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

▶
imup imup a maple package for uniformity improved
SMART_READER_LITE
LIVE PREVIEW

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

Dec 02, 2023 281 likes •714 views

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,

slide-1
SLIDE 1

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

slide-2
SLIDE 2

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 2 / 30

slide-3
SLIDE 3

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 3 / 30

slide-4
SLIDE 4

Angular Speed Uniformity

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 4 / 30

slide-5
SLIDE 5

Angular Speed Uniformity

Given a parameterization p = (x, y) : [0, 1] → R2, let θp = arctan y′ x′ , ωp =

  • θ′

p

  • ,

µp = 1

0 ωp(t) dt,

σ2

p

= 1

0 (ωp(t) − µp)2 dt.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 4 / 30

slide-6
SLIDE 6

Angular Speed Uniformity

Given a parameterization p = (x, y) : [0, 1] → R2, let θp = arctan y′ x′ , ωp =

  • θ′

p

  • ,

µp = 1

0 ωp(t) dt,

σ2

p

= 1

0 (ωp(t) − µp)2 dt.

Definition (Angular Speed Uniformity)

The angular speed uniformity up of a parameterization p is defined as up = 1 1 + σ2

p/µ2 p

when µp = 0. Otherwise, up = 1.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 4 / 30

slide-7
SLIDE 7

Angular Speed Uniformity

ωp =

  • θ′

p

  • = |x′y′′ − x′′y′|

x′2 + y′2 , σ2

p =

1 (ωp(t) − µp)2 dt, up = 1 1 + σ2

p/µ2 p

up ∈ (0, 1]; When up = 1, ωp is uniform; ωp = κ · ν, where κ is the curvature and ν is the speed at a point; up measures the goodness of a parameterization p.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 5 / 30

slide-8
SLIDE 8

Examples: Angular Speed Uniformity

“Bad” up1 ≈ 0.482 “Good” up2 ≈ 0.977

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 6 / 30

slide-9
SLIDE 9

Arc-angle Parameterization

Definition (Arc-angle Parameterization)

If up = 1, then p is called a uniform parameterization or an arc-angle parameterization.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 7 / 30

slide-10
SLIDE 10

Arc-angle Parameterization

Definition (Arc-angle Parameterization)

If up = 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 up = 1.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 7 / 30

slide-11
SLIDE 11

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

slide-12
SLIDE 12

Arc-angle Reparameterization

Question

How to compute an arc-angle reparameterization p∗ if p is not an arc-angle one?

Theorem

Let ψp(t) = 1 µp t ωp(t) dt and rp = ψ−1

p , then up◦rp = 1, i.e. p ◦ rp is an arc-angle reparameteriza-

tion of p. Such rp is called a uniformizing parameter transformation.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 8 / 30

slide-13
SLIDE 13

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

slide-14
SLIDE 14

Rational Approximation of Arc-angle Reparameterization

Q: Is arc-angle parameterization rational? A: The answer is “No” except for straight lines.

Problem

Given: p ∈ Q(t)2 Find: a rational p∗ such that up∗ ≈ 1 or equivalently a rational r such that up◦r ≈ 1

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 9 / 30

slide-15
SLIDE 15

Rational Approximation of Arc-angle Reparameterization

Q: Is arc-angle parameterization rational? A: The answer is “No” except for straight lines.

Problem

Given: p ∈ Q(t)2 Find: a rational p∗ such that up∗ ≈ 1 or equivalently a rational r such that up◦r ≈ 1

Two Approaches

One-piece rational functions of high degree e.g. Weierstrass approximation Piecewise rational functions of low degree e.g. Piecewise M¨

  • bius transformation ✔
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 9 / 30

slide-16
SLIDE 16

Piecewise M¨

  • bius Transformation

Notation

Let T = (t0, . . . , tN), S = (s0 . . . , sN), α = (α0, . . . , αN−1) where 0 = t0 < · · · < tN = 1, 0 = s0 < · · · < sN = 1, and 0 < α0, . . . , αN−1 < 1.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 10 / 30

slide-17
SLIDE 17

Piecewise M¨

  • bius Transformation

Notation

Let T = (t0, . . . , tN), S = (s0 . . . , sN), α = (α0, . . . , αN−1) where 0 = t0 < · · · < tN = 1, 0 = s0 < · · · < sN = 1, and 0 < α0, . . . , αN−1 < 1.

Definition (Piecewise M¨

  • bius Transformation)

A map m is called a piecewise M¨

  • bius transformation if

m(s) =        . . . mi(s), if s ∈ [si, si+1]; . . . where mi(s) = ti + ∆ti (1 − αi)˜ s (1 − αi)˜ s + (1 − ˜ s)αi and ∆ti = ti+1 − ti, ∆si = si+1 − si, ˜ s = (s − si)/∆si.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 10 / 30

slide-18
SLIDE 18

Remarks

m(s) is C0 continuous and thus called C0 piecewise M¨

  • bius trans-

formation. When N = 1, it degenerates to an α-M¨

  • bius transformation.

If m satisfies m′

i(si+1) = m′ i+1(si+1),

it becomes a C1 piecewise M¨

  • bius 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

slide-19
SLIDE 19

Rational Approximation of Arc-angle Reparameterization

Sub-Problem A

Given: p ∈ Q(t)2 which is not a straight line, N the number of pieces Find: a C0 N-piecewise M¨

  • bius transformation m such that up◦m is
  • ptimal
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 12 / 30

slide-20
SLIDE 20

Rational Approximation of Arc-angle Reparameterization

Sub-Problem A

Given: p ∈ Q(t)2 which is not a straight line, N the number of pieces Find: a C0 N-piecewise M¨

  • bius transformation m such that up◦m is
  • ptimal

Sub-Problem B

Given: p ∈ Q(t)2 which is not a straight line, ¯ u an object uniformity Find: a C1 piecewise M¨

  • bius transformation m such that up◦m is close

to ¯ u

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 12 / 30

slide-21
SLIDE 21

Rational Approximation of Arc-angle Reparameterization

Sub-Problem A

Given: p ∈ Q(t)2 which is not a straight line, N the number of pieces Find: a C0 N-piecewise M¨

  • bius transformation m such that up◦m is
  • ptimal

Sub-Problem B

Given: p ∈ Q(t)2 which is not a straight line, ¯ u an object uniformity Find: a C1 piecewise M¨

  • bius transformation m such that up◦m is close

to ¯ u Assumption: the angular speed ωp is nonzero over [0, 1]

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 12 / 30

slide-22
SLIDE 22

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 13 / 30

slide-23
SLIDE 23

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

  • ptimal C0 piecewise reparameterization, CASC 2012

Yang, Wang, and Hong: Improving angular speed uniformity by C1 piecewise reparameterization, ADG 2012

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 14 / 30

slide-24
SLIDE 24

C0 Piecewise Reparameterization

Let T be an arbitrary but fixed sequence. Then the globally optimal α and S are computed by αi = (αi)T = 1 1 +

  • Ci/Ai

and si = (si)T = i−1

k=0

√Mk N−1

k=0

√Mk , where Ai = ti+1

ti

ω2

p(t) · (1 − ˜

t)2 dt, Bi = ti+1

ti

ω2

p(t) · 2 ˜

t(1 − ˜ t) dt, Ci = ti+1

ti

ω2

p(t) · ˜

t2 dt, Mk = ∆tk

  • 2
  • AkCk + Bk
  • ,

˜ t = (t − ti)/∆ti.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 15 / 30

slide-25
SLIDE 25

C0 Piecewise Reparameterization

Let T be an arbitrary but fixed sequence and mT denote the optimal

  • transformation. Then

up◦mT = µ2

p

φ2

p

, where φp =

N−1

  • i=0
  • ∆ti
  • 2
  • AiCi + Bi
  • .
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 16 / 30

slide-26
SLIDE 26

C0 Piecewise Reparameterization

Let T be an arbitrary but fixed sequence and mT denote the optimal

  • transformation. Then

up◦mT = µ2

p

φ2

p

, where φp =

N−1

  • i=0
  • ∆ti
  • 2
  • AiCi + Bi
  • .

max up◦mT s.t. 0 < t1 < · · · < tN−1 < 1 ⇔ min φp s.t. 0 < t1 < · · · < tN−1 < 1

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 16 / 30

slide-27
SLIDE 27

C0 Piecewise Reparameterization

Let T be an arbitrary but fixed sequence and mT denote the optimal

  • transformation. Then

up◦mT = µ2

p

φ2

p

, where φp =

N−1

  • i=0
  • ∆ti
  • 2
  • AiCi + Bi
  • .

max up◦mT s.t. 0 < t1 < · · · < tN−1 < 1 ⇔ min φp s.t. 0 < t1 < · · · < tN−1 < 1 ⇑ Zoutendijk’s method of feasible directions

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 16 / 30

slide-28
SLIDE 28

C0 Piecewise Reparameterization

Let T be an arbitrary but fixed sequence and mT denote the optimal

  • transformation. Then

up◦mT = µ2

p

φ2

p

, where φp =

N−1

  • i=0
  • ∆ti
  • 2
  • AiCi + Bi
  • .

max up◦mT s.t. 0 < t1 < · · · < tN−1 < 1 ⇔ min φp s.t. 0 < t1 < · · · < tN−1 < 1 ⇑ Zoutendijk’s method of feasible directions Procedure: (1) construct φp; (2) locally optimize T; (3) globally

  • ptimize α and S; (4) construct optimal m and p∗ = p ◦ m.
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 16 / 30

slide-29
SLIDE 29

C1 Piecewise Reparameterization

  • 1. Construct a near optimal C1 piecewise M¨
  • bius transformation as

follows.

1.1. Compute a partition T of the unit interval [0, 1] by solving ω′

p(t) = 0

for t over (0, 1). 1.2. Choose si = ti ωp dt

  • µp (1 ≤ i ≤ N − 1)

to obtain S. 1.3. Compute the exact optimal α using T and S. 1.4. Construct a C1 piecewise M¨

  • bius transformation from (T, S, α).
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 17 / 30

slide-30
SLIDE 30

C1 Piecewise Reparameterization

  • 2. Iterative process

◮ Traditional approach: construct

p1 = p ◦ m1 = ⇒ p2 = p1 ◦ m2 = ⇒ · · · = ⇒ pn = pn−1 ◦ mn = p ◦ m1 ◦ · · · ◦ mn such that up1 < up2 < · · · < upn until upn is a desirable uniformity.

◮ Alternative approach: refine T at each iteration by solving

ω′

p · ∆ti ·

  • αi˜

t + (1 − αi)(1 − ˜ t)

  • − 2 ωp · (1 − 2 αi) = 0

for t, where T and α are sequences computed before.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 18 / 30

slide-31
SLIDE 31

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 19 / 30

slide-32
SLIDE 32

Architecture of ImUp

C0 C1 C0 C1 C1 C0

Figure: ImUp structure

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 20 / 30

slide-33
SLIDE 33

Architecture of ImUp

C0 C1 , ) N

, ( p

, 1 ) ~ u , ( p T S T

  • S

Figure: Module of M¨

  • bius transformation
  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 21 / 30

slide-34
SLIDE 34

Public Interface

Function Calling Sequence AngularSpeed AngularSpeed(p, m) Uniformity Uniformity(p, m) MoebiusTransformation MoebiusTransformation(p, opt, N|¯ u) ReparameterizationN ReparameterizationN(p, N) ReparameterizationU ReparameterizationU(p, ¯ u) ImUpPlotN ImUpPlotN(p, N, Npt) ImUpPlotU ImUpPlotU(p, ¯ u, Npt)

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 22 / 30

slide-35
SLIDE 35

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 23 / 30

slide-36
SLIDE 36

Plot with C0 Piecewise Reparameterization

Let p =

  • t3 − 6 t2 + 9 t − 2

2 t4 − 16 t3 + 40 t2 − 32 t + 9, t2 − 4 t + 4 2 t4 − 16 t3 + 40 t2 − 32 t + 9

  • ,

N = 3 and Npt = 40. Original p C0 reparameterization p∗ ωp and ωp∗

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 24 / 30

slide-37
SLIDE 37

Plot with C1 Piecewise Reparameterization

Let p =

  • t3 − 6 t2 + 9 t − 2

2 t4 − 16 t3 + 40 t2 − 32 t + 9, t2 − 4 t + 4 2 t4 − 16 t3 + 40 t2 − 32 t + 9

  • ,

¯ u = 0.99 and Npt = 40. Original p C1 reparameterization p∗ ωp and ωp∗

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 25 / 30

slide-38
SLIDE 38

Experimental Results

Table: Experimental data with ReparameterizationN and ReparameterizationU. d = degree, u = uniformity, N = number of pieces, T = time (seconds). Curve d Original Reparameter- Reparameter- Reparameter- u izationN(p, 1) izationN(p, 3) izationU(p, 0.9) u T u T u N T C1 8 0.808 0.808 0.047 0.886 0.625 0.991 6 0.265 C2 12 0.960 0.960 0.016 1.000 0.203 0.960 1 0.016 C3 6 0.906 0.919 0.031 1.000 0.219 0.906 1 0.016 C4 6 0.796 0.796 0.031 0.911 0.437 0.996 8 0.250 C5 6 0.879 0.879 0.015 0.961 0.328 0.997 4 0.110 C6 50 0.647 0.706 1.422 0.973 8.203 0.971 5 10.578 C7 70 0.181 0.418 3.157 0.550 20.204 0.960 6 31.390 C8 100 0.184 0.184 6.188 0.989 119.141 0.970 2 95.828 C9 120 0.682 0.683 9.109 0.999 66.813 0.994 2 37.188 C10 150 0.253 0.479 29.765 – > 3000 – – > 3000

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 26 / 30

slide-39
SLIDE 39

Outline

1

Problem

2

Methods

3

Implementation

4

Examples and Experiments

5

Summary

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 27 / 30

slide-40
SLIDE 40

Summary

Introduce the definition of angular speed uniformity for parameteriza- tions of plane curves; Prove the existence of arc-angle (maybe irrational) parameterizations; Propose two methods to improve the uniformity of angular speed; Present a software package for computing piecewise rational reparam- eterizations with improved uniformities.

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 28 / 30

slide-41
SLIDE 41

Future Work

Investigate efficient methods for computing a rational approximation

  • f the arc-angle reparameterization for any given rational parameter-

ization whose angular speed may become zero over [0, 1]; Generalize the existing methods to parametric space curves and par- ametric surfaces...

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 29 / 30

slide-42
SLIDE 42

Thanks for your attention!

  • J. Yang (BUAA/NCSU)

ASCM 2012 27 October, 2012 30 / 30