Bounding the Distance between a Loop Subdivision Surface and Its ...
Bounding the Distance between a Loop Subdivision Surface and Its Limit Mesh Zhangjin Huang Guoping Wang HCI & Multimedia Lab Peking University, P.R. China
Loop subdivision surface Loop subdivision surface (Loop 1987)
a generalization of quartic three-directional box spline surface to triangular meshes of arbitrary topology
C 2 continuous except at extraordinary points
the limit of a sequence of recursively refined (triangular) control meshes
initial mesh
step 1
limit surface
Linear approximations Linear approximations of a Loop surface are used in surface rendering, numerical control tool-path generation, etc Two linear approximations:
control mesh
(refined) control mesh limit mesh: obtained by pushing the control points of a control mesh to their limit positions
It inscribes the limit surface limit mesh
Problems of limit mesh approx. Approximation
How to estimate the error (distance) between a Loop surface and its limit mesh?
Subdivision
error
depth estimation
How many steps of subdivision would be necessary to satisfy a user-specified error tolerance?
Distance (error) Each surface patch S corresponds to a limit triangle F in its limit mesh
F is formed by connecting the three corners of S
Distance (error) between a patch S and the limit mesh is defined as max S(v, w) − F (v, w)
( v , w )∈Ω
S (v, w) : Stam’s parameterization of S F (v, w) : linear parameterization of F Ω : the unit triangle
Ω = {(v, w) v ∈ [0,1], ω∈ [0,1 − v]}
S _ F
Control mesh of a Loop patch Assume: the initial control mesh has been subdivided at least once
Each triangle face contains at most one extraordinary point
The control mesh of a Loop patch S of valence n consists of n+6 control points
{Pi : i = 1, 2,…, n + 6}
P1: the only extraordinary point of valence n
Control mesh of a Loop patch of valence n
Second order norm Each interior edge Pi P j in the control mesh can be associated with a mixed second difference (MSD): Pi + P j − Pk − Pm
The second order norm M of the control mesh of Loop patch S is defined as the maximum norm of n+9 MSDs:
Our goal To derive a bound on the distance between a Loop patch S(v, w) and its limit triangle F(v, w) as max S(v, w) − F (v, w) ≤ β ( n) M
( v , w )∈Ω
: the second order norm of the control mesh of S β ( n) : a constant only depends on valence n
M
Regular patch S (v, w) is a quartic box spline
patch defined by 12 control points: 12
S(v, w) = ∑ pi N i (v, w), (v, w) ∈ Ω i =1
N i (v, w) : quartic box spline
basis functions
S (v, w) is also a quartic tri-
Control points of a quartic box spline patch
angular Bézier patch: 15
S(v, w) = ∑ bi Bi (v, w), (v, w) ∈ Ω i =1
Bi (v, w) : quartic Bernstein
polynomials b i : Bézier points of S
Bézier points of a quartic triangular Bézier patch and Bernstein polynomials
Regular patch Limit triangle is F = {b1 , b11 , b15 } , and its linear parameterization is F (v, w) = ub1 + v b11 + wb15 , (v, w) ∈ Ω
(u , v, w) : barycentric system of coordinates for the unit triangle, u = 1 − v − w
By the linear precision property of the Bernstein polynomials, F (v, w) can be expressed into the quartic Bézier form: 15
F(v, w) = ∑ bi Bi (v, w), (v, w) ∈ Ω
Here,
i =1
b ( j+k )(2j+k +1) +k +1 = F( 4j , k4 )
Regular patch For (u , v) ∈ Ω , S(v, w) − F (v, w) =
15
∑ (b i =1 15
i
− b i ) Bi (v, w)
≤ ∑ bi − bi Bi (v, w) i =1
By solving some constrained minimization problems, bi − bi can be bounded as bi − bi ≤ δ i M ,1 ≤ i ≤ 15
Here, M is the second order norm of S , and
δ1 = δ11 = δ15 = 0
δ 2 = δ 3 = δ 7 = δ10 = δ12 = δ14 = 14
δ 4 = δ 6 = δ13 = 13
δ 5 = δ 8 = δ 9 = 125
Regular patch Thus,
15
S(v, w) − F(v, w) ≤ ∑ δ i Bi (v, w)M = D(v, w) M i =1
15
2 2 Here, D(v, w) = ∑ δ i Bi (v, w) = v + w − v − vw − w is called the i =1 distance bound function of S(v, w) with respect to F (v, w) .
D (v, w) = 13 = D ( 13 , 13 ) , we have Since (max v , w )∈Ω
S(v, w) − F (v, w) ≤ 13 M
Extraordinary patch S can be partitioned into an infinite
sequence of regular triangular subpatchs: {S km : k ≥ 1, m = 1, 2,3}
corresponding to tiles Ω km , k ≥ 1, m = 1, 2,3 k F is partitioned into sub-triangles F m , k ≥ 1, m = 1, 2,3 accordingly. k For each subpatch S m , k m
S (v, w) − F (v, w) ≤ Dmk (v, w) M ≤ β mk (n) M k m
Dmk (v, w) : distance bound function k k β m ( n) = max Dm (v, w)
( v , w )∈Ω
Extraordinary patch k β ( n ) = max β Let m ( n ) , we have k ≥1, m =1,2,3
max S(v, w) − F (v, w) ≤ β ( n) M
( v , w )∈Ω
For 3 ≤ n ≤ 50, the following fact is found: k Maximum of β m ( n), k ≥ 1, m = 1, 2,3 appears in the domain Ω = Ω ∩ {(v, w) v + w ≥ 14 }
To compute the values of β (n) , only four subpatchs S11 , S12 , S12 , S 22 are needed to be analyzed.
Subdivision depth estimation Theorem Given an error tolerance ε > 0, after k=
min λl + j 0 ≤ j ≤ λ −1 j
steps of subdivision on the control mesh of a patch S , the distance between S and its level-k control mesh is smaller than ε . Here rj (n) max{β (n), 13}M ⎤ ⎡ l j = ⎢log1/ rλ ( n ) , 0 ≤ j ≤ λ − 1, λ ≥ 1 ⎥ ε ⎢ ⎥ and rj (n) is the j-step convergence rate of
second order norm.
For regular Loop patches,
M⎤ ⎡ k = ⎢log 4 ⎥ 3ε ⎥ ⎢
Comparison The distance between a Loop patch S and its control mesh can be bounded as: max S(v, w) − F (v, w) ≤ Cλ (n) M , λ ≥ 1
( u ,v )∈Ω
Comparison results of C3(n) and β(n), 3≤n≤10
A limit mesh approximates a Loop surface better than the corresponding control mesh in general.
Comparison Set M = 1 , error tolerance ε = 0.0001 . Comparison results of subdivision depths for control mesh and limit mesh approximations are as follows:
Limit mesh approximation has a 20% improvement over control mesh approximation in most of the cases.
Conclusion and remark
We propose an approach to estimate the bound on the distance between a Loop surface and its limit mesh. A subdivision depth estimation formula for limit mesh approximation is derived. The upper bounds derived here are optimal for regular patches, but not optimal for extraordinary patches.
Thank you!
Loop subdivision surface Loop subdivision surface (Loop 1987)
a generalization of quartic three-directional box spline surface to triangular meshes of arbitrary topology
C 2 continuous except at extraordinary points
the limit of a sequence of recursively refined (triangular) control meshes
initial mesh
step 1
limit surface
Linear approximations Linear approximations of a Loop surface are used in surface rendering, numerical control tool-path generation, etc Two linear approximations:
control mesh
(refined) control mesh limit mesh: obtained by pushing the control points of a control mesh to their limit positions
It inscribes the limit surface limit mesh
Problems of limit mesh approx. Approximation
How to estimate the error (distance) between a Loop surface and its limit mesh?
Subdivision
error
depth estimation
How many steps of subdivision would be necessary to satisfy a user-specified error tolerance?
Distance (error) Each surface patch S corresponds to a limit triangle F in its limit mesh
F is formed by connecting the three corners of S
Distance (error) between a patch S and the limit mesh is defined as max S(v, w) − F (v, w)
( v , w )∈Ω
S (v, w) : Stam’s parameterization of S F (v, w) : linear parameterization of F Ω : the unit triangle
Ω = {(v, w) v ∈ [0,1], ω∈ [0,1 − v]}
S _ F
Control mesh of a Loop patch Assume: the initial control mesh has been subdivided at least once
Each triangle face contains at most one extraordinary point
The control mesh of a Loop patch S of valence n consists of n+6 control points
{Pi : i = 1, 2,…, n + 6}
P1: the only extraordinary point of valence n
Control mesh of a Loop patch of valence n
Second order norm Each interior edge Pi P j in the control mesh can be associated with a mixed second difference (MSD): Pi + P j − Pk − Pm
The second order norm M of the control mesh of Loop patch S is defined as the maximum norm of n+9 MSDs:
Our goal To derive a bound on the distance between a Loop patch S(v, w) and its limit triangle F(v, w) as max S(v, w) − F (v, w) ≤ β ( n) M
( v , w )∈Ω
: the second order norm of the control mesh of S β ( n) : a constant only depends on valence n
M
Regular patch S (v, w) is a quartic box spline
patch defined by 12 control points: 12
S(v, w) = ∑ pi N i (v, w), (v, w) ∈ Ω i =1
N i (v, w) : quartic box spline
basis functions
S (v, w) is also a quartic tri-
Control points of a quartic box spline patch
angular Bézier patch: 15
S(v, w) = ∑ bi Bi (v, w), (v, w) ∈ Ω i =1
Bi (v, w) : quartic Bernstein
polynomials b i : Bézier points of S
Bézier points of a quartic triangular Bézier patch and Bernstein polynomials
Regular patch Limit triangle is F = {b1 , b11 , b15 } , and its linear parameterization is F (v, w) = ub1 + v b11 + wb15 , (v, w) ∈ Ω
(u , v, w) : barycentric system of coordinates for the unit triangle, u = 1 − v − w
By the linear precision property of the Bernstein polynomials, F (v, w) can be expressed into the quartic Bézier form: 15
F(v, w) = ∑ bi Bi (v, w), (v, w) ∈ Ω
Here,
i =1
b ( j+k )(2j+k +1) +k +1 = F( 4j , k4 )
Regular patch For (u , v) ∈ Ω , S(v, w) − F (v, w) =
15
∑ (b i =1 15
i
− b i ) Bi (v, w)
≤ ∑ bi − bi Bi (v, w) i =1
By solving some constrained minimization problems, bi − bi can be bounded as bi − bi ≤ δ i M ,1 ≤ i ≤ 15
Here, M is the second order norm of S , and
δ1 = δ11 = δ15 = 0
δ 2 = δ 3 = δ 7 = δ10 = δ12 = δ14 = 14
δ 4 = δ 6 = δ13 = 13
δ 5 = δ 8 = δ 9 = 125
Regular patch Thus,
15
S(v, w) − F(v, w) ≤ ∑ δ i Bi (v, w)M = D(v, w) M i =1
15
2 2 Here, D(v, w) = ∑ δ i Bi (v, w) = v + w − v − vw − w is called the i =1 distance bound function of S(v, w) with respect to F (v, w) .
D (v, w) = 13 = D ( 13 , 13 ) , we have Since (max v , w )∈Ω
S(v, w) − F (v, w) ≤ 13 M
Extraordinary patch S can be partitioned into an infinite
sequence of regular triangular subpatchs: {S km : k ≥ 1, m = 1, 2,3}
corresponding to tiles Ω km , k ≥ 1, m = 1, 2,3 k F is partitioned into sub-triangles F m , k ≥ 1, m = 1, 2,3 accordingly. k For each subpatch S m , k m
S (v, w) − F (v, w) ≤ Dmk (v, w) M ≤ β mk (n) M k m
Dmk (v, w) : distance bound function k k β m ( n) = max Dm (v, w)
( v , w )∈Ω
Extraordinary patch k β ( n ) = max β Let m ( n ) , we have k ≥1, m =1,2,3
max S(v, w) − F (v, w) ≤ β ( n) M
( v , w )∈Ω
For 3 ≤ n ≤ 50, the following fact is found: k Maximum of β m ( n), k ≥ 1, m = 1, 2,3 appears in the domain Ω = Ω ∩ {(v, w) v + w ≥ 14 }
To compute the values of β (n) , only four subpatchs S11 , S12 , S12 , S 22 are needed to be analyzed.
Subdivision depth estimation Theorem Given an error tolerance ε > 0, after k=
min λl + j 0 ≤ j ≤ λ −1 j
steps of subdivision on the control mesh of a patch S , the distance between S and its level-k control mesh is smaller than ε . Here rj (n) max{β (n), 13}M ⎤ ⎡ l j = ⎢log1/ rλ ( n ) , 0 ≤ j ≤ λ − 1, λ ≥ 1 ⎥ ε ⎢ ⎥ and rj (n) is the j-step convergence rate of
second order norm.
For regular Loop patches,
M⎤ ⎡ k = ⎢log 4 ⎥ 3ε ⎥ ⎢
Comparison The distance between a Loop patch S and its control mesh can be bounded as: max S(v, w) − F (v, w) ≤ Cλ (n) M , λ ≥ 1
( u ,v )∈Ω
Comparison results of C3(n) and β(n), 3≤n≤10
A limit mesh approximates a Loop surface better than the corresponding control mesh in general.
Comparison Set M = 1 , error tolerance ε = 0.0001 . Comparison results of subdivision depths for control mesh and limit mesh approximations are as follows:
Limit mesh approximation has a 20% improvement over control mesh approximation in most of the cases.
Conclusion and remark
We propose an approach to estimate the bound on the distance between a Loop surface and its limit mesh. A subdivision depth estimation formula for limit mesh approximation is derived. The upper bounds derived here are optimal for regular patches, but not optimal for extraordinary patches.
Thank you!
