Properties of the Blossom. Existence and Uniqueness. ⢠Every Degree n Polynomial P(t) has a Unique Blossom p(u1. ,...,un. ) Power Law. ⢠Each Parameter u1.
Global Knot Insertion Algorithms Scott Schaefer Department of Computer Science Texas A&M University
Ron Goldman Department of Computer Science Rice University
Part I: Introduction to Knot Insertion
Knot Insertion Algorithm Input
€
•
T = {t1,K,tν +n } -- knot sequence
•
P = {P0 ,K,Pν } -- control points
•
Γ = {τ1,K,τ µ +n } -- new knot sequence -- Γ ⊃ T
€ Output €
•
€ Q = {Q0 ,K,Qµ } -- new control points
Constraint € •
€
ν
µ
k=0
k =0
∑ N k,n(t | T )Pk = ∑ N k,n(τ | Γ)Qk
B-spline Curve
P1
• P2
• •
S(τ )
€
€
• P3
P0 •€
€
€
Knot Insertion
P1
•
€
€
€
Q1
Q0
€
Q2
• P2
•
S(τ ) €
•=€P
€
0
€
Q3
•
Q4 = P3
Theorems Existence •
All splines are B-splines.
Convergence •
The control polygons generated by knot insertion converge to the B-spline curve for the original control polygon as the knot spacing approaches zero.
Corner Cutting •
Knot insertion is a corner cutting procedure.
Applications of Knot Insertion
•
Rendering
•
Intersection
•
Conversion from B-spline to Piecewise Bezier Form
•
Proof of the Variation Diminishing Property
Types of Knot Insertion Algorithms
Local Knot Insertion •
T = {t1,K,tν +n }
•
Γ = {t1 ,K,tk , uk,1 ,K,uk,d k , t k+1,K,tv+n }
€ €
€ €
Global Knot Insertion •
T = {t1,K,tν +n }
•
Γ = {t1 , u1,1 ,K,u1,d1 , t2 ,K,tv+n −1, uv+n−1,1,K,uv+n−1,dv+ n−1 ,tv+n }
Examples of Knot Insertion Algorithms
Local Knot Insertion Algorithms
Global Knot Insertion Algorithms
•
Boehm’s Algorithm
•
Chaikin’s Algorithm
•
Oslo Algorithm
•
Lane-Riesenfeld Algorithm
•
Sablonniere’s Algorithm
•
Goldman-Warren Algorithm
•
Factored Knot Insertion
•
Schaefer’s Algorithm -- NEW
Myths Global Knot Insertion •
Works Only for Uniform Knot Sequences -- Knots in Arithmetic Progression -- Lane-Riesenfeld (1973) tk +1 = tk + α -- Knots in Geometric or Affine Progression -- Goldman-Warren (1993) tk +1 = β t k + α
€ •
Does not Apply to Arbitrary Knot Sequences
€ Blossoming •
Provides Only New Proofs of Already Known Results -- No New Results -- No New Insights
Part II: Local Knot Insertion Algorithms
Knot Insertion Algorithms from Blossoming
•
Boehm’s Algorithm -- Inserts one new knot at a time
•
Oslo Algorithm -- Computes one new control point at a time
•
Sablonniere’s Algorithm -- Local change of basis algorithm
•
Factored Knot Insertion -- Forward differencing for knot insertion
Blossoming Symmetry •
p(u1 ,...,un ) = p(uσ (1) ,...,uσ (n) ) for any permutation σ of {1,...,n}
Multiaffine €
•
€ € p(u1 ,...,(1− α )uk + αwk ,...,un ) = (1− α ) p(u1,...,u k ,...,un ) + αp(u1,...,wk ,...,un )
Diagonal €
•
p(t,...,t { ) = P(t) n
Dual Functional Property €
€ €
•
P(t) = ∑k N k,n (t)Pk ⇒ Pk = p(t k +1,K,tk +n )
(B-spline Curves)
•
P(t) = ∑k Bkn (t)Pk ⇒ Pk = p(a,K,a 123,b,K,b 123 )
(Bezier Curves)
n−k
k
Properties of the Blossom
Existence and Uniqueness • Every Degree n Polynomial P(t) has a Unique Blossom p(u1 ,...,un )
Power Law
€ € • Each Parameter u1,..., un Appears to at Most the First Power • Equivalent to Multiaffine Axiom
Examples (Existence) Monomials P(t) = 1 ⇒
p(u1 ,u2 ,u3 ) = 1
P(t) = t ⇒
p(u1 ,u2 ,u3 ) =
P(t) = t 2
u1 + u2 + u3 3
⇒ p(u1 ,u2 ,u3 ) =
u1u2 + u2 u3 + u3u1 3
P(t) = t 3 ⇒ p(u1 ,u2 ,u3 ) = u1u2 u3 Cubics P(t) = a3 t 3 + a2 t 2 + a1t + a0 p(u1 ,u2 ,u3 ) = a3 u1u2 u3 + a2
u1u2 + u2 u3 + u3 u1 u + u + u3 + a1 1 2 + a0 3 3
Blossoming Diagrams -- Multiaffine Property
p(t 2 ,t 3,u)
t4 − u t 4 − t1 €
€
€
€
u − t1 t 4 − t1
p(t1, t 2,t 3 )
p(t 2 ,t 3,t 4 )
Normalized
€
€
u=
t 2 t 3u
€
t4 − u
u − t1
€ t1t2 t3
t2 t3t4
Unnormalized €
t 4 − u € u − t1 t −u u − t1 t1 + t 4 ⇒ p(t2 ,t3 ,u) = 4 p(t1 ,t2t 3 ) + p(t2 ,t3 ,t 4 ) t4 − t1 t4 − t1 t4 − t1 t 4 − t1
p(t2 ,t3 ,u) =
t4 − u u − t1 t −u u − t1 p(t1,t2t3 ) + p(t2 ,t3 ,t4 ) ⇔ t2t 3u = 4 t1t 2t3 + t2t3t 4 t 4 − t1 t4 − t1 t 4 − t1 t4 − t1
Blossoming Diagrams -- Multiaffine Property
t2 Ltn u u − t1
tn+1 − u € t1 Lt n
t2 Ltn+1
€
€
t −u u − t1 u = n+1 t1 + t n+1 tn+1 − t1 tn+1 − t1
€
€ t −u u − t1 p(t2 ,K,tn ,u) = n+1 p(t1,K,tn ) + p(t2 ,K,tn+1 ) t − t t − t n+1 1 n+1 1 €
€
€
t2 Ltn u =
tn+1 − u u − t1 t1 Ltn + t2 Ltn+1 tn+1 − t1 tn+1 − t1
Boehm’s Algorithm
New Control Points
t4 − u
€
t1t2 t3
t 4t5u
t3t 4u
t2 t3u u − t1 t5 − u
t −u u − t2 6
t2 t3t4
t3t 4t5
Original Control Points Original Knot Sequence:
€ New Knot Sequence:
€ €
K,t1,t 2 ,t 3 ,t 4 ,t5 ,t 6 ,K K,t1,t 2 ,t 3 ,u,t 4 ,t 5 ,t6 ,K
u − t3 t 4t5t6
Oslo Algorithm New Control Point
u1u2u3 t 4 − u3
u3 − t3
t3u1u2
t 4u1u2 u2 − t 2 t5 − u2
t 4 − u2
t2 t3u1
u2 − t3
t3t 4u1
t 4t5u1
t 4 − u1
u1 − t1 t5 − u1
t −u u1 − t2 6 1
t1t2 t3
t2 t3t4
t3t 4t5
u1 − t3 t 4t5t6
Original Control Points Original Knot Sequence: New Knot Sequence:
€
€
K,t1,t 2 ,t 3 ,t 4 ,t5 ,t 6 ,K K,t1,t 2 ,t 3 ,u1,K,u d ,t 4 ,t5 ,t 6 ,K
Part III: Global Knot Insertion Algorithms
Chaikin’s Algorithm: Quadratic B-splines -- Uniform Knots
3P0 + P1 4
€
P0
€ P 0
€
€
€
€
€
€
P1 + 3P2 4
3P2 + P3 4
P2 + 3P3 4
L
€
P0 + P1 € 2
P0
3P1 + P2 4
P0 + 3P1 4
P1
P1 €
€
€
P1 + P2€ 2
P1
€
P2
P2
Split and Average € €
€
€ P2 + P3 2
P3
€
€ P3
P2
€
€ €
L P3
L
Lane-Riesenfeld Algorithm: Cubic B-splines -- Uniform Knots 4P0 + 4P1 8
€
€
€ P 0
€
3P0 + P1€ 4
P0
P0 + 3P1€ 3P1 + P2€ P1 + 3P2 4 4 4
€ P0 + P1 € 2
€ P0
€
P0 +6P1 + P2 4P1 + 4P2 P1 + 6P2 + P3 4P2 + 4P3 8 8 8 8
€
€
P1 + P2 € 2
P1 €
P1 €
€
P1
€
€3P + P 2 3 4
P2
P2
Split and Average € €
€
€
€ P2 + P3 2
L
P2 + 3P3 4 €
P3
€ € P3
P2
€
€ €
L
L P3
L
Proofs of Lane-Riesenfeld Algorithm 1. Continuous Convolution of B-spline Basis Functions J. LANE and R. RIESENFELD (1980): A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 2, pp. 35–46. 2. De Boor Recurrence (Induction) R. GOLDMAN and J. WARREN (1993): An extension of Chaikin’s algorithm to B-spline curves with knots in geometric progression, CVGIP: Graphical Models and Image Processing, Vol. 35, pp. 58-62. 2′. De Boor Recurrence (Induction) ⇒ Oslo Algorithm H. PRAUTZSCH (1984): A short proof of the Oslo Algorithm, Computer Aided €
Geometric Design, Vol. 1, pp. € 95-96.
New Proof of Lane-Riesenfeld Algorithm 3. Blossoming E. VOUGA and R. GOLDMAN (2006): Two Blossoming Proofs of the Lane-Riesenfeld Algorithm -- Dagstuhl Conference on Geometric Modeling 2005, to appear in Computing.
€
Lane-Riesenfeld Algorithm -- Blossoming Interpretation Quadratic B-splines
1 2 ,1
01
€
€ 01 €
11 €
01
12 €
12
€
€
112 ,2
1,112
23 €
22 € 12
€
€
212 ,3
2,212
34
33 23
23 €
3,312
€
34
Uniform Knots
€
€
€
€
€
€
€
34
Lane-Riesenfeld Algorithm Quadratic B-splines 1. Double the Control Points 2. Convert to Piecewise Bezier Form 3. Insert New Knots Using Boehm’s Knot Insertion Algorithm
Lane-Riesenfeld Algorithm -- Quadratic B-splines
u0t1
€
t0t1
t0t1 €
€
€
€
t1u1 t1t1 €
€ t0t1
€ €
€
t1t2 €
t1t2 €
€
t2u2
u1t2
u=
t2t2 € t1t2
€
t2t 3 €
Arbitrary Knots
u2t 3 t2t 3 €
t3 u 3
t3t 3 t2t 3 €
tk+1 − u u − tk t + t k+1 € k € tk+1 − tk t k+1 − tk €
t 3t 4
t3t 4
€
t 3t 4
Schaefer’s Algorithm -- Quadratic B-splines
u0t1
€
t0t1
t0t1 €
€
u0t1 €
t0t1
€
€
t1u1
u1t2
t1t2 €
t2 u 2
u1t2 €
t1t2 t2t 3 t1t2 €Arbitrary € € Knots
u2t 3 t2t 3
€
L
u2t 3 €
t2t 3
L t 3t 4 €
€
€
€
€
u= €
tk+1 − u u€ − tk t + t k+1€ € k tk+1 − tk t k+1 − tk
L
Quadratic B-splines Schaefer’s Algorithm •
Inserts New Knots in First Round
•
Not Necessary to Convert to Piecewise Bezier Form
•
Faster than Lane-Riesenfeld -- Half the Work in the Second Round
Knot Insertion: Quadratic B-splines
t2t 3 t2u 2 €
€ €
u2t 3
t3u 3
t3t 4 u3t 4
€
€ t4 u 4
€
u1t2
t4 t 5
t1t2
€
€ € €
Lane − Riesenfeld:
= t k t k+1
Schaefer:
= u k t k+1
€ €
€
€
Lane-Riesenfeld Algorithm -- Cubic B-splines
? 112 , 2,212
1,112 ,2
€
3 4 ,1,2
€
€
012
012 €
€
4 ,1,2 €
€
012
€
€9
€
2,212 ,3
7 4 ,2,3
€
€ 13 4
3,312 ,4 L
3,4,17 4 L
,2, 3 3,11 4 ,4 € €
123 €
234 345 L 2,212 ,3 3,312 ,4 € € € 123 234 234 345 345 € € € €
€
€
1,112 ,2 € €
? 212 , 2,312
123 €
Uniform Knots
€
€
€
€
L
Lane-Riesenfeld Algorithm -- Cubic B-splines
?
€
€ 9 3 ,1,2 ,1,2 4 4 € €
012 €
€ 012
€
1,112 ,2 €
012 €
€
€(94 ,1,2) = 1€(1,112 ,2) + 1 (1,2, € 3)
2 2 1 1 ( 74 ,2, 3) = (1,2,3) + (2,212 ,3) 2 2 €
€ €
2,212 ,3
112 ,2,212
1,112 ,2
7 4 ,2,3
123 €
123 €
13 4 ,2, 3
2,212 ,3
123 €
234 234
234
1 1 2 1 (1,12 ,2) + (1,2, 3) + (2,212 ,3) 4 4 4 € 1 € 32 € 1 ⇒ ? = (1,112 ,2) + (1,2, 3) + (2,212 ,3) 4 4 3 3 1 3 ? = (1,112 ,2) + (112 ,2, 3) = 112 ,2,2 12 4 4 ?=
(
)
€
Lane-Riesenfeld Algorithm -- Quartic B-splines
??
€
€
1223
1123
0123 €
€ 0123 €
€
1233
1223 €
0123 €
€
€
??
??
?
€
€
2234
1234 €
1234 €
2334
?
€
2344 € €
2334 €
1234 €
?? 3445 3345
3455 3456
2345 3445 €
2345 €
2345 €
3456
Uniform Knots
€
€
€
€
€
€
€
3456
Lane-Riesenfeld Algorithm •
Build Algorithm for Next Degree Atop Algorithm for Previous Degree
•
Append One Additional Round of Averaging
•
Harder and Harder to Prove by Blossoming
€
Schaefer’s Algorithm -- Cubic B-splines
t1u1t 2
t0t1t2 €
€
€ t0t1t2 € €
t1u1t 2 u1t2 t 3 t2u 2t 3 € € €
u0t1t2 €
€
u2t 3t 4
L
€
t3u 3t 4
L
€
u0t1t2 €
t1t2 t 3 u1t2 t 3 t2t 3t 4 u2t 3t 4 t t t 3 4 5 L € € € t1t2 t 3 t1t2 t 3 t2t 3t 4 t2t 3t 4 t3t 4 t5€ t3t 4 t5 € € € Arbitrary Knots €
t0t1t2 € €
u1t2 u2 t2u 2t 3 u2t 3u 3 t3u 3t 4
€ €
u −tj tk − u u= tj + tk t − t t − t € k j€ k j €
€
€
L
Schaefer’s Algorithm Cubic B-splines •
Build Atop Algorithm for Quadratic B-splines
•
Append Next Original Knot to Each of the Blossoms on the First Two Stages -- Example: u0t1 → u0t1t2
•
Promote Every Other Point to the Next Stage with No Additional Computation
€ New Knots Using the Multiaffine Property of the Blossom • Introduce •
Easy to Prove by Blossoming
€
Alternative Algorithm -- Cubic B-splines
t1u1t 2 u1t2 u2 t2u 2t 3 u2t 3u 3 t3u 3t 4 L u0t1t2 €
€
u2t 3t 4 t3u 3t 4 L €
t2t 3t 4 t3t 3t 4 t3t 4 t5 L € € t1t2 t 3 t2t 3t 4 t2t 3t 4 t3t 4 t5 t3t 4 t5 L € € Arbitrary Knots €
t0t1t2 €
t1t1t 2 t1t2 t 3 € €
t0t1t2 t0t1t2 € €
t1t2 t 3 €
€ €
t1u1t 2 u1t2 t 3 t2u 2t 3 € € €
€
t2t 2t 3 €
€
€
€
€ €
u=
€
u −tj tk − u tj + tk tk − t j€ t k − t j €
€
Problem Observation •
Schaefer’s Algorithm does not Reduce to the Lane-Riesenfeld Algorithm when the Knots are Uniformly Spaced.
Questions •
Does there Exist a Global Knot Insertion Algorithm that Reduces to the Lane-Riesenfeld Algorithm for Uniform Knots?
•
If Such an Algorithm Exists, is it Unique?
Lane-Riesenfeld Algorithm: Cubic B-splines -- Arbitrary Knots t2u2t 3
u2t 3u3
v3 − u2 v3 − w2
t4 − t 3 € t4 − t2
u1t2 u2
€t t v 12 2
€
€ €
€ €
€
t0t1t2
t3 − t2 t3 − t1
t3 − u2 € t −t 3 2 €
t1u1t2
t3 − u1 t3 − t 0 €
vk = t k +1 − €
€t t w 23 2 t2 − u1 €t2 − t1
t1t2t 3
€ t1t2t 3
t2t3v3
t3t4 w3
L
t4 − u3 t3 − u2 €t − t € 4 3 t3 − t 2 €t u t € 2 2 3 t2t3t 4 t3u 3t 4 t4 − u2 € t4 − t1
€
€
L
t1t2t 3
(tk +1 − u k−1 )(t k+1 € − uk ) (t€ k +1 − tk ) €
€ € t2t3t 4
€
t2t3t 4
t€5 − u3 t5 − t 2
L t3t4 t5
(u − t )(t − u € ) wk = uk − € k k−1 k k −1 . (t − t ) k€ k−1 €
€
L
Lane-Riesenfeld Algorithm: Cubic B-splines -- Arbitrary Knots t2u2t 3
u1t2 u2
α2 t1t2€ (t0 + 3t 3 ) / 4
L
u2t 3u3
1− α2 1− γ 2
γ2 α3
€t t v 23 2
€ t2t3 (t1 + 3t 4 ) / 4
€
1− α 3 t3t4 v3
€
L
€ 1−€β2 β2 € 1 € 1€ β3 1− β3 € 2 2 € € € € t1t2 (t0 + t 3 ) / 2 t2t3 (t1 + t 4 ) / 2 t1t2t 3 t2t3t 4 t3t4 (t2 + t 5 ) / 2 € € € € 1 1 €1 1 € 1 € 2 2 2 2 2 € € € € t0t1t2 t1t2t 3 t1t2t 3 t2t3t 4 t2t3t 4 t3t4 t5
L
€
€ €
€ €
€
€ €
€
€ €
€
€
L
Parameters for Lane-Riesenfeld Algorithm
€
€
αk =
4(tk +1 − u k−1 )(t k+1 − uk ) (t k+1 − t k−2 )(t k+1 − t k−1 )
βk =
γk =
4(uk − vk ) t k−1 + 3t k +2 − 4vk
vk = t k −1 +
€
€
2(uk − t k−1 )(u k−1 − t k−1 ) (tk +2 − t k−1 )(t k+1 − t k−1 )(1− α k )
βk (tk +2 − t k−1 ) . 2
Summary Global Knot Insertion Algorithms •
Exist for Arbitrary Knot Sequences
•
Are Easily Derived from Blossoming
Open Problems 1. Find a Global Knot Insertion Algorithm for Arbitrary Degree that Reduces to the Lane-Riesenfeld Algorithm for Uniform Knots.
2. Find the Simplest Knot Insertion Algorithm for Arbitrary Degree that Reduces to the Lane-Riesenfeld Algorithm for Uniform Knots. •
How Complicated are the Labels Along the Edges?
