An Elementary Proof of the Equivalence between 2D and 3D Classical Snakes and Geodesic Active Contours
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An elementary proof of the equivalence between 2D and 3D classical snakes and geodesic active contours Gilles Aubert — Laure Blanc-Féraud. N˚ 3340 Janvier 1998 THÈME 3. ISSN 0249-6399. apport de recherche.
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(13) . . Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France) Téléphone : 04 93 65 77 77 - International : +33 4 93 65 77 77 — Fax : 04 93 65 77 65 - International : +33 4 93 65 77 65 à partir du 05/01/1998 Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65.
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(32) =;-=?>A@_@ @ O${ > ) t S ;yz=?>A@ = ,T =z;-=?>A@@ @ { > vlw w vu w w v w l w w w =)' @ =(' O @ 9=(' S @ X=(' O @ S 1=(' S @ sd=(' O @. ´¾zrkvx{mZl½N®Ywb¬ ¥ikz tz¦ yi¬'m mZ ®,y%m »¬ ikzyiz w u©4z h4yik®km8¥|tXk,®kwbvxvuYt4{G´¾¸{m8v {k´Vvxtu{RikTm8z|r,rk m Ttxi,t¨wR¸t mZy#txz|{Nrz4tx{2¸ui,© {R¬°tui,mvumPmPvxum 3  ! 2
(33) P! _ I mh4i,z¸m$txl wwrz|r{ztu kikmPmPw]v{by%´ ®kvxtuikm1m;$kvx#{T#%{b!´ zQ*'tu¸i,®Ym¨yi2´¾{Ntx¥|i,¥{RwR¬ tp zrkW p¥|mt ;"!Q«¯_mw]zmry%®kvxmNT¬Lm8uik{R¬ÅtuiYwRt4tuikmPvum S S ; y =z>A@ =
(34) T =;-=?>A@_@ @ { >& t ; #y # =?>A@ O { C> | t = T,=z;$##=z>A@_@ @ O { > v w v w v w w w w w w &;'! Q =(' S (@ u=(' O @ )Xk ! g$! k e+c *a , #3k ! aPbocAboc~g[em Cm b-e ! _k ePc t. S. xthikwbm²Tzwbrk¯_8{Rtxikm$m¨zrkzr mP,G®,l:w®k¥|z|l t¦{RmPv 5]_^. ?!. «u»®,yz ymP kxm¸vxzz|NmPmP¥|¦: þ© {rtxikm$vuziGt¸} i,wr, ¸z m$wbr, txikmX¥m%´ tu}~iYwbr, ¸z mX{´. psrtxikz umPy%tuz{rÀ¬'m km,rkmrk{txwbtuz{r, wr, iG¦T_{btxikmPumPP© º¨mrk{tu%m .\txikm¸mtL{b´îtuikm$z|lwmmZ mZ[© 46m$wNuu®kl mtxi,wRt/. s1032 ; 2 G; 2 ! Q[GJqÀz ,rkz|tum${v y{®krG2,t{wbv4¯ktu¥i,mm8W ; z|rG2 tumPz4r,ukz ztmP¦ymz|l¬ zwb¸N5m m5T¬'SR© mwx¸®kl m¨tuikm´¾{N¥|¥{R¬ zrk iG¦T_{btxikmPuzJp . 4 T B V6!MPOL M u®,yitui,wbt =87oG:9,@ L T,=87EG;9,@ !< v =zV =>.i@ C w w ?A@8@CB
(35) D 8EG8FH D IJF . v w. T,=87oG:9,@ X s |3K w. . ¬ i,mpsvxr*m z|{ lzwb'Ntxm ikmkvx {zy%mZtwbuur,z|yr,m¨,´¾®,wbr,r\y#txmZz| {NrÃm:© z$ m ,rkmP *w¨Y{Nz|rGtr=L7oG:9
(36) @ u®,yi«txi,wRt T,=87EG;9,@ s |3K°©psr txtxikikzm8'zwl v¸txwzym¥|mNTN¬'z|´VmXwr,r,m mP {rkw¥¦Il z|{´ vxtxmXik,mvxvummZy%mz¸m$z¸ tx m,w r,r,z tx{z|{NrrÃrk© m4*GwRmtuzwxm u®kN l ¸m ®Ytuyi,i2wbt4txi,w]wRty%®,vuNmw ; v !M.6zw Lwr2mP kmX{´ v N OPN v GQ RTS*G:U%S$G:U%S% RTS%V FWGS 8 v RWSCs S 88 v U%SCs. é é ç* %ø . F.
(37) . C *
(38) . ¬ z|tui º¨mrk{tum. T
(39) L= 7o;G 9,@ w
(40) T =L7oG;9,@ w w w T
(41) L= 7o;G 9,@ w w . SS G AN G T,=87 y G:9 y @ s. S. s. . SS. w. SO. s. . w. z ´ { =_=87EG;9,@G; v @ R S z ´ { =_=87EG;9,@G; v @ U%S z ´ {{ ===8=87o7 yG:G:9,9 @y G@; G; v @v @( PU U S S. C C ¸P . . G. =87EG;9,@! M O { = =87EG;9,@G; v @ RWS =87E;G 9,@ ! M O R S { ==87oG:9,@G; v (@ U =87E;G 9,@ ! M O { = =87EG;9,@G; v @ U S . S. . . .
(42) . z®kvxm <pmkwbl k¥m{b´w]y®kvxm3; v ¬ z txiÀwNuu{y%zwRtxmP 2um%t S iYwbr,ps r¥m{v¬' mm8v' txm{ Yrku_mmPw:y%z|´¾´¾®k¦Ir,mZy# txz|{NmZr L¯T ¦(B%PMmZ7vx{:L RwM¥|®kZ# mZ'uvx®,wbytui2ikmPtxvLi,wRtxti,p wbr¯T¦Izr,rkz|tumRwb¥®kmPP¬ ikzyiz'mPwN¸zmv'tu{ =?F @ sdI ¸ 88 =?@ suF ¸ZÇ ? Zo = @»z4vxm®,¥wv'l {Nrk{btx{rkzy mPyvumZwuz|rk ¸Pn =87EG;9,@[L = T
(43) =L7oG:9
(44) @ @Lz z|,xyikz|t %8¬ z|tui z,xyikz|t %8y%{Nr,twbrGt ¸ w w t¦Tkzyw¥þm kwbl k¥m{b´¸®,yi´¾®kr,y%tuz{r z zmPr¯T¦ =?@ s S_Z S © 46m8mPG®,wb¥¥¦Irk{btxm p l=;r@[s = T,=z;r@ @ ¸ ! w w l=87EG;9,@ s = T,=87oG:9,@ @ ¸P w w 2. . . . . ¾ðé å . .
(45)
(46)
(47) . %Hk k (k
(48) . 5]?\. Åh4;Èi,z|¯Tm=tu¦Ii,k{z|vx®tx{Tt${bw´k¥|{Gvxvxy¨um ¥¥|z|mPmZ{rkî´»b{Ntxr8mPirktumikvm4swby%¥z {N tr,¦N=?>Ak¸tu@[¬'vxs®,m y#ytxwz|{N r«r; w{bx´,¸=z®kw¨@ l y%{ ®kmvx©=txi,º¨m wR;mt$Y#rk¬ # m z xtxwRi tuz¸w ; ´¾¦Gy z=?rk@ w?s !%F© m% t ;' 46! mQY¯Yvxm4¸tXw,Nz|wNvxmwr l]ym®ktuvxvxz%Pmm © v w y w ;$# = @ su;-= S = @_@ ¸ ¬'mi,wem ;# y = @ s I3 ¸P w w wr, DÇ 'z4¬ vxz tutumrAwN'´¾{¥¥|{R¬J p q O =;r@[s t l=z; # = @_@ { ¸P v ¬ ; i,# =m vx@¨m w! r, *s"tu{« v Sy%{Nw ; r,¸y tu=?vx®,@ w y#{ t _+¨wÀ¥|mPy%rk®,bvutxNiÂm ;{´ ## ;]=?©Å@¨xh4wRiktxzmÀ¸´¾rk¦Tmz|r,T tIl¸tum=; #y%# {N=#r,u@z@¸txs ]zr ; vxm=#Ywb@ vwb l m%©2txvuº¨8z %mm2r,tu{biktumAm%$±y%®,tuvuikNmm #y # w w u{¥®txz|{Nr{´îtui,m 'z &mvxmrGtxzw¥ÃmPG®,wbtuz{r p $ y =?@ s l=z; # =$o=?@@ ~Çb $ =zF @ s F yPy%{Nvx zrk:tu{]tui,m8iG¦T_{btxikmPuz ¸ $ DÇb 'i,w w]®kr,zG®km8u{¥®tuz{r $ ! 5S¨¬ z txi $ y V0Fk©º¨m ,r,m s($o=?@ ~Ç 4. . . . ;. txikmrÀ¬'mi,wem wr, . . ~ÇÇ ~Çbn . ;#y # =z@ s $
(49) y =z@ ;# y =#$ =z@_@ s($
(50) y?=?@[s l =z;$##=z@_@ w w w q O = ;r@[s t*),+.v p. xk¬'m${N¯txwz|r®,uz|r, ~Çbn. s($ *S = !5@. q O z= ;r@[s 1 t 0 v. é é ç* %ø. ## =?@[su; # =#$ =z@_@. { s$Wy?=?@ { is l =z;$# =$o=?@_@@ { is l =;/## =?@@ { w. º¨mr,{btuzrk /. . . O =;. @ F . ## =?@@ { is 2t 0. v43. . ~ÇR . O =z;. ## =?@_@ { . O =;. ## =?@@ | ; #y # =?@ O65 { w w. ! . ~Ç.
(51) . C *
(52) . 6m8i,weNm$uik{R¬ rÃk¸txwv¸txz|rk]´¾vx{l '; !Q«tuiYwRt tui,mvxm$m z¸tx w y%®,vuNm3;$##%!Q*u®,yitui,wbt 4. ~Çb »¬Lmi,weNmXtx{mkwbl zrkm$tuikmt¬'{ywumP / XI8wr, %/ O I©. t S l=;-=?@ ;y=z@ { i s * t 0 v w w v 1=)' S @u=)' O @. psrÀ{Nvx mPv'tu{y%{Nr,y%¥®, m$tuiYwRt . / XI. 3. . O =z;$##=z@_@ | ; # y # ?= @ O65 { w w. . psrtxikz yPwumkuz|rYy%m$tuikm8zrGtumPvwbr, k wvum$rk{Nrr,mNwbtuzmNG¬'mi,wem p. q O = ;r@[s 2 t 0 O z= ; ## =z@_@ | ; #y # =?@ O65 { . w w v 3 S O z= ; z= @_@ | ;y =? @ O 5 { X=(' @ t ## ## w S v 3 w. i,mr,ym. )t. v. S. / OXI. ~Çb . l=;-=?>A@_@ ; y =z>A@ { > )= ' S @ w w. =)' S (@ X=)' O @ . hwb²Tzrk:tuikm8zr,l:®kl {RNmvC;! Q ¬'m kmP ®,ym . ~Ç . . x©. xth4 kikmi,m],zrk z|rGz|tuytuzwNmP{¸vurÅmRw{z¥& ´= r,/Cw{brÅG tI mP@'w :¬ Nm«z|mPtuwi,wbu{¦·r,® Âtw txwbikrTtum«i,¦AmiTw¦T, ,Yvu {mPz txGtuz|ikz{N{mZr,®Y¸wbmZ ¥{Ny{rk{r mrG©tuvxz|h4¯,¸i,®txwbtumztu{mZwr\ Âz|l ´¾z{rÅv-z:,q tuwO {6vx©:wm hîTv{ÀtuwbmPkwr,iy ·iknkztxm©ikNmAm© y%tx4*ik®kzvxPm\þm wN¬'u; m:u®k# ®,# l u=?m]m@8tu´¾{Nik{r mv y¥wvuz|t¦ÀtuiYwR1t T«y%{NrGtxwbzr,${r,¥|¦«{Nrkmy%{NrNtx{®kvZ© yy%{N vx kz|rktu{ DR}P %î mPrk{btxm N2¸®,yi*tuiYwRt5N O'N v © 46m m kwbl z|r,mtui,m t¬'{2ywN¸mZX¬ i,m%tui,mv¨tuikm:z|rGtumPvxumPy%tuz{r ; ## S S z¨ml t¦À{Nv¨rk{tP© mPyPwb¥¥îtxi,wRt q O s( v 0 O =z; ## =?@_@ | ; # y # =z@ O 5 { ©psrÀmZwyiÀywumT¬'m mvxzmw:¥{R¬'mv4¯Y{N®kr, ´¾{3 v q O wJ p 3 w w q O =z;r@( X=)' S @ |
(53) =LN @»¬ z|tui S 8 =HN @ suF ~Çb Zo 2z|vtXyPwum p(Q !9; ## S S 4Åz txik{®kt¥{NS x¨{b´LNmrkmPvxw¥|z|t¦_¬'m]wNuu®kl mtxi,wRt53sl;/##= /3@x©146m:m Gtxmr, 6;$##{Nr =zFHG /CK¯T¦ m Yrkz|r, ; ## u®,yitui,wbJt p ;$##*=z@=z ´ !6D FHG /5K ; # S # =z@ s 5s ;$## = /3@z|´[! = /CG I K Tzr,y%. m =; #S # @ y s F]{6r = /CG I K~Tz|tz mZwu¦tu{ Nm%t q O =;r@[s t S O =z; #S # =z@_@ | = ; #S # @ y =z@ O 5 { W=0=_I$= /3@ O =H@ Dn w w v 3 . . . . . . ¾ðé å . .
(54) .
(55)
(56) . '®t ! S S txikmrÀ¬'m$iYwem8wyPy%{v zrktx{ 4wbr, Zn p . h4ikmr ÄmPr,y%m. q O z= ;r@( t S v. 3. . O =z;. O = ,T =H$@ @( O = NI @ w w . #S # =?@_@ | w. =z;. q O z= ;r@( )= ' S @. . O = @[s. . #S # @ y =z@ w O 5 { = =_I = /3@ O = NI @. . = =_I$= /3@ O = NI @ . . Dnk. XºTmPmPyrk{{br,tx Àm5yP5ws um ;pi;$## # = # /3 @»w4S S zrstuikm8kvxmTz{®,4yPwumkwbrY S$ m ,rkmP ÀwN. DnNÇ. . { =H$G S@ s { = $G S S @. wbr, . ;S*=?@ s ; ## =z@ | A= L S = /CG I K ; S ; S =z@ s ; S S z= i@ ! (= /C!7G D I FHK G /CK. z km,rkmZ {r7D FWG /5K ©[4*m8m Gtxmr, {rtuikm8zrGtumvxRwb¥ î,¸zrk z ´ »z ´ ÄmPr,y%mNk¯T¦tuikm km ,rkz|tuz{rÀ{´ ; S wbr, rk{tuzrk]tui,wbt ; S y su; #y # ¬'m8,i weNm ;$# #. t. 0 v. 3. . . Dnn . O = ; S =?@_@ | ; S y =?@ O[{ s w w O =;S*=?@@ | ;#y # =z@ O 5 { | =_I = /3@ O =HAS@ w w t. v. S. wbr, ¬'m8 mP ®Yy%m8zrwuz|l z¥wv4¬'we¦2wN4z|rtui,m8kvumPTz|{N®, ywN¸m¨tui,wbt. . t. v. S. . t*0 O =z;S*=z@_@ | ; #y # =?@ O65 { . w w v43 O = ; S =?@_@ | = ; S y =z@ O { T= =_$I = /3@ O = NI @ w w q O =;r@. Dnb . uth4i,ikwm8r ; z &_# mP# vumPzr±r,y%tum$ik¯Ym m,t¬Lvx¸mPtmr2txmtuvxiklm©A¥m%h4´ t¸ik} i,mIw´¾r,{ 2¥¥|{Ru¬ z kzm$rkÀ{´ tumZCnyikLrkzwbyr,w ¥Lyw¥y®k¥»®,z Lwtxi,vumIwRt ¬'mPvum¨zi,mZwe 6Nmz|r±; {NS vx vwRmPtxv8iktum{ v é é ç* %ø .
(57) Z. C *
(58) . mP¸z¸l tuzz|l¥wbwRv4tum¨tx{ tui,Cnkm$ 'z &z|rmvxtxmikr,zy m yPwvu0 m©[3 46Om=z; mS*,=zr,@_m@ t= ¬L{ z O rN=tx; mvx# R# wb=?¥4@ 5 {{ r7»DwFWr,G 2/5KÃu{ mPtxrk{:{b{Ntx¯mP txwz|rwbwbrIr,z rkmPG®,pw¥|z|t¦ O . . . O s !6D FHG /5K ; s !6D FHG /5K ; . . # # =?@(! S O # # =?@(! S . . . . . . .
(59) .
(60) . Tzr,y%m O wbrY . z|N®kvxm8ÇHpmkwb-l k¥m-{b ´V y%-®k vxm1; v W;$##wr, U; S ´¾{Nvul w],wv¸txz txz|{Nr{´& k h&#ì'm$iYwem p 2. . . t. 0. v. O =; S ?= @_@ sXt. . O =z; S z= @_@ | t. .
(61). Dn. O =; S ?= @@. .
(62) . ! . pswbrGwb{Nzrvx mPmZvy%tx{N{*l]y_w{N¥uymP®k ¥wR{Rtum2mPtuv=iktxmikvxz|rGmmtxmum%vtxwbJ¥p {´ tuikmvxz|NiNtu}~iYwbr, ±uz km2{b´ C n ! #mZwyi±{r,m2l ®Yt ¯Ym . . . . . O s $! O ;=?@! s $! ; =?@! . §'w¥y®k¥®,4{RmPv . t.
(63). . O. O =z;S*=?@@ { is t.
(64) . -. . S. SG. O =z;S*=z@_@ { | t. 6m8rk{R¬ªmP¸tuzl wbtummPwNyizrGtumPvwb¥{RmPv'mZwyiÀ¸®,¯,¸mt 4. Dn . s I G HG.
(65) . . . t. Op. . O =;S=?@_@ { |. Dn . O =;S*=?@@ {
(66) . . ¾ðé å . .
(67) N.
(68)
(69) . N©¨Nmv SO ¬'m8i,weNm t. º¨®kmtx{ # w.
(70) . O =z; S z= @_@ { sXt . -. t.
(71) 3. -. O =; S ?= @@. . =.
(72). O =; S ?= @@ { T - =. . O =;/## =?@@ { o| . . .
(73). O =z; S ?= @_@ { t. . z¥wv - ¬'we¦¬'m{¯twbzr. Çk©'psrAw uz|l. . t.
(74). . . O =z;$## =?@_@ ; S ?= @ =7;$# # z= @ w w w ;$##*=z@(! S O U; S =?@ !. . ´¾vx{l ¬ ikzyiÀ¬'m8 mP ®Yy%m,vumZywb¥¥z|rk:tuiYwRt t. Dn . O =z;$##=z@_@ { 5. p.
(75) . O =;$# #=?@@ { o| . -. O =; S ?= @@ { t.
(76). . wbr, . . . Dn . SS p. CN . /1=LR S | U S @. O =z;$##=z@_@ { | . . . C,. / U S. . . . . n,©4h4 i, ,m z|z|r,rGItxmw:vr,wbm¥þ¬¶{RN_m{v zrGt O zl {vxm8y%{l k¥m¯YmZyw®,¸mR;$## =?@! S O wr, U; S =?@! S © . 6mywr¬ vxz|tum 4. t.
(77) . O =z;S*=z@_@ { is t. 6m8i,weNm 4. t.
(78) . t. . O =z; . =z@_@W= O = ;
(79) w. t. é é ç* %ø. ## =?@G;S*=?@?K. ; =?@ s S O D ;. t. .
(80) . ## =?@@ { o| t
(81) . O =;.
(82) . . O = ;< z= @_@ { T=. . O =;S=?@_@ { =. ## =?@_@ { . O =z; . O = ;< ?= @@ { . CNn . # # =z@_@ { / ;6 =6; ## / U S w. O =z;S*=z@_@ =. . w. . w. O = ;- ?= @@ i{ [s.
(83) O = w T,=z;S=z@_@ w @ = O = w T
(84) = ;-=?@@ w @ { . |. C . CGÇ .
(85) eÇ. C *
(86) . u®,yPyy%i{Nvxtu i,zwbrktJ]p tu{ D wr, Zn 4wbrY ¯T¦h we¦T¥{v4´¾{vxl ®,¥w:tui,mvxmmztwy%{Nr,twbrGt. . t. h4i,mr. .
(87) . O =; S ?= @_@T=. O = ;< z= @_@ [{ . . O =;S=?@_@ { t O =; # # z= @_@ { o| / U S[| y /
(88)
(89) . t. zr,wb¥¥¦I{RmPv O ¬'mi,wem8uik{R¬ r2txi,wRt 2. t.
(90). O =z;S*=?@@ { t. . ¬ i,mvxmOS s / U%S | §'w¥y®k¥®,4{RmPv 46m8kvx{y%mmZ wN'´¾{v . . t. v. F. . .
(91) -. O =z;S*=z@_@ { | t.
(92) . . ;$## =?@!.
(93) -. O =z; S z= @_@ { sXt . t. .
(94) 3. .
(95) O =; S =?@@ - =. . C . t. . O =;S=?@_@ { |. . w. t.
(96) -. . w. O =; S ?= @_@ { t.
(97) -. . w. . CN . O =z;$## =?@_@ 5o{ . S wr, U; S =?@! - S S u {]¬ z|tuiÀiT¦TY{tuikmZ¸z Z T,=z;$## =?@_@ T,=z; S =z@_@ ! w. CN . O =;S*=?@@ {
(98) . O =;/## =?@@ { o| . z|r,ym zl]{Nrk{btx{rkzy$ mPyvumZwuz|rk]¬'m$iYwem . CN N. ## =?@_@ { | SO. O =z;. N©¨Nmv S ¬'m8i,weNm t. A!. O. O =z;S*=?@@ { is t .
(99) . N S L. y/. .
(100). C. y / N . . . S. b . O =;$# # =z@_@ { . z!. ¾ðé å . . y.
(101) Zn.
(102)
(103) . Çk© 2,{v4txikmuwl m$vxmPwN¸{Nr,'¬LmNm%t t . n,©¨Nmv ¬'m8i,weNm . t. ;.
(104) . . O =;S=?@_@ { t. . # # =z@_@ { . z! . O =z;S*=z@_@ { s t O =; ## =?@@ { o|
(105)
(106) t O =;S*=?@@ = O =z; ## =?@_@ 5 {
(107) 3. Ç. z! . O =; S ?= @@ { t O =;$# # =z@_@ { o| y'/
(108)
(109) . zr,wb¥¥¦I{r ¬Lmi,weNm$uik{R¬ rtui,wbt . 2. t.
(110) . . O =z; S ?= @@ { t.
(111) . . h4ikmvxm%´¾{Nvum$´¾vx{l D n ! # ¾ 'wbrY z!R =¬'mmt t. v. 0. wbr, ´¾vx{l D nb Lwr, . O =; S ?= @_@ { t. . ?!. ! p. v. 0. sd=_I. = 3/ @ = SS @ |POS . O =z;$## =?@_@ { | y /. O =z;$## =?@@ { | OS | y /. . t10 O =z;$##=z@_@ | ;#y # ?= @ O65 { 0t S v43 w w v. mt¸tuzrk S. O =; . ## =?@=wr, ;S*=z@»¯YmPz|r,z|r S ¬'mi,wem t.
(112) . . G 4yPwbr¯Ym8¬ vxz tutumPrwN S. . z!. R . N. z!. N. O = ; S =?@@ | ; S y =z@ O { T = w w I =_I$= /3@ = N @ = SO = y / . | y / N ?! q O =z;r@( =)' S W@ = ?! . bn . N. N. z! ! . b . z!. z!. . zrGtumP'vxu¦ÀmPy%y%tu{z{®,rk¥|;zrk# # tx ikm S vxS mP#u,®k¬L¥|txm8 {ND¯nNtxÇwz|rwbr,p 4{´Vtuikm Yvx¸t¨wbrY tuikm umPy{r, ÀyPwum C wyPy%{Nvx zrk tx{tuikm q O =z;r@ u=)' S @W= =I$= /3@ O = NI @G S z!b . . . . . é é ç* %ø . . .
(113) P. C *
(114) . txi,wRtz t. b :wbr, 2¯T¦ txwb²Tzrktxikmz|r ,l ®kl {RNmv ;'! Q«zr. S. ; y =z>A@ = T
(115) =;-=?>A@@ @ { > u=(' S @ v w w w w =)' S @ u=)' O @ NL F / OXI. =. . . =I$= 1 / @ O = NI @G . S. z!. . wz!br, I#© ¬'m¨y{r,y¥|®Y mXtxi,wRt ¯T¦I {Nz|rk h4i,wbtXwyikzmNmPLtui,m8kvu{T{´ zr2txikmywN¸m N© 5 !%: 5 R
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(153) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France).
(154). . ISSN 0249-6399.
(155)
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