An Adaptive Subdivision Method Based on Limit Surface Normal
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1 An Adaptive Subdivision Method Based on Limit Surface Normal Zhongxian Chen, Xiaonan Luo, Ruotian Ling Computer Application Institute Sun Yat-sen University, Guangzhou, China Abstract Subdivision surfaces have become a standard technique for free shape modeling. But subdivision schemes are costintensive at higher levels of subdivision, making adaptive methods indispensable for visualization and shape modeling. In this paper, we present an adaptive subdivision method based on limit surface normals. The subdivision surfaces generated by the proposed method approximate the limit surfaces better than those generated by traditional methods[1]. Furthermore, the proposed method can be easily applied to any triangular subdivision scheme with C 1 or higher degree of continuity. 1. Introduction Subdivision schemes have become a powerful tool for modeling and animation. For most popular subdivision schemes, the number of faces increases by a fixed factor in each step of subdivision, resulting in a heavy computational load at higher level of subdivision. In fact, for most surfaces in modern graphical applications, there are regions that become reasonably smooth after few steps of subdivision and only certain areas with higher curvature change need more subdivision steps to become smooth. Adaptive subdivision aims to provide a rule to judge whether a given face in the mesh should be subdivided in the next step of subdivision. In this paper, we introduce an adaptive subdivision method whose error criterion is based on the deviation angles between the normal of a face and the limit normals of the vertices in it. We use a method based on difference subdivision scheme[5] to calculate the limit normals, making the proposed method easily applicable to any triangular subdivision scheme with C 1 or higher degree of continuity. By calculating the symmetrical Hausdoff distance, we compare the proposed adaptive method with Armesh s[1] method and see that the proposed method generates surfaces that approximate the limit surface better. Furthermore, crack prevention and view-dependent adaptation are also introduced in this paper. 1.1 Previous Work Some methods have been developed for adaptive subdivision. Amresh et al.[1] introduce an adaptive subdivision method for Loop s subdivision scheme. In his method adaptive refinement is guided by the angle difference between the normal of a face and that of all its neighboring faces. But the surfaces generated by Amresh s method do not approximate the limit surface well enough. Meanwhile, Kobbelt[6] has developed an adaptive method for his 3 subdivision using a combination of dyadic refinement, mesh balancing and red-green triangulation. And, based on the measurement of planarity at corners and edge midpoints of the next subdivision level, Zorin et al.[13] developed an adaptive scheme with some additional constraints on the vertices calculated by his adaptive scheme. 1.2 Paper Organization In this paper, we only implement the proposed adaptive method on Loop s subdivision scheme[8], but it is straightforward and easy to apply this method to other subdivision schemes with at least C 1 continuity. In Section 2, we first have a brief review of Loop s subdivision scheme. Section 3 presents the proposed adaptive subdivision method in details. In Section 4, we will compare the proposed adaptive method with that of Amresh and give some implementation results. 2 Loop Subdivision Scheme Loop subdivision scheme is an approximating subdivision scheme for triangular meshes. The scheme is based on triangular splines and produces C 2 -continuity over regular meshes. Performing one subdivision step on a triangle will result in four sub-triangles. The vertices of these sub-faces can be classified into vertex and edge points of the parent
2 face. They are computed using the so-called vertex mask and edge mask as shown in Fig.1, where k is the valence of the central vertex and β can be chosen as: β = 1 k (5 8 ( cos(2π k ))2 ) (1) namely the control normal, is computed. Second, an angle α is specified to indicate the size of the cone. The computation of limit normals and the specification of α will be discussed with details in 3.2 and 3.3 respectively. Figure 1. Masks of Loop subdivision:(a) Vertex mask (b) Edge mask. 3 Adaptive Method In this section we describe the proposed adaptive subdivision method in details. We first present the error criterion of the method in 3.1. Then the calculation of limit normal is described in 3.2. Finally, view-dependent adaptation and crack prevention are discussed in 3.3 and 3.4, respectively. 3.1 Error Criterion For modern graphics applications, input meshes can consist of a large number of faces. If, at each step of subdivision, each face in the mesh is uniformly split into a fixed number of sub-faces, the number of faces increases exponentially and quickly exceeds the memory limitations. One way to solve this problem is to find out and subdivide the locations of the mesh which approximate the limit surface most unfavorably, while avoid making unnecessary refinement on other locations of the mesh. In this subsection, we discuss the error criterion of the proposed adaptive method, which aims to measure the approximation quality of a facet with respect to the limit shape and determine whether it should be subdivided. To measure whether a mesh adequately represents the limit surface s shape, the cone of normals[1] can be used(see Fig.2). The construction of the cone of normals proceeds in two steps. First, the limit normal of a vertex, Figure 2. The cone of normal over two subdivision steps. Here the superscripts denote the subdivision step. (a) Initial mesh (b) Mesh after one subdivision step (c) Mesh after two subdivision steps. In Fig.2, N v is the limit surface normal at vertex v and N j i is the normal of adjacent face f i after the j-th subdivision step. It can be seen that as the subdivision processes, the normals of the adjacent faces around v becomes closer and closer to the control normal, indicating that the location around v is approximating the limit shape better and better. Finally, the normals of adjacent faces are all contained by the cone of normals. Therefore, we can use the deviation angles between the normal of a face and the limit normals of the vertices in it to measure the approximation quality and determine the divisibility of a face. In other words, given a face f with vertices v 1, v 2, v 3, we will subdivide f in the next step of subdivision if min{ N V i N f N Vi N f } < cos(α), i = 1, 2, 3, (2) where N Vi is the limit normal of v i and N f is the normal of f. Otherwise the face will be left not subdivided. Here, α serves as a threshold and can be pre-specified by the user.
3 3.2 Calculation of Limit Normal As depicted by Kawaharada in his recent paper[5], the exact limit surface normal at a vertex can be efficiently calculated if the subdivision scheme yields surfaces with at least C 1 continuity. In this part, we introduce a new method to calculate the limit surface normals for Loop s subdivision scheme based on Kawaharada s theories. First, we generate the subdivision matrix S k of Loop s scheme using a vertex v of valence k, and its 1-ring neighborhood(see Fig.3). The subdivision matrix of the scheme is as follows: S k = 1 kβ β β β β β... β 3/8 3/8 1/8... 1/8 3/8 1/8 3/8 1/8... 3/8 1/8 3/8 1/ /8 1/8... 1/8 3/8 (3) Define p j as (v j, vj 1, vj 2,..., vj n) T and p j+1 as (v j + 1, v j 1 + 1, vj 2 + 1,..., vj n + 1) T, the subdivision scheme can be written as: Then we define a matrix as = p j+1 = S k p j (4) Using a matrix D k = S k 1, we get v j+1 v j v j+1 1 v j+1. = v j 1 D k vj. v j+1 k v j+1 v j k vj Let d j denote the vector consisting of the elements v j 1 v j,vj 2 vj,...,vj k vj. Each element of dj is called a first difference vector. Note that the sum of each row of S k is 1, because Loop subdivision scheme is affine invariance. Thus, the element v j does not affect elements vj 1 vj,vj 2 v j,...,vj k vj. So we can get a sub-matrix of D k as D k, such that d j+1 = D k d j. It can be seen that D k subdivides the first difference vectors, and thus D k is called the first difference subdivision matrix of Loop subdivision. Here, let d j x denote the column vector which is a set of x elements of d j. Similarly, let d j y, d j z denote the column (5) (6) Figure 3. V and its 1-ring neighborhood. Here j denote the j-th step of the subdivision. vectors corresponding to y elements, z elements. And we define a matrix ΛD k such that ΛD k (u 1 u 2 ) = D k u 1 D k u 2, (7) where is the wedge product and u 1, u 2 R k. Then, we have ΛD k (d j y d j z) = D k d j y D k d j z. (8) Now, we define N j = (d j y d j z, d j z d j x, d j x d j y). Then, N j+1 = ΛD k N j. (9) Note that a row of N j is a cross product of v j i vj and v j l vj, that is, a normal on the neighborhood of vj. We can see that ΛD k, namely the normal subdivision matrix of Loop subdivision, subdivides the normals of the faces formed by first difference vectors. Furthermore, by taking wedge product on each two rows of D k, ΛD k can be easily calculated. It has been proved in [5] that, if the subdivision scheme is C 1 continuous, all rows of N point to the same direction, that is the direction of the normal at v. To calculate the direction of the limit normal, we first decompose ΛD k into V 1 AV, where A is the Jordan normal form and V is a regular matrix. The Jordan decomposition here can be performed by mathematical tools such as Matlab. Let Λ i, i = 1, 2... be eigenvalues of ΛD k such that Λ i Λ i+1. Suppose Λ 1 lies in the a-th row of A, then the exact limit surface normal of v, namely N(v ), can be calculated by: N(v ) = k 1 i=1 j=i+1 k i 1 (d i d j) V 1, l = j i+ (k m) la m=1 (1) where is the cross product, d i = v i v, and V 1 la is the element lying in the l-th row of the column Va 1. Here, a
4 way to make the calculation more efficient is to pre-tabulate the elements of Va 1 for all valences in a two-dimension floating point array before calculation. Then only by changing the elements in the array, we can apply the method to other subdivision schemes. 3.3 View-dependent Adaptation The purpose of view-dependent adaptation is to assign higher accuracy to visually important surfaces while leave the regions with little visual influence to the viewer not subdivided, and finally reduce graphics load. Silhouettes and contours are particularly important visual cues for object recognition. Detecting faces along object silhouettes and allocating more detail to those faces can therefore disproportionately increase the perceived quality of an object[12]. On the other hand, subdividing the backfacing regions is no more than a waste of time and space. In the proposed adaptive method, we use the difference angle between the control normal of viewing cone and the normal of a face to measure its visual importance. As can be seen in Fig.4, a viewing cone that originates from the view point is created with an angel and a control normal namely N view. Given a face f, we denote its normal as N f and the difference angle between N view and N f as β, where β π. It can be seen that face f is backfacing only when β < π/2. If β = π/2, that is N f is orthogonal to N view, the face is potentially on the silhouette. As β increases from π/2 to π, the visual importance of the face gradually decreases. where f(β) is as follows: { π/α β < π f(β) = 2 π k cos(π β) 2 β π (12) Here, k is a dilatant factor. For most cases in our experiment, it yields good-looking results when k = 1. In general cases, when N view is in opposite direction to z-axis, f(β) can be simplified as f(z), { π/α z < f(z) = (13) k z z where z is the z-element of normalized N f. 3.4 Crack Prevention When two adjacent faces have different subdivision levels, a crack appears between them. The cracks in the mesh not only lead to irritating visual artifacts, but also bring serious trouble for mesh processing and limit normal computation. To avoid the cracks, some refinement should be made on the faces that are not subdivided. Based on the proposed error criterion, we can partition a given mesh into two sub-meshes, one consisting of divisible faces that will be subdivided in the next step of subdivision and the other consisting of indivisible faces that will be left not subdivided. We denote the mesh for indivisible faces by M ind. Based on the divisibility of their neighboring faces, the faces in M ind can be further classified into fours groups. The refinement for the faces in different groups is illustrated in Fig.5. Indivisible faces with three indivisible neighbors are left undivided as shown in Fig.5(a). The refinement for indivisible faces with one, two and three divisible neighboring faces is shown in Fig.5(b), (c) and (d) respectively. Note that indivisible faces with three divisible neighbors are subdivided as divisible faces. 4 Analysis of Results Figure 4. Viewing cone. Here N view is the control normal of the viewing cone and N f is the normal of a face. β is the difference angle between N view and N f. Therefore, in order to enhance perceived quality and avoid unnecessary subdivision, we tune Inequation(2) to be: min{ N V i N f } < cos(α f(β)), i = 1, 2, 3, (11) N Vi N f In this section, we first present the approximation analysis of the proposed adaptive method and compare it with Amresh s[1] adaptive method(because Amresh s method is based on Loop subdivision). And then some implementation results of the proposed method are given to demonstrate its visual quality. 4.1 Analysis of Approximation Quality In this part, we analyze the approximation quality of the proposed adaptive method by calculating the Symmetrical Hausdoff Distance(SHD)[2] between the limit surface which is generated by uniform Loop subdivision, and the surface generated by the adaptive method. To compare with
5 of the SHD of surfaces generated by the proposed method to that of the surfaces generated by Amresh s method. As shown by the results, the SHDs of the surfaces generated by the proposed method are smaller, indicating that the proposed adaptive method generates surfaces that approximate the limit surfaces well, even better than those generated by Amresh s adaptive method. Table 1. Approximation Quality of the proposed method and Amresh s method Figure 5. Refinement for indivisible faces. Amresh s adaptive method, in this part we do not take into account any view-dependent property that is mentioned in 3.3. Let us first define the distance d(p, S ) between a point belonging to a surface S and a surface S as: Models Surfaces of proposed method Surfaces of Amresh s method SHD Ratios Face# SHD(%) Face# SHD(%) (%) Bunny(a) 32, , Teddy(b) 113, , Shark(c) 1, , Cup(d) 34, , Radish(e) 4, , Cusp(f) 4, , d(p, S ) = min p S p p 2, (14) where. 2 denotes the Euclidean norm. From this definition, the Hausdoff distance between S and S, denoted by d(s, S ), is given by: d(s, S ) = max p S d(p, S ). (15) Since this distance is in general not symmetrical, i.e. d(s, S ) d(s, S), the computation of one-sided error can lead to significantly underestimated distance values. Thus we define symmetrical Hausdoff distance d s (S, S ) as follows: d s (S, S ) = max[d(s, S ), d(s, S)]. (16) Then, we can compare adaptive methods by calculating the symmetrical Hausdoff distances between their generated surfaces and the limit surface, on the premise that the numbers of the facets in their generated surfaces are nearly equal. A number of models have been used to test the proposed method and compare it with Amresh s method, some of the results are listed in Table 1. Here, the symmetrical Hausdoff distance is given as percentage of the diagonal of the limit surface s mesh bounding box. To directly compare these two adaptive methods, we calculate the ratio Figure 6. Initial models used in Table Implementation Results First, to demonstrate the error criterion and limit normal calculation, we present two examples without considering any view-dependent property. That is, we set f(β) 1 in Inequation 11 for these two examples. In Fig.7, the 142-facet mannequin head model is subdivided using uniform Loop subdivision and the proposed adaptive subdivision method, respectively. After three steps
6 of subdivision, the number of generated triangles are 988 and 3954, respectively. As can be seen, although the adaptive result consists of fewer triangles, its visual effect is also pleasing when compared with the uniform result. In Fig.8, using threshold π/18 and π/36 respectively, the 684-facet shoe model is adaptively subdivided for two steps, with 846 and 186 sub-facets generated. It indicates that the number of generated facets is significantly influenced by the magnitude of threshold. Figure 8. (a)initial mesh; (b and c)mesh generated after two steps of adaptive subdivision, with threshold π/18 and π/36, respectively. Figure 7. (left)mesh after three steps of uniform Loop subdivision; (right)mesh after three steps of the proposed adaptive subdivision, with threshold π/36. Now, we test the proposed adaptive method with viewdependent adaptation. In Fig.9, both the sphere and bunny are subdivided by uniform Loop subdivision and the proposed adaptive method respectively. The models on the left are the results of uniform Loop subdivision, and those in the middle and on the right are surfaces generated by the proposed adaptive method, in different views. As can be seen, more refinement is made near silhouettes and backfacing regions are not subdivided. For the sphere model, the uniform result consists of 576 facets, while the viewdependent adaptive result consists of only 188 facets. For the bunny model, the numbers of generated facets are 1328 and 6626, respectively. Acknowledgments The work described in this article is supported by the National Science Fund for Distinguished Young Scholars (No.: ) and the Key Project (No.: 65333) of the National Natural Science Foundation of China. References [1] A. Amresh, G. Farin, and A. Razdan. Adaptive subdivision schemes for triangle meshes. Arizona State University, October 2. [2] P. Cignoni, C. Rocchini, and R. Scopigno. Metro : measuring error on simplified surfaces. Computer Graphics Forum, 17(2): , June [3] H. Hoppe. Progressive meshes. In SIGGRAPH 96 Conference Proceedings, pages 99 18, August [4] H. Hoppe. View-dependent refinement of progressive meshes. In SIGGRAPH 96 Conference Proceedings, [5] H. Kawaharada. c k -continuity of stationary subdivision schemes. The University of Tokyo, January 26. [6] L. Kobbelt. 3 subdivision. Computer Graphics Proceedings, pages , 2. [7] L. Kobbelt, K. Daubert, and H. Seidel. Ray tracing of subdivision surfaces. In Rendering Techniques 98 (Proceedings of the Eurographics Workshop), pages 69 8, [8] C. Loop. Smooth subdivision surfaces based on triangles. Master s thesis, Utah, Universtity of Utah, Department of Mathematics, [9] H. Muller and R. Jaeschke. Adaptive subdivision curves and surfaces. Proceedings of the Conference on Computer Graphics International 1998 (CGI-98),, pages 48 58, June [1] L. Shirman and S. Abi-Ezzi. The cone of normals technique for fast processing of curved patches. Computer Graphics Forum (Proceedings of Eurographics 93), 12(3): , 1993.
7 [11] J. Stam. Evaluation of loop subdivision surfaces. In SIG- GRAPH 98 Conference Proceedings on CDROM, July [12] J. Xia and A. Varshney. Dynamic view-dependent simplification for polygonal models. Visualization, [13] D. Zorin, P. Schroder, and S. Sweldens. Interactive multiresolution mesh editing. SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pages , August Figure 9. (left)uniform Loop subdivision surfaces; (middle and right)view-dependent adaptive subdivision surfaces, in front view and side view respectively.
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