangles that interpolate base triangles at different levels. Fig. 10 (d) is an original model with faces. Fig. 10 (a) is the base mesh with 4000 faces generated by our proposed method. Fig. 10 (b) and (c) are the refined meshes by tessellating PN triangles once and three times using the base meshes in Fig. 10 (a) respectively. We can see that (c) is quite similar to original mesh (d). We can use (c) as (d) Model391168f (e) QEM 5000f (f) Ours 5000f (g) Model 25445f (h) QEM 2000f (i) Ours 2000f Fig. 9. Some examples: (a), (d), (g) are the original mesh, (b), (e), (h) are simplified by QEM, (c), (f), (i) are simplified by our method an approximation of (d), but (c) only needs to store the base mesh (a) with 4000 faces and get the refinements (b) and (c) automatically from the base mesh. V. CONCLUSIONS AND FUTURE WORKS We present a new method for surface mesh simplification based on the representation of meshes by curved PN triangles. Curved PN triangles are defined naturally for any input triangular mesh and final simplified base meshes. A new error metric that measures distances between simplified curved PN triangles and original curved PN triangles has been proposed. Important surface features can also be preserved well by a feature weighted error metric for surface simplification. Compared with traditional QEM based mesh simplification algorithm, our new proposed method can achieve lower errors with the same number of base triangles in the simplified surfaces. Surface meshes represented by curved PN triangles can also be transferred to triangular meshes with smooth silhouette or more details by surface tessellation. So, our proposed algorithm can be used in fields like data reduction or real time rendering in computer games. 603

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