Estimating Criteria for for Fitting Fitting B B- -spline Curves - - PowerPoint PPT Presentation

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Equipe MGII GraphiCon98 GraphiCon98 Moscow - Russia Moscow - Russia September 7-11, 1998 September 7-11, 1998 Estimating Criteria for for Fitting Fitting B B- -spline Curves spline Curves: : Estimating Criteria Application to

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

GraphiCon’98 Moscow - Russia September 7-11, 1998 GraphiCon’98 Moscow - Russia September 7-11, 1998

Estimating Criteria Estimating Criteria for for Fitting Fitting B B-

  • spline Curves

spline Curves: : Application Application to to Data Compression Data Compression

Eric SAUX, Eric SAUX, Marc DANIEL Marc DANIEL Institut de Recherche en Informatique de Nantes, France Institut de Recherche en Informatique de Nantes, France

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Outline Outline

  • Survey

Survey of

  • f the present

the present compression compression methods methods

  • Strategies based

Strategies based on polygonal

  • n polygonal curves

curves

  • Strategy based

Strategy based on

  • n spline curves

spline curves

  • A

A new new compression compression strategy strategy

  • Data

Data fitting with fitting with B B-

  • splines

splines

  • Criteria

Criteria for for estimating estimating data approximation data approximation

  • Reduction

Reduction technique technique

  • Result comparison

Result comparison

  • Conclusion

Conclusion

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SLIDE 3

International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Outline Outline

  • Survey

Survey of

  • f the present

the present compression compression methods methods

  • Strategies based

Strategies based on polygonal

  • n polygonal curves

curves

  • Strategy based

Strategy based on

  • n spline curves

spline curves

  • A

A new new compression compression strategy strategy

  • Data

Data fitting with fitting with B B-

  • splines

splines

  • Criteria

Criteria for for estimating estimating data approximation data approximation

  • Reduction

Reduction technique technique

  • Result comparison

Result comparison

  • Conclusion

Conclusion

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Survey of the present methods Survey of the present methods

  • Stategies based

Stategies based on polygonal

  • n polygonal curves

curves

  • The intersecting cones method

The intersecting cones method (E. (E. Arge Arge & M. & M. Daehlen Daehlen) ) Inputs Inputs A polygonal A polygonal curve curve P = P = ( ( p p0

0,……,

,……,p pn

n )

) A A tolerance tolerance ε ε ≥ ≥ 0 A polygonal A polygonal curve curve Q = Q = ( ( q q0

0,……,

,……,q qm

m )

) with with m m < < n n such such as as d d( (P,Q P,Q) ) ≤ ≤ ε ε Output Output General General strategy strategy Intersection of Intersection of cones cones

  • riginated at
  • riginated at p

pi

i

qi pr Cr Dr qi+1=Dr I Uε(pr)

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Survey of the present methods Survey of the present methods

  • Douglas

Douglas and and Peucker’s method without tolerance Peucker’s method without tolerance Rearrange the Rearrange the points of points of P P General General strategy strategy P’ = P’ = ( ( p’ p’0

0,……,

,……,p’ p’n

n )

) P = P = ( ( p p0

0,……,

,……,p pn

n )

)

1 1 2 2 1 3 2 1 3 4

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

Subscripts of points of P ' Tolerance

ε

Multi-scale analysis

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Survey of the present methods Survey of the present methods

  • Stategies based

Stategies based on

  • n spline curves

spline curves

General General strategy strategy Inputs Inputs A polygonal A polygonal curve curve P = P = ( ( p p0

0,……,

,……,p pn

n )

) A A tolerance tolerance ε ε ≥ ≥ 0 An An interpolating interpolating B B-

  • spline

spline f f on

  • n

( )

k n i i

t T

+ =

=

Output Output An An approximating approximating B B-

  • spline

spline g g with with m < n and m < n and τ τ ⊂ ⊂ T T such such as as ( )

k m i i + =

τ = τ ε ≤ −

*

g f

Computation of Computation of weights weights

+ +

* i i

g f − = ω Reconstruction of Reconstruction of the approximating curve the approximating curve

+ + + +

Selection Selection of

  • f knots to be removed

knots to be removed t ti

i can be removed

can be removed if if

ε ≤ ωi

  • The knot removal strategy

The knot removal strategy of T.

  • f T. Lyche

Lyche & K. & K. Morken Morken

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Survey of the present methods Survey of the present methods

  • The knot removal strategy

The knot removal strategy of M.

  • f M. Eck

Eck & J. & J. Hadenfeld Hadenfeld Inputs Inputs Output Output The same The same as in T. as in T. Lyche Lyche and K. and K. Morken’s strategy Morken’s strategy where is where is an an interpolating interpolating B B-

  • spline

spline

( ) ( )

∑

=

=

n i T k i i

t N C t f

, ,

Removal Removal of

  • f knot

knot t ti

i

General General strategy strategy Approximating Bspline Approximating Bspline g gi

i on

  • n μ

μ = = T T -

  • {

{t ti

i}

}

( ) ( )

∑

− = μ

=

1 , , n j k j j i

t N A t g " "forward forward" construction " construction

  • f
  • f

with with

I j

A

i I

g " "backward backward" construction " construction

  • f
  • f

with with

II j

A

i II

g Position of Position of control control point point A Aj

j in

in

+ +

] , [

II j I j A

A AI

2

A II

2

C3

C2

A2

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Outline Outline

  • Survey

Survey of

  • f the present

the present compression compression methods methods

  • Strategies based

Strategies based on polygonal

  • n polygonal curves

curves

  • Strategy based

Strategy based on

  • n spline curves

spline curves

  • A

A new new compression compression strategy strategy

  • Data

Data fitting with fitting with B B-

  • splines

splines

  • Criteria

Criteria for for estimating estimating data approximation data approximation

  • Reduction

Reduction technique technique

  • Result comparison

Result comparison

  • Conclusion

Conclusion

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Data

Data fitting with fitting with B B-

  • splines

splines

Problem Problem Given Given a polygonal a polygonal curve curve P P = ( = ( p p0

0,……,

,……,p pn

n ),

), find find a a curve curve as close as possible of as close as possible of P P

( ) ( )

∑

=

=

m i T k i i

t N Q t f

, ,

Least Least squares squares fitting fitting: : find control find control points points Q Qj

j so that

so that

( ) ( )

∑

=

− ζ

n j j j

p f

2 is

is minimum minimum Householder Householder Choice Choice

  • rder
  • rder k

k parameters parameters ζ i knot vector knot vector T T

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Choice

Choice of a

  • f a parameterization method

parameterization method

uniform uniform h hi

i = constant

= constant cumulative cumulative chord length chord length

i i i

p p h − =

+1

centripetal centripetal

i i i

p p h − =

+1

General General expression expression proposed by Lee proposed by Lee

( )

1 ,

1 1

≤ ≤ − − = ζ = ζ

∑ ∑

= + = +

e p p p p

n j e j j i j e j j i

  • uniform with

uniform with e e = 0 = 0

  • cumulative

cumulative chord length with chord length with e e = 1 = 1

  • centripetal with

centripetal with e e = 0.5 = 0.5

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Choice

Choice of a

  • f a knot vector

knot vector

uniform uniform constant constant spacing spacing from from an extension of De an extension of De Boor Boor formula for interpolation formula for interpolation for for i i = 0,……, = 0,……,m m-

  • k

k ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ + − ζ + + ζ = ζ = = = ζ = = =

+ + + + + −

1 ...... ...... ......

1 2 1 1

2 1

l l t t t t t

l i l i k i n k m m k

Result Result Best results Best results in approximation in approximation with with a a centripetal centripetal parameterization parameterization a a knot vector from the knot vector from the extension extension

  • f De
  • f De Boor

Boor formula formula

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression method A new compression method

Two other parameterizations Two other parameterizations Foley Foley and and Nielson’s parameterization Nielson’s parameterization : : geometrical properties geometrical properties (angles) (angles) Intrinsic Hoschek’s parameterization Intrinsic Hoschek’s parameterization Improvement Improvement of

  • f Hoschek’s method

Hoschek’s method Best Best global approximation global approximation Best Best convergence convergence speed speed Common problem Common problem Oscillation Oscillation problem problem

f(t) pi pi+1 pi+2 f (ζi) f (ζi+1) f (ζi+2)

f(t) pi pi+1 pi+2 f 1(ζi) _ f 1(ζi+1) _ f 1(ζi+2) _ f(t) pi pi+1 pi+2 f (ζi) f (ζi+1) f (ζi+2) _ _ _

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression method A new compression method

  • Criteria

Criteria for for estimating estimating data approximation data approximation

is the fitting is the fitting B B-

  • spline curve

spline curve

( ) ( )

∑

=

=

m i T k i i

t N Q t f

, ,

  • k

k is the order is the order

  • T

T = ( = ( t t0

0,…,

,…,t tk

k-

  • 1

1,

,t tk

k,……,

,……,t tm

m,

,t tm

m+1 +1,…,

,…,t tm

m+k +k )

) is the knot vector is the knot vector f f belongs to linear space belongs to linear space S Sk

k,T ,T

Definitions Definitions

( )

{ } { }

d j m i Q Max Max f N

ji

,......, 1 ; ,......, ; = = =

∞

( ) ( )

1

2 2

+ = ∑

=

m Q f N

m i i

  • Definitions

Definitions of

  • f norms

norms : : N N∞

∞ et

et N N2

2

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression method A new compression method

is the interpolating is the interpolating B B-

  • spline curve

spline curve of initial data

  • f initial data p

pi

i

( ) ( )

∑

=

=

n i T i i

t N R t g

~ , 2 ,

g g belongs to linear space belongs to linear space

T

S

~ , 2

  • rder
  • rder = 2

= 2

  • is the knot vector

is the knot vector

( )

k m m m k k

t t t t t t T

+ + −

= ~ ,..., ~ , ~ ,......, ~ , ~ ,..., ~ ~

1 1

the same order the same order degree elevation degree elevation (H. (H. Prautzsch Prautzsch) )

  • the same knot vector

the same knot vector subdivision subdivision algorithms algorithms ( (Boehm Boehm, Oslo, , Oslo, improved improved Oslo) Oslo)

  • To

To use use N N∞

∞ and

and N N2

2 on

  • n f

f-

  • g

g, B , B-

  • spline curve

spline curve f f and and g g should should have have Polygonal Polygonal curve curve

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

Band Band criterion criterion for B for B-

  • spline

spline approximation approximation

g L ε ε

Result Result Band Band criterion criterion for data approximation for data approximation

( )

ε = −

∞

g f N

pi pi+1 f(ζi) f(ζi+1) 2ε L

Result Result

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Local estimation

Local estimation criteria criteria

Aim Aim To To have a have a local local approach approach of approximation and estimation

  • f approximation and estimation

+ +

Find the best curve Find the best curve segment [ segment [ p pi

i ,

, p pi

i+1 +1 ]

]

pi pi+1 f(ζi) f(ζi+1) pi pi+1 f(ζi) f(ζi+1)

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy + +

Use Use the convex hull the convex hull of B

  • f B-
  • splines

splines

( ) ( ) f t Q N t

j j k T j i k i

=

= − +

∑

, , 1

Qi-k+1 Qi ti ti+1

+ +

Transform this best curve Transform this best curve segment segment into its into its " "Bézier Bézier" " representation representation

pi pi+1 f(ζi+1)=Qj _ _ f(ζi)=Qj-nb+1 _ _ ζi _ ζi+1 _

Apply the Hausdorff metric Apply the Hausdorff metric on

  • n both

both polygonal polygonal curves curves and and

( )

1

,

+

=

i i i

p p P

( )

j nb j j

Q Q Q ,

,...... 1 + −

=

Result Result If If then the then the best curve best curve segment segment is at the most is at the most at at ε ε distance distance from from [ [ p pi

i ,

, p pi

i+1 +1 ]

]

( )

ε = =

j i H

Q P d ,

NOT ENOUGH NOT ENOUGH

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Reduction

Reduction technique technique

General General strategy strategy

  • Best choices

Best choices

  • Uniform knot vector

Uniform knot vector

  • Intrinsic Hoschek’s parameterization applied

Intrinsic Hoschek’s parameterization applied to to a a centripetal one centripetal one

  • Principle

Principle

  • A bissection

A bissection method method on

  • n the number

the number of

  • f

control control points points

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

  • Results comparison

Results comparison

  • n compression rate
  • n compression rate

Results Results

Method Knots Strategy Compression Arge Daehlen 68 % Douglas Peucker 56 % Lyche Morken 13 % De Boor Eck Hadenfeld 51 % B-spline Fitting (e=0.5) 22 % Fittting (Hoschek e=0.5) 61 % Fitting (e=0.5) 49 % Polygonal Uniform

Initial Initial isobathymetric line isobathymetric line Tolerance so that Tolerance so that no no visual difference visual difference

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

Results Results

Method Knots Strategy Compression (Smooth curves) Compression (Complex curves) Arge Daehlen 74 % 61 % Douglas Peucker 65 % 51 % Lyche Morken 24 % 5 % De Boor Eck Hadenfeld 58 % 38 % B-spline Fitting (e=0.5) 34 % 11 % Fittting (Hoschek e=0.5) 68 % 49 % Fitting (e=0.5) 59 % 40 % Polygonal Uniform

Initial Initial smooth line smooth line Initial Initial complex line complex line

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

Results Results

Method Knots Strategy Computation time (Smooth curves) Computation time (Complex curves) Arge Daehlen 0.1 ’’ 2 ’’ Douglas Peucker 0.4 ’’ 3 ’’ Lyche Morken 47’ 08’’ 1h 22’’ De Boor Eck Hadenfeld 40’ 31’’ 59’ 21’’ B-spline Fitting (e=0.5) 1’ 27’’ 3’ 27’’ Fittting (Hoschek e=0.5) 9’ 33’’ 13’ 24’’ Fitting (e=0.5) 1’ 52’’ 4’ 45’’ Polygonal Uniform

  • Results comparison

Results comparison

  • n computation time
  • n computation time
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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

A new compression strategy A new compression strategy

Useful Useful in in embarked embarked cartographic cartographic information information systems systems

  • Advantage when

Advantage when

  • zooming

zooming in in

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Outline Outline

  • Survey

Survey of

  • f the present

the present compression compression methods methods

  • Strategies based

Strategies based on polygonal

  • n polygonal curves

curves

  • Strategy based

Strategy based on

  • n spline curves

spline curves

  • A

A new new compression compression strategy strategy

  • Data

Data fitting with fitting with B B-

  • splines

splines

  • Criteria

Criteria for for estimating estimating data approximation data approximation

  • Reduction

Reduction technique technique

  • Result comparison

Result comparison

  • Conclusion

Conclusion

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

Conclusion Conclusion

We We have have build build Norms Norms N N∞

∞ and

and N N2

2

Local estimation Local estimation criteria criteria We We have have improved improved Hoschek’s Hoschek’s technique technique Which has allowed Which has allowed us us to to introduce introduce Good reduction Good reduction rates rates

  • Raisonable

Raisonable computation computation costs costs

  • A

A new reduction strategy new reduction strategy

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International Conference Graphicon 1998, Moscow, Russia, http:// International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/

Equipe MGII

GraphiCon’98 Moscow - Russia September 7-11, 1998 GraphiCon’98 Moscow - Russia September 7-11, 1998

Estimating Criteria Estimating Criteria for for Fitting Fitting B B-

  • spline Curves

spline Curves: : Application Application to to Data Compression Data Compression

Eric SAUX, Eric SAUX, Marc DANIEL Marc DANIEL Institut de Recherche en Informatique de Nantes, France Institut de Recherche en Informatique de Nantes, France