ImUp ImUp: A Maple Package for Uniformity-Improved - PowerPoint PPT Presentation

ImUp ImUp: A Maple Package for Uniformity-Improved Reparameterization of Plane Curves Jing Yang LMIB – Beihang University Dongming Wang CNRS – Universit´ e Pierre et Marie Curie Hoon Hong North Carolina State University 27 October, ASCM 2012 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 1 / 30

Outline Problem 1 Methods 2 Implementation 3 Examples and Experiments 4 Summary 5 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 2 / 30

Outline Problem 1 Methods 2 Implementation 3 Examples and Experiments 4 Summary 5 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 3 / 30

Angular Speed Uniformity J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 4 / 30

Angular Speed Uniformity Given a parameterization p = ( x, y ) : [0 , 1] → R 2 , let arctan y ′ � , � � θ p = x ′ , ω p = � θ ′ p � 1 � 1 0 ( ω p ( t ) − µ p ) 2 dt. σ 2 µ p = 0 ω p ( t ) dt, = p J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 4 / 30

Angular Speed Uniformity Given a parameterization p = ( x, y ) : [0 , 1] → R 2 , let arctan y ′ � , � � θ p = x ′ , ω p = � θ ′ p � 1 � 1 0 ( ω p ( t ) − µ p ) 2 dt. σ 2 µ p = 0 ω p ( t ) dt, = p Definition (Angular Speed Uniformity) The angular speed uniformity u p of a parameterization p is defined as 1 u p = 1 + σ 2 p /µ 2 p when µ p � = 0 . Otherwise, u p = 1 . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 4 / 30

Angular Speed Uniformity � 1 � = | x ′ y ′′ − x ′′ y ′ | 1 ( ω p ( t ) − µ p ) 2 dt, σ 2 � � θ ′ � ω p = , p = u p = x ′ 2 + y ′ 2 p 1 + σ 2 p /µ 2 0 p u p ∈ (0 , 1] ; When u p = 1 , ω p is uniform; ω p = κ · ν , where κ is the curvature and ν is the speed at a point; u p measures the goodness of a parameterization p . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 5 / 30

Examples: Angular Speed Uniformity “Bad” “Good” u p 1 ≈ 0 . 482 u p 2 ≈ 0 . 977 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 6 / 30

Arc-angle Parameterization Definition (Arc-angle Parameterization) If u p = 1 , then p is called a uniform parameterization or an arc-angle parameterization . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 7 / 30

Arc-angle Parameterization Definition (Arc-angle Parameterization) If u p = 1 , then p is called a uniform parameterization or an arc-angle parameterization . Example The parameterization p = (cos t, sin t ) is an arc-angle parameterization since u p = 1 . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 7 / 30

Arc-angle Reparameterization Question How to compute an arc-angle reparameterization p ∗ if p is not an arc-angle one? J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 8 / 30

Arc-angle Reparameterization Question How to compute an arc-angle reparameterization p ∗ if p is not an arc-angle one? Theorem Let � t ψ p ( t ) = 1 ω p ( t ) dt µ p 0 and r p = ψ − 1 p , then u p ◦ r p = 1 , i.e. p ◦ r p is an arc-angle reparameteriza- tion of p . Such r p is called a uniformizing parameter transformation . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 8 / 30

Rational Approximation of Arc-angle Reparameterization Q: Is arc-angle parameterization rational? A: The answer is “No” except for straight lines. J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 9 / 30

Rational Approximation of Arc-angle Reparameterization Q: Is arc-angle parameterization rational? A: The answer is “No” except for straight lines. Problem p ∈ Q ( t ) 2 Given: a rational p ∗ such that u p ∗ ≈ 1 or equivalently a rational r such Find: that u p ◦ r ≈ 1 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 9 / 30

Rational Approximation of Arc-angle Reparameterization Q: Is arc-angle parameterization rational? A: The answer is “No” except for straight lines. Problem p ∈ Q ( t ) 2 Given: a rational p ∗ such that u p ∗ ≈ 1 or equivalently a rational r such Find: that u p ◦ r ≈ 1 Two Approaches One-piece rational functions of high degree e.g. Weierstrass approximation Piecewise rational functions of low degree obius transformation ✔ e.g. Piecewise M¨ J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 9 / 30

Piecewise M¨ obius Transformation Notation Let T = ( t 0 , . . . , t N ) , S = ( s 0 . . . , s N ) , α = ( α 0 , . . . , α N − 1 ) where 0 = t 0 < · · · < t N = 1 , 0 = s 0 < · · · < s N = 1 , and 0 < α 0 , . . . , α N − 1 < 1 . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 10 / 30

Piecewise M¨ obius Transformation Notation Let T = ( t 0 , . . . , t N ) , S = ( s 0 . . . , s N ) , α = ( α 0 , . . . , α N − 1 ) where 0 = t 0 < · · · < t N = 1 , 0 = s 0 < · · · < s N = 1 , and 0 < α 0 , . . . , α N − 1 < 1 . Definition (Piecewise M¨ obius Transformation) A map m is called a piecewise M¨ obius transformation if  . . .    m ( s ) = m i ( s ) , if s ∈ [ s i , s i +1 ]; .  .  .  where (1 − α i )˜ s m i ( s ) = t i + ∆ t i (1 − α i )˜ s + (1 − ˜ s ) α i and ∆ t i = t i +1 − t i , ∆ s i = s i +1 − s i , ˜ s = ( s − s i ) / ∆ s i . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 10 / 30

Remarks m ( s ) is C 0 continuous and thus called C 0 piecewise M¨ obius trans- formation . When N = 1 , it degenerates to an α -M¨ obius transformation. If m satisfies m ′ i ( s i +1 ) = m ′ i +1 ( s i +1 ) , it becomes a C 1 piecewise M¨ obius transformation. Different choices of T, S, α produce different m ( s ) . Thus m is represented as ( T, S, α ) . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 11 / 30

Rational Approximation of Arc-angle Reparameterization Sub-Problem A p ∈ Q ( t ) 2 which is not a straight line, Given: N the number of pieces a C 0 N -piecewise M¨ Find: obius transformation m such that u p ◦ m is optimal J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 12 / 30

Rational Approximation of Arc-angle Reparameterization Sub-Problem A p ∈ Q ( t ) 2 which is not a straight line, Given: N the number of pieces a C 0 N -piecewise M¨ Find: obius transformation m such that u p ◦ m is optimal Sub-Problem B p ∈ Q ( t ) 2 which is not a straight line, Given: u an object uniformity ¯ a C 1 piecewise M¨ Find: obius transformation m such that u p ◦ m is close to ¯ u J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 12 / 30

Rational Approximation of Arc-angle Reparameterization Sub-Problem A p ∈ Q ( t ) 2 which is not a straight line, Given: N the number of pieces a C 0 N -piecewise M¨ Find: obius transformation m such that u p ◦ m is optimal Sub-Problem B p ∈ Q ( t ) 2 which is not a straight line, Given: u an object uniformity ¯ a C 1 piecewise M¨ Find: obius transformation m such that u p ◦ m is close to ¯ u Assumption: the angular speed ω p is nonzero over [0 , 1] J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 12 / 30

Outline Problem 1 Methods 2 Implementation 3 Examples and Experiments 4 Summary 5 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 13 / 30

State of Art Yang, Wang, and Hong: Improving angular speed uniformity by reparameterization (revised version under review) Yang, Wang, and Hong: Improving angular speed uniformity by optimal C 0 piecewise reparameterization, CASC 2012 Yang, Wang, and Hong: Improving angular speed uniformity by C 1 piecewise reparameterization, ADG 2012 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 14 / 30

C 0 Piecewise Reparameterization Let T be an arbitrary but fixed sequence. Then the globally optimal α and S are computed by √ M k � i − 1 1 k =0 α i = ( α i ) T = and s i = ( s i ) T = √ M k , � N − 1 � 1 + C i /A i k =0 where � t i +1 � t i +1 t ) 2 dt, ω 2 p ( t ) · (1 − ˜ ω 2 p ( t ) · 2 ˜ t (1 − ˜ A i = B i = t ) dt, t i t i � t i +1 � � t 2 dt, ω 2 p ( t ) · ˜ � C i = M k = ∆ t k 2 A k C k + B k , t i ˜ t = ( t − t i ) / ∆ t i . J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 15 / 30

C 0 Piecewise Reparameterization Let T be an arbitrary but fixed sequence and m T denote the optimal transformation. Then N − 1 u p ◦ m T = µ 2 � p � � � � , where φ p = ∆ t i 2 A i C i + B i . φ 2 p i =0 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 16 / 30

C 0 Piecewise Reparameterization Let T be an arbitrary but fixed sequence and m T denote the optimal transformation. Then N − 1 u p ◦ m T = µ 2 � p � � � � , where φ p = ∆ t i 2 A i C i + B i . φ 2 p i =0 ⇔ max u p ◦ m T min φ p s.t. 0 < t 1 < · · · < t N − 1 < 1 s.t. 0 < t 1 < · · · < t N − 1 < 1 J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 16 / 30

C 0 Piecewise Reparameterization Let T be an arbitrary but fixed sequence and m T denote the optimal transformation. Then N − 1 u p ◦ m T = µ 2 � p � � � � , where φ p = ∆ t i 2 A i C i + B i . φ 2 p i =0 ⇔ max u p ◦ m T min φ p s.t. 0 < t 1 < · · · < t N − 1 < 1 s.t. 0 < t 1 < · · · < t N − 1 < 1 ⇑ Zoutendijk’s method of feasible directions J. Yang (BUAA/NCSU) ASCM 2012 27 October, 2012 16 / 30