Ternary Butterfly Subdivision

Transcription

1 Ternary Butterfly Subdivision Ruotian Ling a,b Xiaonan Luo b Zhongxian Chen b,c a Department of Computer Science, The University of Hong Kong b Computer Application Institute, Sun Yat-sen University c Department of Computer Science, Rensselaer Polytechnic Institute Abstract This paper presents an interpolating ternary butterfly subdivision scheme for triangular meshes based on a 1-9 splitting operator. The regular rules are derived from a C 2 interpolating subdivision curve, and the irregular rules are established through the Fourier analysis of the regular case. By analyzing the eigen-structures and characteristic maps, we show that the subdivision surfaces generated by this scheme is C 1 continuous up to valence 100. In addition, the curvature of regular region is bounded. Finally we demonstrate the visual quality of our subdivision scheme with several examples. Key words: Subdivision surfaces, interpolating, triangular mesh, 1-9 splitting 1 Introduction Subdivision surfaces are valued in geometric modeling applications for their flexibility. Since 1978 [1, 5], subdivision has been an active research area. Advances in computer memory have made subdivision methods practical in the late 1990s and this has prompted a large amount of new research work [2, 9, 16, 25]. The flexibility primarily comes from the fact that objects to be subdivided can be of arbitrary topology and thus can be represented in a form that makes them easy for designing, rendering and manipulating. Simply speaking, a subdivision surface is defined as the limit of a sequence of meshes. Each mesh in the sequence is generated from its predecessor using a group of topological and geometric rules. Topological rules are used to produce a finer mesh from a coarse one while geometric rules are designed to compute Corresponding Author: rtling@cs.hku.hk (Ruotian Ling) Preprint submitted to Elsevier 20 April 2009

2 the positions of vertices in the new mesh. These two groups of rules constitute a subdivision scheme. Subdivision can be distinguished into two classes: interpolating schemes [7, 28] and approximating schemes [1, 5, 17, 22]. If the old vertices are changed during refinement, the subdivision algorithm is considered to be approximating, otherwise it is interpolating. Although approximating algorithms yield limit surfaces with higher continuity, interpolating algorithms enjoy some obvious advantages which the approximating ones do not have: interpolating schemes are more efficient for the applications requiring interpolating specified vertices. Furthermore it is easy to generate multi-resolution surfaces by using interpolating schemes [26]. For a long time, interpolating schemes failed to generate surfaces with higher continuity. Although a 6-point interpolating scheme for a curve with C 2 continuity has been proposed in [24], it is not practical to extend this scheme to a surface because too many vertices would be included in a mask. Hassan et al. reported an interpolating ternary subdivision scheme for curves which achieves C 2 continuity [8]. The most desirable property is that only 4 points are needed to generate a new vertex. Starting from this curve case, Dodgson et al. [4] and Li et al. [13] designed interpolating schemes for triangular meshes and quadrilateral meshes respectively, but Li et al. s scheme goes further in constructing surfaces with extraordinary vertices. It is well known that rules of a subdivision scheme for irregular meshes are particularly important if the scheme is to be used in practical applications [14, 28], but unfortunately, irregular rules are not investigated in Dodgson et al. s work, making it difficult to refine a coarse mesh with irregular vertices, which is often encountered in practice. In this paper, we first slightly modify Dodgson et al. s scheme, and then extend the regular rules to meshes with arbitrary topology through Fourier analysis. Based on the eigen-structures and characteristic maps, we show the C 1 continuity of subdivision surfaces for both regular and irregular regions up to valence 100. Due to the new 1-9 splitting operator, the face number increases by power of 9 in each step of the proposed subdivision. For this reason, we also explore the ability of adaptive subdivision. Finally some examples are presented to show the visual quality of the proposed subdivision scheme. 2 Related Work In 2002, Hassan et al. introduced interpolating ternary subdivision curves [8] shown in Fig. 1. The advantage of this new scheme is that it yields C 2 continuous limit curves. This subdivision scheme inserts two E-vertices into 2

3 each edge of the given control polygon respectively at 1/3 and 2/3 parametric positions. A newly inserted vertex of the interpolating ternary subdivision scheme is computed as follows: q 1 = a 0 p k i 1 + a 1p k i + a 2p k i+1 + a 3p k i+2 (1) where a 0 = µ a 1 = µ a 2 = µ a 3 = µ, (2) with free parameter µ. The mask for q 2 is symmetric to that of q 1. When 1 9 µ 1 15, the scheme generates C2 limit curves [6]. Fig. 1. Interpolating ternary subdivision curve: mask for newly inserted vertex q 1 Motivated by this subdivision curve, two subdivision schemes for surfaces have been proposed [4, 13] consequently. Both of their ideas are to generalize the curve setting to surface configurations such that the regular rules can be derived by solving a system of equations. Although the irregular rules of [4] have been first investigated in [15], their result does not guarantee that the limit surfaces near extraordinary vertices are C 1 continuous. Furthermore, the eigenvalue analysis of irregular subdivision matrices in [15] takes only the 1- neighborhood of an extraordinary vertex into consideration. But according to the rules outlined in that paper, the smallest similar stencil should be 2- neighborhood, so that the eigenvalues provided in that paper cannot fully demonstrate the property of limit surfaces. 3 Regular Subdivision Masks Our ternary subdivision introduces two types of new vertices: E-vertices, parametrically on the mesh edges, and F-vertices, parametrically at the face center. The regular rules of our subdivision scheme are similar to [4] except for a small modification. The regular masks for face vertex (F-vertex) and edge vertex (E-vertex) are presented in Fig. 2 and Fig. 3 respectively. 3

4 Fig. 2. Regular subdivision mask for a F-vertex Q F (the black dot). Fig. 3. Regular subdivision mask for an E-vertex Q E (the black dot), in which we suggest ǫ and ν be zero. As mentioned above, the rules of Dodgson et al. s scheme are derived from the ternary subdivision curve. The underlying surface masks reduce to the curve masks when the given control mesh collapses to a polyline along one of the three directions of the triangular mesh, so the weights in the masks must satisfy the following constraints: κ + 2θ = a 0 δ + ǫ = a 0 2η + 2θ = a 1 γ + α + δ = a 1,. (3) η + 2κ = a 2 ξ + β + γ = a 2 2θ = a 3 ν + ξ = a 3 By applying the conditions in Eq. (3), the mask with free parameters µ, ν, ǫ can be obtained. ξ = 1 ( 2 + 6µ) ν 36 θ = 1 ( 1 + 3µ) δ = 1 ( 2 6µ) ǫ κ = 1 36 ( 12µ) η = 1 (14 + 6µ) 36, γ = 1 36 (4) + ǫ + ν β = 1 (12 24µ) ǫ 36 α = 1 ( µ) ν 36. (4) In order to simplify the analysis, we suggest that two weights, ǫ and ν, be zero. Then the shape of the E-mask is reduced to Dyn et al. s butterfly scheme [7]. 4

5 By analyzing the subdivision matrix with ǫ = ν = 0, we can compute the eigenvalues by Mathematica 5.0 or other mathematic tools: 1, 1 3, 1 3, 1 9, 1 9, 1 9, µ, 1 5µ 6, 1 3µ, 1 9µ, 1 9µ µ 12, 1 5µ 6, 1 9µ 18, 1 5µ 6, 1 3µ 36, 1 3µ, 9, 1 3µ, If µ is set to be 1, the seventh eigenvalue is minimized and the 7th, 8th, 9th 11 and 10th eigenvalues are then equal to 1. The following eigenvalues are all 11 less than 1 1. For our proposed regular masks, we let µ be in this paper Masks Near Extraordinary Vertices 4.1 Decomposition of regular masks In order to simplify the derivation of irregular masks, we investigate the decomposition of the regular masks first. Fig. 4 and Fig. 5 show the process of decompositions. Three 1-neighborhood smaller masks are produced from the regular F-mask while two smaller masks are generated from the E-mask. The newly inserted vertices in the smaller masks should satisfy the following constraints because the smaller masks are decomposed from the regular ones. Q F = 1 3 (qf 1 + qf 2 + qf 3 ) (5) Q E = 1 2 (qe 1 + qe 2 ) So the relation between the weights in 1-neighborhood masks and those in the regular masks is obtained: χ 1 + σ 2 = 2α σ 1 + χ 2 = 2β τ = 3θ φ 1 + φ 2 = 2γ 2ζ = 3κ, ψ 1 = 2δ. (6) ρ + 2υ = 3η ψ 2 = 2ξ ι 1 = 2ǫ ι 2 = 2ν 5

6 Fig. 4. Decomposition of the regular F-mask, where q1,2,3 F (the black dots) are the F-vertices in 1-neighborhood masks. Fig. 5. Decomposition of the regular E-mask, where q1,2 E E-vertices in 1-neighborhood masks. (the black dots) are the 4.2 Extensions of irregular 1-neighborhood masks Now we can formulate the matrix representation of these 1-neighborhood masks. Let us consider the 1-neighborhood centered at the vertex p 0 of valence 6. The adjacent vertices of p 0 can be partitioned into 6 blocks, B i = (p i0, p i1 ), (i = 1, 2, 3, 4, 5, 6), (7) and B = (B 1, B 2, B 3, B 4, B 5, B 6 ) T. (8) Note that all p i0 (i = 1, 2, 3, 4, 5, 6) denote vertex p 0. Such a trick helps to generate a succinct circulant representation of subdivision matrix. The labeling system is illustrated in Fig. 6. After one round of the ternary butterfly subdivision, a 2-neighborhood, which completely depends on its 1-neighborhood predecessor, is created around p 0. The new configuration of vertices can be 6

7 partitioned into 6 blocks also: B i = ( p i0, p i1, p i2, p i3 ), (i = 1, 2, 3, 4, 5, 6), (9) and B = ( B 1, B 2, B 3, B 4, B 5, B 6 ) T. (10) Fig neighborhood stencil for newly inserted vertices in the 2-neighborhood and an illustration of the labeling system. The relationship between B i and B i can be depicted by a subdivision matrix A. B = AB, (11) where A is a circulant matrix, A 1 A 2 A 3 A 4 A 5 A 6 A 6 A 1 A 2 A 3 A 4 A 5 A 5 A 6 A 1 A 2 A 3 A 4 A =, (12) A 4 A 5 A 6 A 1 A 2 A 3 A 3 A 4 A 5 A 6 A 1 A 2 A 2 A 3 A 4 A 5 A 6 A 1 and 7

8 1/6 0 1/6 0 1/6 0 χ 1 /6 σ 1 χ 1 /6 φ 1 χ 1 /6 ψ 1 A 1 = χ 2 /6 σ 2, A 2 = χ 2 /6 φ 2, A 3 = χ 2 /6 ψ 2, ρ/6 υ ρ/6 υ ρ/6 ζ 1/6 0 1/6 0 1/6 0 χ 1 /6 ι 1 χ 1 /6 ψ 1 χ 1 /6 φ 1 A 4 = χ 2 /6 ι 2, A 5 = χ 2 /6 ψ 2, A 6 = χ 2 /6 φ 2. ρ/6 τ ρ/6 τ ρ/6 ζ (13) And then we can apply the Fourier transform [13, 14] to the sequence A 1, A 2, A 3, A 4, A 5, A 6, where A j is a matrix of 4 rows. For the first 3 rows of A j, the transform follows the traditional fashion: Â k = 1 j 6 A j ω (j 1)(k 1), (14) where ω = e 2π 6 i, k = 1, 2,..., 6. To avoid complex numbers in the Fourier coefficients and to keep the symmetric structure of the weights in masks, we slightly modify to the Fourier transform in the fourth row: Â k = ω 1 k 2 1 j 6 A j ω (j 1)(k 1), (15) where ω = e 2π 6 i, k = 1, 2,..., 6. The six Fourier coefficients obtained are: Â 1, Â2, Â3, Â4, Â5, Â6. (16) Based on these coefficients, now we can extend the 1-neighborhood of valence 6 to valence n. A circulant matrix is uniquely determined by the Fourier coefficients of its element sequence. In order to guarantee that the subdivision matrix A (n) of a mask with a central vertex of valence n is an extension of the matrix A, we only need to assume A (n) = Circulant(A (n) 1, A (n) 2,..., A (n) n ), (17) where Â(n) 1, Â(n) 2,..., Â(n) n are the Fourier coefficients. Now the problem of the reconstruction of the irregular masks is converted to the computation of Â(n) k. We cannot utilize the Fourier coefficients in Eq. (16) directly because of the modification in Eq. (15). We assume that Ã(n) k, k = 1, 2,..., n, is a 2 4 matrix 8

9 sequence such that the sequence satisfies (a) if n = 3, à (3) 1 = Â1, Ã(3) 2 = Â2, Ã(3) 3 = Â6. (18) (b) if n = 4, à (4) 1 = Â1, Ã(4) 2 = Â2, Ã(4) 3 = Â4, Ã(4) 4 = Â6. (19) (c) if n 5, à (n) 1 = Â1, Ã(n) 2 = Â2, Ã(n) 3 = Â3, à (n) k = Â4, (k = 4,..., n 2), à (n) n 1 = Â5, Ã(n) n = Â6. (20) Let denote the first 3 rows of matrix Ã(n) k row of Ã(n) k, so à (n) k = R 4 and let R 4 represent the fourth. (21) Now the Fourier coefficients can be derived from Ã(n) k, where  (n) k = R 4 ω k 1 2. (22) The Fourier coefficients Â(n) k of subdivision matrix A (n) can be obtained by Eqs. (18-22). According to the inverse Fourier transform A j = 1 n 1 k n  k ω (k 1)(j 1), (23) where ω = e 2π n i, j = 1, 2..., n. From the subdivision matrix A (n), we get the weights in Fig. 7. More details of the Fourier transform can be found in Appendix A. For a mask with a central vertex of valence n, its weights are: (a) if n = 3, 9

10 w0 F = ρ w1 F = 1(2(υ + ζ + τ) + 2 3(υ τ) cos π) 3 3 w2,3 F = 1 (2(υ + ζ + τ) + 2 3(υ τ) cos 2πk 3π ) 3 3 w0 E1 = χ 1 w1 E1 = 1 3 (3σ1 + 4φ 1 ι 1 ) (24) w2,3 E1 = 1 3 ((σ1 + 2φ 1 + 2ψ 1 + ι 1 ) + 2(σ 1 + φ 1 ψ 1 ι 1 ) cos 2π(k 1) ) 3 w0 E2 = χ 2 w1 E2 = 1 3 (3σ2 + 4φ 2 ι 2 ) w2,3 E2 = 1 3 ((σ2 + 2φ 2 + 2ψ 2 + ι 2 ) + 2(σ 2 + φ 2 ψ 2 ι 2 ) cos 2π(k 1) ) 3 (b) if n = 4, w0 F = ρ w1 F = 1(2(υ + ζ + τ) + 2 3(υ τ) cos π) 4 4 w2,3,4 F = 1 (2(υ + ζ + τ) + 2 3(υ τ) cos 2πk 3π ) 4 4 w0 E1 = χ 1 w1 E1 = 1 4 (4σ1 + 2φ 1 + 2ψ 1 2ι 1 ) (25) w2,3,4 E1 = 1 4 ((σ1 + 2φ 1 + 2ψ 1 + ι 1 ) + (2σ 1 + 2φ 1 2ψ 1 2ι 1 ) cos 2π(k 1) 4 +(σ 1 2φ 1 + 2ψ 1 ι 1 ) cos 4π(k 1) ) 4 w0 E2 = χ 2 w1 E2 = 1 4 (4σ2 + 2φ 2 + 2ψ 2 2ι 2 ) (c) if n 5, w E2 2,3,4 = 1 4 ((σ2 + 2φ 2 + 2ψ 2 + ι 2 ) + (2σ 2 + 2φ 2 2ψ 2 2ι 2 ) cos 2π(k 1) 4 +(σ 2 2φ 2 + 2ψ 2 ι 2 ) cos 4π(k 1) 4 ) w0 F = ρ w1 F = 1 (2(υ + ζ + τ) + (2 3υ 2 3τ) cos π + (2υ 4ζ + 2τ) cos 2π ) n n n w F k 2 = 1 n (2(υ + ζ + τ) + (2 3υ 2 3τ) cos 2πk 3π n + (2υ 4ζ + 2τ) cos 4πk 6π n ) 10

11 w0 E1 = χ 1 w1 E1 = 1 n (nσ1 + (12 2n)φ 1 (12 2n)ψ 1 + (6 n)ι 1 ) wk 2 E1 = 1 n ((4φ1 + 2ι 1 ) + (6φ 1 6ψ 1 ) cos 2π(k 1) + (2φ 1 6ψ 1 + 4ι 1 ) cos 4π(k 1) ) n n w0 E2 = χ 2 w1 E2 = 1 n (nσ2 + (12 2n)φ 2 (12 2n)ψ 2 + (6 n)ι 2 ) wk 2 E2 = 1 n ((4φ2 + 2ι 2 ) + (6φ 2 6ψ 2 ) cos 2π(k 1) + (2φ 2 6ψ 2 + 4ι 2 ) cos 4π(k 1) ) n n (26) Fig. 7. (a) Irregular mask for F-vertex; (b) irregular mask for E-vertex; (b) irregular mask for another E-vertex. However, there are 14 free parameters in Eq. (6) but only 10 equations. We will show how to get the free parameters in the next section. 5 Subdivision Rules In the previous sections, algorithms for both regular and irregular regions have been designed but several free parameters in the masks are still left undetermined. The E-vertices and F-vertices of our scheme near extraordinary vertices are not computed in the way that the traditional subdivision schemes are, so further discussions are presented in this section to clarify and summarize the subdivision rules. 11

12 5.1 Topological rule In one step of our refinement, a F-vertex is inserted in each face while two E-vertices are generated on each edge. By connecting these new vertices with old ones, 9 new small faces are produced from each triangle. The splitting process is illustrated in Fig. 8. This topological rule is called the 1-9 splitting operator. Fig splitting operator. 5.2 Geometric rules For an interpolating scheme, it is sufficient to design rules for newly inserted vertices because the old vertices from previous levels keep their original positions. There exist two types of new vertices, vertices in regular regions and those in extraordinary regions, which should be treated differently. For new vertices in regular regions, masks presented in Fig. 2 and Fig. 3 are applied to compute their positions in our proposed method. Note that we set µ = 1, ǫ = 0 and ν = 0 in all our analysis. 11 The 1-neighborhood masks constructed in Section 4 are not sufficient for new vertices in extraordinary areas, although this is a popular approach to handle extraordinary situations [10, 14, 28]. To ensure that the irregular masks are an extension of the regular case, i.e, the regular mask can be considered as a special case of irregular masks, a small modification is done to the algorithm. This is also the way that [13] handles the irregular situations. The modified approach is outlined as follows: For every triangle that has at least one extraordinary vertex, we compute values for the F-vertex using three 1-neighborhood masks centered at three vertices of the triangle, respectively, and then take the average. The F-mask is shown in Fig. 7 (a) with the weights in Eq. (4). For every edge that connects at least one extraordinary vertex, we compute values for the E-vertex using two 1-neighborhood masks centered at two endpoints of the edge respectively, and then take the average. The E-masks are shown in Fig. 7 (b) (c) with the weights in Eq. (4). 12

13 Then the smallest similar stencil of extraordinary vertex is its 2-neighborhood. In other words, all vertices of its 2-neighborhood generated by one step of subdivision are computed using only vertices of its current 2-neighborhood. Moreover, no other smaller neighborhood has this property as shown in Fig. 9. Now we can establish the subdivision matrix A (n) by applying the geometric rules to the 2-neighborhood displayed in Fig. 9. Because we want to achieve C 1 continuous limit surfaces for arbitrary meshes, the 4 leading eigenvalues of A (n) must satisfy the constraint: 1 = λ 1 > λ 2 = λ 3 > λ 4. (27) By examining the eigenvalues of A (n) and evaluating the visual quality, we determine the free parameters in Eqs. (24) (25) (26). The proposed parameters given in Eq. (28) yield C 1 continuous limit surfaces near extraordinary vertices. That will be numerically verified in the next section. Fig. 9. The smallest similar stencil for 2 consecutive levels of subdivision, where the grey lines are in the j th level and dark lines are in the j + 1 th level. χ 1 = 2 χ 2 = 7 τ = σ ζ = 1 1 = 1 σ 3 2 = , φ 1 = 4, φ 2 = 2. (28) ρ = ψ 1 = 14 ψ 2 = 8 υ = ι 1 = 0 ι 2 = Boundaries To compute an F-vertex, all vertices in the stencil have to be used. For triangles near the boundary some of those vertices may not exist. This happens when one or more vertices of the triangle are on the boundary. To address this problem, mirror vertices [12, 16], lying outside of the mesh, are imported. A 13

14 typical configuration is shown in Fig. 10. The mirror vertices p 7, p 8, p 9 are computed as follows: p 7 = p 3 + p 4 p 0 p 8 = p 4 + p 5 p 1, (29) p 9 = p 5 + p 6 p 2 where p i (i = 0,..., 6) are the real vertices in the stencil on the boundary. By inserting these mirror vertices into the stencil, an F-vertex can be obtained. Obviously, Fig. 10 is not the only configuration, but all of other configurations can be handled using the idea of mirror vertices. Fig. 10. Mirror vertices in dashed circles of a stencil near the boundary. For an E-vertex on the boundary, the ternary interpolating subdivision scheme for curves is used directly to compute its position. This approach contributes to the stitching of 2 meshes which share the same boundary [9, 21]. Because only the vertices on the boundary are involved for computing new E-vertices, if two meshes share a boundary, their subdivided meshes will share a boundary, too. 6 Smoothness Analysis In most cases, interpolating subdivision surfaces cannot be generalized to a known parametric CAGD surface. Several tools have been designed to help us with determining the limit surface s properties. Typical tools for C 1 continuity include subdivision matrix and characteristic map [20, 27]. In this section, the C 1 continuity of our ternary butterfly scheme is numerically verified. Note that valence 6 is the regular setting and the others are irregular configurations. As pointed out in [20], a necessary condition for C 1 continuity is 1 = λ 1 > λ 2 = λ 3 > λ 4, where λ i is the i th largest eigenvalue of the subdivision 14

15 Table 1 Four leading eigenvalues of the subdivision matrix S (n) for valence n = 3,...,10. n λ 1 λ 2 λ 3 λ 4 n = n = n = n = n = n = n = n = matrix A (n). We have calculated the eigenvalues of subdivision matrixes A (n) for n = 3,..., 100. Table 1 shows the four leading eigenvalues of valence 3,...,10 while λ 2, λ 3 and λ 4 are depicted in Fig. 11. It can be observed that the proposed scheme satisfies the necessary condition for C 1 continuity in both regular and irregular regions, at least for valence 3,...,100, a range wide enough for practical applications. Fig. 11. The magnitudes of λ 2, λ 3 and λ 4 of the subdivision matrix S (n) for valence n = 3,...,100. However, this is still not sufficient to show the C 1 continuity of the scheme. It remains to demonstrate the regularity and injectivity of the characteristic maps. A characteristic map [14, 20] of a subdivision matrix is defined as the planar limit surface whose initial control mesh consists of the two eigenvectors of the eigenvalues λ 2 and λ 3. Namely, x- and y-coordinates of a vertex in the control mesh come from the two eigenvectors respectively. The z-coordinates of all vertices are set to be zero. We have calculated the characteristic maps for valence 3,..., 100, all of which are well behaved. Some of these characteristic maps are illustrated in Fig. 12 to show the ini- 15

16 Fig. 12. Visualization of the characteristic maps of valence 3,...,8. The first and the third rows are the initial control meshes and the second and forth rows are the results after 4 rounds of the ternary butterfly subdivision. tial control meshes and the corresponding subdivided meshes after 4 steps of refinement. The above result leads to the conclusion that our ternary butterfly subdivision scheme generates C 1 continuous surfaces for both regular and irregular regions around vertices with a valence no larger than 100. As pointed out in [18, 19], if the six leading eigenvalues of the subdivision matrix have the structure: 1, λ, λ, λ 2, λ 2, λ 2, (30) then the scheme generates limit surfaces with bounded curvature. Note that when the valence is 6, the six leading eigenvalues of our subdivision matrix satisfies the structure described in Eq. (30), so we conclude that bounded curvature is achieved in those regular regions of limit surfaces. But that is not necessarily true at extraordinary vertices. 16

17 7 Adaptive Algorithm Due to the 1-9 splitting topological rule which is employed in our proposed scheme, the number of faces grows by the power of 9 in each round of refinement, which is faster than other schemes [6, 28]. Hence, for efficient applications, we try to refine the control mesh only in those regions where the quality of the surface is not desirable and use a coarser mesh in the region with fewer geometric details. This strategy is called adaptive subdivision [11, 13]. The main problem of adaptive subdivision is to distinguish the divisible regions, which is usually with high curvature, from those indivisible regions. We use the dihedral angle as the threshold θ T to determine the divisibility of a face. If all the angles between a face and its neighbors are less than θ T, we mark it as the indivisible region. But the indivisible regions adjacent to those divisible regions must be treated specially. If all the indivisible regions remain the same after one round of refinement, holes [12] would occur in the subdivided mesh because the shared edges are not split. We use the technique described in Fig. 13 to avoid this problem. The indivisible regions near those divisible regions are split according to the degree of indivisibility. If a face is considered to be indivisible, the degree of indivisibility is defined as the number of its indivisible neighbors. Our adaptive algorithm can be briefly summarized as follows: (1) Set M div = φ and M ind = φ, where M div is the set of divisible faces and M ind is the set of indivisible faces. (2) Compute the normal vector of each face. (3) Compute all angles between the normal vectors of two adjacent faces. Note that each face has 3 adjacent neighbors, so 3 angles, θ a, θ b and θ c, can be calculated for each face. (4) Examine the face F of the mesh. If all of θ a, θ b and θ c are less than the threshold θ T, add F to M ind, otherwise add F to M div. (5) Repeat Step (4) until all faces are checked. (6) Subdivide M div using the proposed uniform scheme. (7) For indivisible faces adjacent to those divisible faces, determine their degree of indivisibility and then split these faces with the technique described in Fig. 13. Specifically, if an indivisible face is surrounded by three divisible faces, we divide it into 9 smaller faces by the rules described in Section 5. If further refinements are applied to the triangles already adaptively subdivided, i.e., the center triangles in Fig. 13 (b) and (c), we can first remove the long sub-triangles in the center followed by a uniform refinement. This method is called red-green triangulation [23] and is effective to remove visual artifacts 17

18 caused by long-triangles. After adaptively subdividing a mesh, we can still ameliorate the long-triangle artifacts by a post-processing step. We employ the edge-flip operation based on maximizing the smallest angle [3] and the dihedral angle threshold in adaptively subdivided regions to improve the mesh quality. Note that the red-green method ensures that a face is adaptively subdivided at most once, so the number of adaptively subdivided faces is small. Because the post-processing step is only applied to the adaptively subdivided regions, the computational cost is relatively low. Fig. 13. Top row: the dark faces are set to be divisible while the others are set to be indivisible. Bottom row: meshes after adaptive subdivision with a central face of degree of divisibility 0 (a), 1 (b), 2 (c), and 3 (d). 8 Experiments To demonstrate the visual quality of the ternary butterfly subdivision surfaces, we have implemented the subdivision algorithm using OpenGL library for visualization. The examples shown in Fig. 14 and Fig. 15 are designed to compare our approach with the traditional butterfly scheme [7, 28]. We use sufficiently dense meshes to approximate their limit surfaces. Fig. 14 (b) (157k faces) and Fig. 15 (b) (472k faces) are generated by uniformly refining the input meshes using the ternary butterfly scheme with 4 and 5 steps, respectively. Fig. 14 (c) (98k faces) and Fig. 15 (c) (524k faces) are generated by uniformly refining the input meshes using the traditional butterfly scheme with 6 and 8 steps, respectively. Due to the use of different masks in the refinement, our limit surfaces are different from that of the traditional butterfly scheme. The light reflection areas of refined examples demonstrate the difference. We can see that the reflection areas generated by our approach are larger. Fig. 16 describes the process of subdivision by presenting a sequence of subdivided meshes. The initial ant mesh consists of 912 triangles and 486 vertices. 18

19 Fig. 14. Comparison of limit surfaces between the ternary butterfly scheme and the traditional butterfly scheme: (a) Initial mesh with 24 faces; (b) limit surface of the ternary butterfly scheme; (c) limit surface of the traditional butterfly scheme Fig. 15. Comparison of limit surfaces between the ternary butterfly scheme and the traditional butterfly scheme. (a) Initial input with 8 faces; (b) limit surface of the ternary butterfly scheme; (c) limit surface of the traditional butterfly scheme. Fig. 16. Ant model: sequence of meshes generated by the ternary butterfly scheme (top row) and their corresponding rendered models (bottom row). (a) Initial input; (b) result generated after one step of refinement; (c) result generated after two steps of refinement. After one step of refinement, the refined mesh has 8208 triangles and 4134 vertices. After two steps of refinement, the refined mesh has triangles and vertices. Obviously, we can see that the number of triangles grows by power of 9. Fig. 17 and Fig. 18 display three examples of uniform subdivision surfaces. All results are produced after two steps of refinement. These examples show that the proposed subdivision scheme is capable of modeling meshes for practical 19

20 Fig. 17. Dolphin model: (a) initial mesh; (b) result generated after 2 steps of refinement. Fig. 18. (a) Initial mesh of a foot bone model; (b) result by applying subdivision to (a); (c) initial mesh of a dinosaur model; (d) result by applying subdivision to (c). Both of the results in (b) and (d) are produced after 2 steps of refinement. Fig. 19. Mannequin head model: (a) initial control mesh; (b) result after 2 steps of adaptive refinement; (c) result after 2 steps of uniform refinement. applications. The ability to deal with complex meshes are demonstrated in Fig. 18. Fig. 19 shows the difference between adaptive subdivision and uniform subdivision. We can see clearly that only those regions with fine details are subdivided. The threshold is set to be π/24 in this example. The adaptively subdivided mesh has vertices and faces, while the uniformly subdivide mesh has vertices and faces. 20

21 Acknowledgments The authors wish to thank Christopher Stuetzle for his help on improving the written English of the paper, Dongming Yan, Feng Sun, Yufei Li, Li Cao and Pengbo Bo for proof-reading. A The details of Fourier transform In Section 4, we have designed the irregular rules for meshes with extraordinary settings. Here, more mathematical details are described to substantiate the derivation. In the regular 1-neighborhood case, which has a central vertex of valence 6, we have constructed a circulant subdivision matrix A in Eq. (12). According to the modification presented in Eq. (15) and Eq. (14), the revised Fourier transform can be employed to get its coefficients, which are listed in the following equations χ 1 σ 1 + 2φ 1 + 2ψ 1 + ι 1 0 σ 1 + φ 1 ψ 1 ι 1 Â 1 = χ 2 σ 2 + 2φ 2 + 2ψ 2 + ι 2, Â2 = 0 σ 2 + φ 2 ψ 2 ι 2, ρ 2υ + 2ζ + 2τ 0 3υ + 3τ σ 1 φ 1 ψ 1 + ι 1 0 σ 1 2φ 1 + 2ψ 1 ι 1 Â 3 = 0 σ 2 φ 2 ψ 2 + ι 2, Â4 = 0 σ 2 2φ 2 + 2ψ 2 ι 2, 0 υ 2ζ + τ σ 1 φ 1 ψ 1 + ι 1 0 σ 1 + φ 1 ψ 1 ι 1 Â 5 = 0 σ 2 φ 2 ψ 2 + ι 2, Â6 = 0 σ 2 + φ 2 ψ 2 ι 2. 0 υ + 2ζ τ 0 3υ + 3τ Since the Fourier coefficients have been calculated and we have set the rules to determine the Fourier coefficients in Eqs. (18-22), the subdivision matrix A (n) can be formulated. Similar to the derivation mentioned above, three unique 21

22 situations have to be discussed separately. (a) For the 1-neighborhood mask with a central vertex of valence 3, the subdivision matrix is: where A (3) = Circulant(A (3) 1, A (3) 2, A (3) 3 ), A (3) 1 = χ 1 3σ 1 + 4φ 1 ι 1 χ 2 3σ 2 + 4φ 2 ι 2, ρ 2(υ + ζ + τ) + 2 3(υ τ) cos π 3 with A (3) k=2,3 = χ 1 a (3) 1 + a (3) 2 cos 2πk 3 χ 2 a (3) 3 + a (3) 4 cos 2πk 3 ρ 2(υ + ζ + τ) + 2 3(υ τ) cos 2πk 1 3, a (3) 1 = σ 1 + 2φ 1 + 2ψ 1 + ι 1 a (3) 2 = 2(σ 1 + φ 1 ψ 1 ι 1 ) a (3) 3 = σ 2 + 2φ 2 + 2ψ 2 + ι 2 a (3) 4 = 2(σ 2 + φ 2 ψ 2 ι 2 ). (b) For the 1-neighborhood mask with a central vertex of valence 4, the subdivision matrix is: where A (4) = Circulant(A (4) 1, A (4) 2, A (4) 3, A (4) 4 ), 22

23 A (4) 1 = χ 1 4σ 1 + 2φ 1 + 2ψ 1 ι 1 χ 2 4σ 2 + 2φ 2 + 2ψ 2 ι 2, ρ 2(υ + ζ + τ) + 2 3(υ τ) cos π 4 A (4) k=2,3,4 = χ 1 a (4) 1 + a (4) 2 cos 2πk + a (4) 4 3 cos 4πk 4 χ 2 a (4) 4 + a (4) 5 cos 2πk + a (4) 4 6 cos 4πk 4 ρ a (4) 7 + a (4) 8 2πk 1 4, and a (4) 1 = σ 1 + 2φ 1 + 2ψ 1 + ι 1 a (4) 2 = 2σ 1 + 2φ 1 2ψ 1 2ι 1 a (4) 3 = σ 1 2φ 1 + 2ψ 1 ι 1 a (4) 4 = σ 2 + 2φ 2 + 2ψ 2 + ι 2 a (4) 5 = 2σ 2 + 2φ 2 2ψ 2 2ι 2 a (4) 6 = σ 2 2φ 2 + 2ψ 2 ι 2 a (4) 7 = 2υ + 2ζ + 2τ a (4) 8 = 2 3υ 2 3τ. (c) For the 1-neighborhood mask with a central vertex of valence n 5, the subdivision matrix is: where A (n) = Circulant(A (n) 1, A (n) 2,..., A (n) n ), A (n) 1 = 1 n 1 0 χ 1 1 χ 2 2 ρ cos π n + a(n) 5 cos 2π n, 23

24 1 0 A (n) k 2 = 1 χ cos 2πk + n a(n) 4πk 8 n n χ cos 2πk + n a(n) 11 4πk, n ρ cos 2πk π + n 14 cos 4πk 2π n and 1 = nσ 1 + (12 2n)φ 1 (12 2n)ψ 1 + (6 n)ι 1 2 = nσ 2 + (12 2n)φ 2 (12 2n)ψ 2 + (6 n)ι 2 3 = 2υ + 2ζ + 2τ 4 = 2 3υ 2 3τ 5 = 2υ 4ζ + 2τ 6 = 4φ 1 + 2ι 1 7 = 6φ 1 6ψ 1 8 = 2φ 1 6ψ 1 + 4ι 1 9 = 4φ 2 + 2ι 2 10 = 6φ 2 6ψ 2 11 = 2φ 2 6ψ 2 + 4ι 2 12 = 2υ + 2ζ + 2τ 13 = 2 3υ 2 3τ 14 = 2υ 4ζ + 2τ Hence, from the subdivision matrices, a bit more derivation yields the weights in Eqs. (24) (25) (26).. References [1] E. Catmull and J. Clark. Recursively generated b-spline surfaces on arbitrary topological meshes. Computer Aided Design, 10(6): , [2] Z. Chen, X. Luo, L. Tan, B. Ye, and J. Chen. Progressive interpolation based on Catmull-Clark subdivision surfaces. Computer Graphics Forum, 27(7): , [3] M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications, 3rd edition. Springer- Verlag,

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