Equipe MGII GraphiCon’98 GraphiCon’98 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 to Data Compression Data Compression Application Eric SAUX, Marc DANIEL Marc DANIEL Eric SAUX, Institut de Recherche en Informatique de Nantes, France Institut de Recherche en Informatique de Nantes, France International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Outline Outline � Survey Survey of of the present the present compression compression methods methods � � Strategies based Strategies based on polygonal on polygonal curves curves � � Strategy based Strategy based on on spline curves spline curves � A new new compression compression strategy strategy � A � � 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 � International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Outline Outline Survey of of the present the present compression compression methods methods � Survey � � Strategies based Strategies based on polygonal on polygonal curves curves � � Strategy based Strategy based on on spline curves spline curves � A new new compression compression strategy strategy � A � � 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 � International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Survey of the present methods Survey of the present methods � Stategies based Stategies based on polygonal on polygonal curves curves � A polygonal curve curve P = P = ( ( p p 0 ,……, p p n ) A polygonal 0 ,……, n ) Inputs Inputs tolerance ε ε ≥ ≥ 0 A tolerance 0 A A polygonal curve curve Q = Q = ( ( q q 0 ,……, q q m ) A polygonal 0 ,……, m ) Output Output ≤ ε ε ) ≤ with m with m < < n n such such as as d d ( ( P,Q P,Q ) � The intersecting cones method The intersecting cones method (E. (E. Arge Arge & M. & M. Daehlen Daehlen) ) � D r General General C r strategy strategy p r Intersection of cones cones Intersection of originated at p p i originated at i q i q i+ 1 =D r I U ε ( p r ) International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
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 points of points of P P General Rearrange the General strategy strategy P = ( P = ( p p 0 0 ,……, ,……, p p n n ) ) P’ = ( P’ = ( p’ p’ 0 0 ,……, ,……, p’ p’ n n ) ) 2 2 3 Multi-scale analysis 0 0 0 1 1 1 Tolerance 2 2 11 3 3 8 6 ε 7 5 4 10 4 12 9 0 0 3 5 7 9 10 11 0 1 2 4 6 8 12 1 1 Subscripts of points of P ' International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Survey of the present methods Survey of the present methods � Stategies based Stategies based on on spline curves spline curves � � The knot removal strategy The knot removal strategy of T. of T. Lyche Lyche & K. & K. Morken Morken � A polygonal curve curve P = P = ( ( p p 0 ,……, p p n ) A polygonal 0 ,……, n ) Inputs Inputs tolerance ε ε ≥ ≥ 0 A tolerance A 0 ( ) + = n k An interpolating interpolating B B- -spline spline f f on on T t An = i i 0 ( ) + τ = τ m k An approximating approximating B B- -spline spline g g An = i i 0 Output Output τ ⊂ ⊂ T − ≤ ε m < n and τ with m < n and T such such as as f g with * + + General General ω = f − i Computation of weights weights g Computation of i * strategy strategy + + Selection of of knots to be removed knots to be removed Selection ω i ≤ ε t i can be removed if if t i can be removed + + Reconstruction of the approximating curve the approximating curve Reconstruction of International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Survey of the present methods Survey of the present methods � The knot removal strategy The knot removal strategy of M. of M. Eck Eck & J. & J. Hadenfeld Hadenfeld � Inputs Inputs The same The same as in T. as in T. Lyche Lyche and K. and K. Morken’s strategy Morken’s strategy n ( ) ( ) ∑ = where is an f t C N t an interpolating interpolating B B- -spline spline where is Output Output i i , k , T = i 0 i on μ = on μ Approximating Bspline g g i = T T - - { { t t i } General Removal of of knot knot t t i Approximating Bspline i } General Removal i − n 1 strategy ( ) ( ) strategy ∑ = i g t A N t μ j j , k , A II = j 0 2 C 3 "forward forward" construction " construction " A 2 i I of g with of with A I j A I "backward " backward" construction " construction 2 C 2 i II of of with with g A j II + + I II Position of control Position of control point point A A j j in in [ A j A , ] j International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII Outline Outline Survey of of the present the present compression compression methods methods � Survey � � Strategies based Strategies based on polygonal on polygonal curves curves � � Strategy based Strategy based on on spline curves spline curves � � A A new new compression compression strategy strategy � � Data Data fitting with fitting with B B- -splines splines � Criteria for for estimating estimating data approximation data approximation � Criteria � � Reduction Reduction technique technique � � Result comparison Result comparison � Conclusion � Conclusion � International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII A new compression strategy A new compression strategy � Data Data fitting with fitting with B B- -splines splines � Given a polygonal a polygonal curve curve P P = ( = ( p p 0 ,……, p p n ), find find a a curve curve Given 0 ,……, n ), Problem Problem m ( ) ( ) ∑ = f t Q N t as close as possible of P P as close as possible of i i , k , T = i 0 Least squares squares fitting fitting: : find control find control points points Q Q j so that Least j so that ( ( ) ) n ∑ 2 is ζ − f p is minimum minimum Householder Householder j j = j 0 parameters ζ i parameters knot vector T Choice Choice knot vector T order k order k International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
Equipe MGII A new compression strategy A new compression strategy � Choice Choice of a of a parameterization method parameterization method � uniform h uniform h i = constant i = constant = − cumulative chord length chord length cumulative h p p + 1 i i i = − centripetal centripetal h p p + 1 i i i General expression expression proposed by Lee proposed by Lee General i ∑ uniform with e e = 0 = 0 � uniform with e � − p p + j j 1 ( ) cumulative chord length with chord length with e e = 1 = 1 cumulative � � = ζ = ζ = ≤ ≤ j 0 0 , e 0 1 0 i n ∑ e − centripetal with e e = 0.5 = 0.5 � centripetal with � p p + j j 1 = j 0 International Conference Graphicon 1998, Moscow, Russia, http://www.graphicon.ru/ www.graphicon.ru/ International Conference Graphicon 1998, Moscow, Russia, http://
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