Optimal Blurred Segments Decomposition in Linear Time
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimal Blurred Segments Decomposition in Linear Time Isabelle Debled-Rennesson — Fabien Feschet — Jocelyne Rouyer-Degli. N° 5334 Octobre 2004. N 0249-6399. ISRN INRIA/RR--5334--FR+ENG. Thème COG. apport de recherche.
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(188) W6& D$5 S Y Y Y ∆(1,b0 ) (M, M 0 ) = 0. ∆(ak ,bk ) (M, M 0 ) = λ (yM 0 − yM ). M0. M. 0. (1,b0 ). 0. 0. (ak ,bk ). 0. (1,b0 ). 0. (ak ,bk ). (1,b0 ). (ak ,bk ) min. (ak ,bk ). max. (1,b0 ). (ak ,bk ). (1,b0 ). Mmin. (1,b0 ). (ak ,bk ). Mmin. 0. k. Mmax. 0. (ak ,bk ). k→+∞. Mmax. (1,b0 ) 0. k. 0. 0. k. k. M. M. M. M. k. $. +$. $. ω−1 b. $. +$. ,$. b. %$. !. b. *. b. b. ÑÁÔаÑÓÒ.
(189) ½.
(190) !#"%$$&'($
(191) )+*,#
(192) .-/0. vertical distance. @. M convex hull . t£q ¢ {w]b osÃpr[kmt£\^ p ¢ suht£s[q t¦su] WYX[t±ih] s[tvkwt¦p@st±i't £ ¦¢ ilkm{(Okw]t¦s!8t£qu¶§rsu ]@[i°kwpokwX[]8xÁp ¦ pO«t£s[q¿s[]«»h] s[tvkwt¦p@sn)prsun)]){ws[t¦suq ¦¢ {w{w]?il]qr\^])s:kwi¶ <C=@?BACAGDEF HCD;JqKLDCMONsUGc PSRF,N"T 5
(193) ½ $
(194) 5 $%
(195) $
(196) ½ &?Æ('*)*+ +yÊ,) 3 v ?S & 6&'("% ½ S
(197) "% C # $ $ vY WYX[]b{w]n)prqrsutvkwt¦p@sDp}x ¦¢ {m{w].il]qr\^])s:kwi¡«t¦kmX?«t£jkwX t£i¡kmX ¢ iY"] ¢ t¦¥O ]s@k$kmpkmXu]bn)pr\^ ¢ k(Okwt¦p@s prx'kwX[] ¥r]{lkwt£n ht±i~k(}sun)]bp}x'kwX[]
(198) np@sj¥r]¯im]k conv(S ) ¶ v #
(199)
(200) ? ., . ^ 8. . !. b. b. . r
(201) m. p mO~,]pq m* "!l.
(202) !. #]p ~ZmOoQws$>mOs~Z !
(203) . rs. m. . . ~Z!# #. $(. Wp>6]D[{w]n)t£im]r²kmX[].¥r]{lkwt£n ht±ilkw}sn]Dprx us[t¦km].n)prsj¥r])¯im]k t±i kwX[]D\O¯ht£\ 8 ])suq}kmX¼prxkmX[] t£s:km]{wim]nkmt£prsp}x C «tvkwX»Ã¥r]){mkmt±n) t£s[]@¶ kt±i
(204) n ]G}{ kwXuOkjgnCprsj¥@]¯htvk~g@²'i ¢ n(X¼t¦s:kw]){(il]Gn2kmt£prsi }{w] ¥@]){mkmt±n) il]qr\^])s:k(i)²Hilp«¡]prs g.X4¥r] kwpn(Xu}{(rnkm]{mt ¨ ]bkwX[]6p@imtvkwt¦p@sui§p}x]¯jkm{w])\ t¦k~gr¶ $¢ kG² ])k ¢ i [])s[prkm]jg }s {w]im]Gn2kmt£¥r] gSkmX[] pO«$]ilk}suX[t£qrXu]ilkt¦s:kw]){(il]Gn2kmt£prs-6prt£s:kp}x «tvkwX¼kmX[] ¥@]){mkmt±n) t¦su]Lq@t¦¥@])s.:Ug x = i }sDjg V(i) kwX[] ht±i~k(}sun)]§6]k~«$])]s L }su U ²jkmX[]sxÁp@{YCrsjg i 6= j ² kwX[] ¢ r[{wrs[q ] t±i pjnOkw]ºt¦suimt±h] ¶·ejp¼t¦kDt£iDilkm{(}t£qrX:kmxÁpr{w«$r{wkwp»il]].kwXuOkkmX[] x ¢ sun2kwt¦p@s V(.) t±ib(L.n)prUsunU4¥rL] )x ¢ sn2kmt£prs'¶¾o]sun]@²ª])¥rC]{mg phn) ]¯jkw{m]\ ¢ \ prxKkmX[]x ¢ sn2kmt£prs&t£ib imp. q p@u ]¯jkm{w])\ ¢ \ prxkwX[]
(205) x ¢ sun2kwt¦p@s°¶WYX[] xÁp £ pO«t¦s[qu{mp@p:ilt¦kmt£prs´n(Xur{w@n2km]{mt ¨ ]Gi phn ])¯:kw{m]\ tvk~g prx'kwX[]bx ¢ sunkmt£prs V(.) ¶ Ê 8Ê + +y,Ê ) $
(206) !
(207) q "#$q6
(208) . C e X 5#" '($
(209) V(.) & ! 6
(210)
(211) 0
(212) $&'($
(213) %B % $
(214) "#
(215) " $
(216) $ ! 46
(217) .$5 7 $ ½ e ½ e $5 i L U C CY $ $5 Ì°])k ¢ inp@suimt£h]{´6p@imt¦kmt£prs i ¢ n(X¼kmXu}ks[]tvkwX[]){ s[pr{ }{w]D¥r]{lkwt£n)]i
(218) prx ¶À&] n)prsuimt±h]){¡kmX[]b]Ghqr]Gi¡prx C n)prs:kwrt¦s[i t£s[q L }su U rsuD«$L] h])s[prUkm]jg = ² + pr{ − kwX[]xyrCn2kYkwXuOk kwX[]i p@]GioprxkwX[]]Ghqr]Gio}{w] ])t¦kmXu]){b[²ilkm{wt±n2k g?p:ilt¦kmt£¥r]
(219) p@{¿ilkm{wt£nk g.s[]q@}kmt£¥r]r¶YWYX[]{m]r{m] @n)rim]i kwpnprsilt±h]){«Xut£n(XÃ}{w]b{m]G ¢ n]Gjgimgj\^\])km{wt¦]Gi$kmpp@s g^nrim]i ¡t¦k§t±i¿n ]G}{¿kmXu}k\^pO¥r] p@sÃkmX[]]hq@]i§hpj]iosup}k§n(Xurs[qr]bkwX[]¥O ¦¢ ]p}x X[])sn] (=, tvkot±io=)i ¢ BDnt£])s:kYkmp\^pO¥r] ¢ s:kmt p@s[]bp}xkmX[]bk~«$p^p@t¦s:kwiY6]n)pr\^]i¥@]){mkm]¯.p}x CV(.)¶ i. i. i. j. j. i. . +*. *. *. i. i. i. i. i. i. . . ÐÐ Ö
(220) Fuë2é2é(ì. i. i.
(221) G. C . ½. #$
(222) "
(223) . $
(224) . jgD\^pO¥jt£s[qrn(³j«$r{wui V(.) t¦sn{w]rim]i¶ jgnp@\}{wt¦s[q§kmXu]oi pr6]i²}t¦kKt£i]Grimg kmp h]Gnt±h]Y«X[]kwX[]){K«$]YXu4¥r]¡kmpb\^pO¥r]$xÁp@{m«Y}{( (+, pr{u+)@n(³j«$r{w^kwpqr])krsÃt¦sun){m]Grim]§t£s?kmX[] ¥O ¦¢ ]bprx V(.) ¶ (+, =) jgD\^pO¥jt£s[qrn(³j«$r{wui V(.) t¦sn{w]rim]i¶ ®sDkm8uX[{m]§p@xÁ\ p ¢ kw{X[nt£i¡rim[]{wipr²}6kmp@Xuim]§tvkw6t¦p@p@s°imtv²}kw«$t¦p@]¿sn)t±ri8sDsuhp}k] ¢ npj]nk~ «$ p ]¯jxykm@{wn2])kw\ i8¢tvx°\«$]¿}sur{m] kmX pj¢ pri$³js[t¦sup}qbkYxÁ
(225) pr{Kq kwp@X[u]o ¥@ ])]{m¯jkmkmt±{wn)]) \ h¢ t£\Ãilkw¶rsun] prx
(226) ¼nprsj¥@]¯ im]k {wilk²Y«$]>prs g Xu4¥r]?kmpºnprsilt±h]){p@t¦s:kwiprs kwX[]>6pr{(h]){Dprx }su¸im]n)prsuª² ])¯jkm{w])\ ¢ \¹¥O £¢ ]ir{m]Ãp@hkwrt¦s[]GºxÁpr{impr\^] x 6p@imtvkwt¦p@sui^p}xbkmXu]´¥r]){mkmt±n]Gi^p}x CC ¶·WYX[t£i ]Gr[i kwpSkwX[{w])]Ãn)rim]i ])t¦kmXu]){^«¡]?Xu4¥@].np ¢ ]-¤Á]Ghqr]@² ¥@]){mkm])¯[©²pr{äÁ¥@]){mkm]¯ª² ]hq@]G©
(227) pr{äÁ¥r]{lkw]¯ª² ¥r]){mkm])¯u© «Xu]){w]bkmX[] u{wilk¿] ]\]s:k¿{m][{w]im])s:kwiYkwX[] pO«$]){ur{lk¿prxkmX[]nprsj¥@]¯.im]kb}su?kwX[]im]np@suÃ] ])\^]s@k {w])u{m]Gil]s@k(iYkmXu] ¢ u]{§ur{lkG¶ k\ ¢ ilkb]s[p}kwt£n)]´kwXuOk§kmX[][{w])¥jt£p ¢ i¿[{mp@p:ilt¦kmt£prs´t±i§t£h]s:kmt±n) kmp kwX[] prsu]bn(X}{(rn2kw]){wt ¨ t¦suq
(228) kmXu] «t£jkwXÃp}xn)prsj¥r])¯il])k 3¦G& 4<¶ kt£in ]r{°kwXuOkkmX[]$]hq@]inpr{w{w]imp@suht£s[qYkmp¿kmX[]$p:ilt¦kmt£prsp}x[]¯jkw{m]\ ¢ \dp}x hq@]}rsu{m]$i t¦¢kwiuurp@{w{lkwt¦ ¦s[ ] q t£s[]i²}k(}³jt¦suqXupr{wt ¨ prs:kw : t£s[]ixÁp@{8bnp ¢ ]
(229) ¤Á¥r]{lkw]¯ª² ¥r]){mkm])¯u©2/¶ =´p@{m]pO¥r]{²4kwX[]]V(.) riwilt£s[q.kmX[{wp ¢ q@X>kwX[]^¥r]){mkm])¯Sh] s[]^]¯[rnk gÃkmX[] pO«¡]{rsu ¢ [6]){ ]}sut¦s[q t¦s[]Gibp}xY}sSp@hkmt£\ 6p ¢ suht£s[q t£s[]bxÁpr{ C ¶K¾¿])sun)]r²[p@hkmt£\ ° t£s[]ir{m] h]G ¢ n]GDxÁ{wpr\ kmX[] 6p@imt¦kmt£prsuiYprxpr[kmt£\^ t¦k~gr¶ . . (+, −). . . . . f. znp [~Z!\ils !#. mOo"i#!\pq rj. $kkwX[t±ip@t¦s:k²@«¡]n)prsuimt£[]){bil])kKp}x6prt£s:kwi ²r ¦¢ {m{w]il]qr\^])s:kt£skwX[] {wilkpjnkwrs@ko«t¦kmX D(a, b, µ, ω) @iYprhkwt¦\ Sp ¢=su{(x ht£s[q , y t¦su),]r¶K0WY≤X[] i ¥@<]){mn}kmt±n) ht±ilkw}sn] p}xkmX[]
(230) n)prsj¥r])¯ X ¢[ £ prx ² ²t±i ] ¢ kwpÃkmX[].¥r]){mkmt±n) ht±i~k(}sun)]Dp}x ¶ÃÀ&].i ¢ up:il]^kwXuOk n)prs:kwrt¦sui b. i. i. k~«$p ¢ [S]{ conv(S ]G}s[t£s[q6)prt£s:kwi² U }s U ²h}sDprsu] pO«¡]{ ]}sut¦Ds[qp@t¦s:kG² L ¶ U ² U Srsu L }{w] ¥@]){mkmt±n]Gip}x conv(S ) ¶ =´p@{m]pO¥r]{²4kwX[]o¥r]{lkwt£n [t£ilkwrsun]prx D n}s^6]§n) n ¢[ Okm]G^OkKkwX[]o6prt£s@k L ¶ Wpnp@\ ¢ km]
(231) rsu?\^rt¦s:k(}t£sÃn)prsj¥r])¯DX ¢[ ¦ i²[«¡] ¢ il] =´] ³j\}s i q@pr{wtvkwX[\ 3 5&4<¶$Ì°])k ¢ i{w]n) ¦ kwXuOk t¦kwi8 qrpr{wt¦kmX[\ t£sun){m]\]s:kw £ gnp@\ ¢ km]GikmX[]on)prsj¥r])¯X ¢[ £ p}x n 6prt£s:kwixÁp@{m\^t£s[q
(232) bilt£\^ ]p gjqrp@su t£s[] t¦sÃkwt¦\^] O(n) urim]?p@sÃ[p ¢ ] ])sh] ¢ ] ¢ ]D¤yh]" ¢ ]G© t£ilk¶Yejt£sun]
(233) 6prt£s:kwior{m] @[h]?«t¦kmX t£sun){m]Grimt¦s[q x ²u«¡] }{w]bq ¢ r{wrs:km])]GkwpX4¥r]bimt£\ ] 6p gjqrprs H t£s[]@¶ e ¢ [p:il]kmXu}k§«$]ru´s[])«6prt£s:k M kmp S ² S = S ∪ {M } ¶oWYXu]){w]}{w]bkmX[{w])]n)@il]Gi
(234) ¤yim])] 8t£qu¶Kr©¶ b. b. b. F. L. L. b. F. L. L. L. 0 b. b. b. ?M? UF. UL P. ?M?. LL. t¦q ¢ {m]
(235) KWYX[]kmX[{w])]
(236) nrim]i$«X[])s´@[ht£s[q6prt£s:k x 6] prsuq@ikmp kmXu])s°²OxÓkw]){^kwX[]S}u t±n)}kmt£prs p}x =´] ³j\^rs¸ qrpr{wt¦kmX[\ 3 5
(237) 4¿kmX[]´¥@]){mkmt±n) [t£ilkwrsuMn]{w])\}t£suikwX[]DDimr\^]r¶.ejpÃkwX[]s[]«im]k
(238) prx$6prt£s:kwi {w])\}t£sui à £¢ {w{m]G-il]qr\^])s:k «tvkwX-kmX[] 8. ?M?. . ÑÁÔаÑÓÒ.
(239) ½. @.
(240) !#"%$$&'($
(241) )+*,#
(242) .-/0. iw}\^]b«t±jkmX´}s.«t¦kmX?kwX[]
(243) imr\^]p@hkmt£\ 6p ¢ suht£s[q t£s[] ¶®s?kwX[]
(244) p}kwX[]){onrim]i² t±i}6pO¥r]bpr{ 6] pO« D ²hkmX[]
(245) n)prsj¥r])¯il])kot£iY\^phht u]Ãrsu.impkmX[] ¥@]){mkmt±n) Dht£ilkwrsun]b\ ¢ i~k6] {m]Gnp@M\ ¢ km]Gª¶ À] u{(i~kilk ¢ hgSkwX[]nrim]«X[]){w] ²t£s:km]{wim]nkmt£prsp@t¦s:k6]k~«$])]s-kmX[]¥@]){mkmt±n) 8 t£s[]^xÁ{wpr\ L rsu U U ²[t±ioilkm{wt£nk gDt£suilt±h] [U U P] ¶. L. F. L. F. L. M. N. P. UF. UF. UL. UL P. LL. LL. M. t£q ¢ {m] [[t¦s[qDs[]« 6prt£s:ko])t¦kmX[]{o}6pO¥r]¤ ]xÓk2©$pr{6] pO«¤Á{wt£qrX:k(© e ¢ [p:il]kwXuOk t±i@[h]G&rpO¥@] ¶Ì']k ¢ ib}[ g =´] ³j\}s qrp@{mt¦kmX[\ö
(246) WYX[]^np@sj¥r]¯>im]k t¦x¿\^phht u] }su«¡M]Ãn) ¦ N kwX[]Ãp@t¦s:k^D])xÁpr{w] M t¦skmX[] ¢ [6]){^ur{lkp}xokwX[]Ã{m]Gi ¢u kwt¦s[qnp@s:¥@]¯ X ¢[ £ ²6im])]!8t£qu¶>¤ ]xÓk2©2¶ N t±i¿s[]Gn]Gimiw}{wt g6]xÁp@{m] U p@{¿t±i U ¶§¿iDn)prsuim] ¢ ]sun]@²ukwX[]¥@]){mkmt±n) u{mpr|~]n2kwt¦p@s¼p}x L t±it¦suimt±h] [N M ] ¶&ejpu²kwX[]¥@]){mkmt±n) ht±i~k(}sun)]prx$kwX[]?s[])« n)prsj¥r])¯im]kilkm{wt±n2k g t£sun){m]Grim]i¶WYX[]³r]gÃp@t¦s:kbt£i§kmX ¢ i§kwp phn)Okw]
(247) kwX[]su])« 6p@imtvkwt¦p@sSprx8]¯jkm{w])\ t¦k~gr¶ kbt£ibn ]r{¿kwXuOk n}s[sup}k§{w])u{m]Gil]s@k¿p:ilt¦kmt£prs´prxK]¯jkm{w])\ t¦k~g?imt£sun] ) ≤ V(L ) ¶ =´pr{w])pO¥@]){G² M hpj]i§s[p}k N u{mpr|~]n2k¥@]){mkmt±n) £ gi~kw{mt±n2k gºt¦silt±h]Srs ]Ghqr]´prx§kwX[] pO«¡V(N ]{^u}{mkDp}x ¶ ¾o]sun]@²¡kwX[]>s[]« 6p@imt¦kmt£prs´p}x])¯:kw{m]\ tvk~g.tvxKs[]Gn]Gimiw}{wt gDp@hkwrt¦s[]G?xÁpr{§prs[]
(248) 6prt£s:k§Ok¿kmconv(S X[]
(249) {wt£qrX:k¿) p}x L t£sÃkmXu] pO«¡]{ }{mk¿p}x conv(S ) ¶§Ì°])k ¢ i§{w]n) ¦ s[pO« kmXu}k phn) ]¯jkm{w])\Dr{m] q p@u ]¯jkm{w])\^xÁp@{okwX[]
(250) x ¢ sn2kmt£prs i ¢ n(XÃkmXOk§«$]
(251) p@s g?Xu4¥@] kmpkw]ilk¿kwX[]n}suht±[}km]
(252) 6prt£s:kwiot£sSim ] ¢ ]sun]rsu>ilkmp@ui¿}kokmX[] u{(i~k V(.) phn) ]¯jkm{w])\ ¢ \ò[n) £ ] C ¶ e ¢ [p:il]^s[pO« kmXu}k t±i@[h]GS6] pO« ¶^Ì°])k ¢ i }[ g =´] ³j\}s- qrp@{mt¦kmXu\?¶WYX[] pO«$]){ }{mk8p}xHkmXu]§np@sj¥r]¯il])k¡t±MiK\^phht ]}s«¡]§i~Dkwt £ n £ N kmXu]¿6prt£s@k¡])xÁpr{w] M t¦skwX[] pO«$]){ur{lk8prx kwX[]
(253) s[]«nprsj¥@]¯X ¢[ £ ²Him])] 8t¦q¶´¤Á{wt¦q@X@k2©2¶ N t±ios[]Gn]iwiw}{wgOkkwX[] ]xÓk¿prx L ¶¾o])sn]r² N n)rs[s[p}k 6]kwX[]os[]« 6p@imt¦kmt£prsp}xª])¯:kw{m]\ tvk~gilt£sun] V(N ) ≤ V(L ) ¶À&]¿[p
(254) s[p}k¡³:supO«º[{w]n)t£im] g«Xu]){w] N ®t±nis?}sD 6phn)6p}}km]bXÃkm]
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