Eurographics Symposium on Geometry Processing (2005) M. Desbrun, H. Pottmann (Editors)
A New Way To Tune Subdivision U.H. Augsdörfer, N.A. Dodgson and M.A. Sabin
[email protected],
[email protected],
[email protected] Computer Laboratory, University of Cambridge, 15 J.J. Thomson Ave, Cambridge CB3 OFD, England
We present a novel method which supports the tuning of a variety of stationary subdivision schemes to give the best possible behaviour near extraordinary vertices. The new tuning method uses a novel set of freedoms and is based on the use of the mask rather then the stencils. In using the mask rather then the stencils we resolve the problem occurring when extraordinary points fall close together. It also gives an obvious count to the number of freedoms available for tuning. We tune the coefficients in the mask, and then re-normalise the linear combinations giving the new vertices by summing the contributions and dividing the total by the sum of the coefficients used. Rather than optimising within the coefficient space, we use the subdominant eigenvalue, λ, as a freedom and impose the condition that the subsubdominant eigenvalues, µ, are its square [DS78]. Therefore, we ensure that the condition µ0 = µ2 = λ2 , necessary for bounded curvature around extraordinary vertices, is always satisfied. This also reduces the number of freedoms and thus simplifies the optimisation.
1 2 1
γ
β
γ
2 4
2
β α
1 2
1
γ β
γ
β
γ
Figure 1: Mask coefficients around regular (left) and extraordinary (right) vertices for the 4-8 subdivision scheme. c The Eurographics Association 2005.
We optimise our choice of coefficients against the Gaussian curvature criterion introduced by Peters and Reif [PR04, KPR04]. They showed that the Gaussian curvature around the extraordinary vertex can be used to detect when a mesh will lead to undesirable curvature behaviour. To built up a shape-in-the-limit chart similar to the one introduced by Karˇciauskas et al. [KPR04], we need to analyse the Gaussian curvature in the spline ring around the extraordinary vertex for a representative set of meshes. The column eigenvector corresponding to the subdominant eigenvalue λ, obtained from the ω = ±1 Fourier subdivision matrix gives the natural configuration, the way in which the neighbourhood of the extraordinary point is laid out within the tangent plane. The column eigenvectors corresponding to the eigenvalue µ from the ω = 0 and ω = ±2 Fourier subdivision matrices give the cup, mc , and saddle components ms1 and ms2 (one with its saddle axis aligned n=5 30
coefficients
As an example, we apply our method to Velho’s 4-8 subdivision scheme [Vel01], but any primal subdivision scheme can be tuned in this manner. The main advantage of the 4-8 scheme as a first example is the small size of its mask and small number of coefficients. The mask for the 4-8 scheme around extraordinary points can be written as shown in Figure 1 from which the stencils of the scheme are readily extracted.
Treating the subdominant eigenvalue as an input we solve for β and γ by requiring that the determinants of both the ω = 1 and ω = 2 Fourier subdivision matrix be zero [DS78]. After solving for β and γ we can solve for α by imposing the condition µ0 = λ2 for the determinant of the ω = 0 Fourier subdivision matrix.
20
α
10 0 0.65
γ β 0.7
λ
0.75
0.8
Figure 2: The relationship between λ and the chosen coefficients for a valency n = 5 of the extraordinary vertex.
U.H. Augsdörfer, N.A. Dodgson & M.A. Sabin / A New Way To Tune Subdivision
A representative set of quadratic input meshes, mi , can be established by combining the three fundamental natural configurations mi = (1−r)mc +r cos(φ)ms1 +r sin(φ)ms2 where the radius, r, ranges from [0, 1] and the angle φ ranges from [0, 2π]. All possible input meshes therefore map into a disk. For each shape, we calculate the Gaussian curvature, K, and identify whether it is elliptic (K > 0) or hyperbolic (K < 0) for all points in the spline ring around the extraordinary vertex in the limit surface. If K changes sign it is hybrid and will lead to artifacts in the limit surface. We built up the shape-in-the-limit charts by plotting a coloured dot for each mesh of the representative set of input meshes: green for elliptic, blue for hyperbolic and red to denote hybrid behaviour of the Gaussian curvature around the extraordinary vertex of the mesh in question. We optimise the scheme’s behaviour around extraordinary vertices by minimising the number of hybrid cases, the red entries in the shape-in-the-limit charts, for a representative set of input meshes for any given λ. By determining the shape-in-the-limit chart for a wide range of λ we can establish which value of λ yields the coefficients α, β and γ which minimise the occurrence of undesirable hybrid Gaussian curvature behaviour. Figure 4 shows the number of hybrid cases as a function of the subdominant eigenvalue, λ, for a natural configuration around an extraordinary vertex of valency n = 5, 6, 7, 8, 9. The graphs clearly show quantisation owing to simply counting hybrid cases from a set of evenly spaced samples; further
15 hybrid count / %
with the coordinate axis and a second with its saddle axis aligned at a 45◦ shift) perpendicular to the tangent plane. Each of these meshes defines a higher order natural configuration, which we refer to in the following as the characteristic mesh.
10 9 5
8 5
0 0.75
7
6 0.8
λ
0.85
0.9
Figure 4: The percentage of meshes in a representative set for which the Gaussian curvature is hybrid shown for different valency (as indicated) as a function of the subdominant eigenvalue λ. n
λ
α
β
γ
4 5 6 7 8 9
0.7071 0.7756 0.8286 0.8669 0.89435 0.91457
4 11.398 20.7511 33.3881 50.1244 71.4656
2 1.5465 1.1744 0.9102 0.7136 0.5853
1 0.6688 0.4092 0.2441 0.1635 0.0882
Table 1: The subdominant eigenvalue and coefficients for the 4-8 Velho subdivision scheme for which the least number of shapes from a representative set of meshes result in a hybrid Gaussian curvature.
work is required to ascertain whether there is a better way to sample the space of quadratic input meshes. The minimum value on each curve is our optimum value of λ for each valency. The optimised coefficients for each valency are listed in Table 1. We intend next to apply this method to subdivision schemes with larger masks such as Loop and Catmull-Clark. We are grateful to EPSRC, grant number GR/S67173/01, for funding this project.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
λ = 0.72, α = 16.6214, β = 0.1834, γ = 7.0867 2.5 2 1.5
References [DS78] D OO D., S ABIN M.: Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design 10, 6 (1978), 177–181. 1
1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
λ = 0.7756, α = 11.398, β = 1.5465, γ = 0.6688 Figure 3: Shape-in-the-limit chart for a mesh with and extraordinary vertex with valency n = 5 are shown together with the natural configuration for two different subdominant eigenvalues λ.
[KPR04] K ARCIAUSKAS K., P ETERS J., R EIF U.: Shape characterization of subdivision surfaces – case studies. Computer Aided Geometric Design 21, 6 (2004), 601– 614. 1 [PR04] P ETERS J., R EIF U.: Shape characterization of subdivision surfaces: basic principles. Comput. Aided Geom. Des. 21, 6 (2004), 585–599. 1 [Vel01] V ELHO L.: Quasi 4-8 subdivision. Computer Aided Geometric Design 18, 4 (2001), 345–358. 1 c The Eurographics Association 2005.
