Support Vector Machines for burnt area discrimination
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Support Vector Machines for burnt area discrimination Olivier Zammit — Xavier Descombes — Josiane Zerubia. N° 6343 Novembre 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6343--FR+ENG. Thème COG.
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(26) Bav`cVde¬ankgHj³ D jacRTV¥kjTVFwyrn^W`bVd{lwyr~wylVvedwi`cVi¨HacRTV Y*\ 2 ¯p`cV(V . khiTrnV - ±kx`ª}TV ¡ljTVF}7B^acRTV¥Vd¢5lwyackgij → − − w .→ x+ © l R ( V n r V x k * ` 7 w i F V ¬ e n a i g ¥ r T j i g n r U y w. c a ? g c a T R W V B R B ^ | { d V c r T { x y w l j " V y w | j } x k ° ` c a T R W V T x k i w d ` 9 Q B R l ( ` I ¨ n a g ( e i g n r c r F V e¬an^ → − b=0 w b e(wH`c`cko¿^acRTV*anrnwikjlkjThW`cV ad¨l©ªV*U+|`tafRlw·HV 6 → ko yi = +1 ©RTke~R!kx`9Vd¢5TkwyV(j5a9acg → ko yi = +1 − − − − w .→ xi + b ≥ > 0 w .→ xi + b ≥ 1 k o ko yi = −1 → − − → − − w .→ xi + b ≤ < 0 yi = −1 w .→ xi + b ≤ 1 Q9RTVF`bVe gHjl`tanrnwikj5a~`kUW{T^acRlwyavanRTV+Uwyrnhikj!acgUWw _kUWk%!(Vi¨I}_V(¡ljTVd}®5^anRTV"}_kx`banwyj|e V;V at©OV(Vdj anRTV9Vd¢5lwyackgijl` → wijl} → ¨kx`Vd¢5lwiiang 2 Q9RTVOacr~wykjTkjTh°`nwyUW{TVd`U"l`ta − → − − → − ||w|| ;VfgHjWacRTV°higBg_}w`c.kx}_xVf+b gyoIan=RTVf−1anT|V°}_V w¡ljl. Vdx}+b B^"=acRT+1 VfVF¢5lwankgHjl` → wijl} → − − − → − w .→ u°`+Uw _kUWk1!dkjThLacRTV?Uwyrnhikj£k`+Vd¢5Tk·wiVdj5a+acgUWkjTkUWk1!dkjlh®xacRl+Vbkj5=HV(−1 r~`bVgioacRTwV. `cx¢5l+wibrcV=UW+1 wirb hHkj2¨&anRTVAgi{_ankUwi{lwyr~wyUWV anV(r~` w wijl} b wircV!hikiVdj<B^anRTV®`cgi_ackgijgyo°acRTV®e(gijBiV °lwi}_r~wanke. Y[Z:PAY\.
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(28) !"#$". q&rcgHhir~wyUWUWkjTh"qrngiTV(U¯ °qqª±©kmanRkjTVdwirOkjTVF¢H|wykmat^e(gijl`bacr~wykj5an` 6 2 − k→ wk (w,b) 2. min. `cT_stVFe¬aang 6. ¯-±. − − yi (→ w .→ xi + b) ≥ 1 , ∀i ∈ h1, N i. 0O^Akj5acrngB}Tle kjTh{;gH`ckackiV 2wyhHrnwijThiV U"Tack{TkV(r~` kafVdwi}l`OacgacRTV*o¿gHg©kjTh}_lwy²{lrcgHTVdU 6. {λi }i∈<1,N >. wH`c`cg_e kxwanVd}A©kacRacRTVWe gHjl`bacr~wykjHa~`(¨ ¯/i±. N 2 − X → − ||→ w || − − − λi .[yi (→ w .→ xi + b) − 1] − L(→ w , b, λ ) = 2 i=1. 9Q RlVTjTkx¢HlV7`cwH}T}_V{;gikj5agyoOacRTV wyhHrnwijThikxwyj L ¨2©Rlke~Rk`+w!UkjTkU+TU © kmanR£rcVF`b{;Vde a*acg (w, b) wijl}w!UWw _kU"TU ©kacR£rnVd`c{|VFe¬a*acg λ ¨}TV acVdrcUWkjTVd`¥anRTV7`cgi_ackgijKgyoªacRTV °&q qQ9RTV(rnV o¿gHrcVH¨³V al` ¡|rn`baUkjTkUWk1!dV*acRTV wyhir~wyjlhikxwyj7©kacR! rnVd`c{|VFe¬a9acg (w, b) 6 ∂L − ∂→ w ∂L ∂b. − = → w−. N X. ¯1H±. − λi y i → xi = 0. i=1. = −. N X. ¯p5±. λi y i = 0. i=1. 0O^!`bll`tankman_ackjThacRTVF`bV"Vd¢5lwankgHjl`fkj5acg Lª¢|¯1H±¬¨;acRTV}_lwi2{TrngiTV(Ukx`vrnVd}_|e Vd}ang7anRTVo¿gHg©kjlh °q&q©kacR!;giTj|}e(gijl`bacr~wykj5an`9wyj|}?wWkjlVdwyr9VF¢H|wykmat^e(gijl`bacr~wykj5a 6 max W (λ) = − λ. `bTTstVde¬aang 6 DÂo²©OVfe(wi. N N N X 1 XX − − λi λj y i y j → xi .→ xj + λi 2 i=1 j=1 i=1. N X. λi yi = 0 , 0 ≤ λi , ∀i ∈ h1, N i. acRTVv`cgi_ankgHjgio|anRTkx`{TrngiTV(U!¨H©ªVv}TV acVdrcUWkjTV9acRlVv`bgHTackgijW{lwyr~wyUWV(acV(r i=1. λ∗. → − w∗ =. N X. − λ∗i yi → xi. → − w∗. o¿rngiU ¯p5± 6 ¯1¦i±. i=1. anRBl` b∗ e(wij;V¥gH_anwikjlVd}o¿rcgHU acRTV ´ wyrnl`cR_ ´*TRTj_ Q2|eAiV(r°¯p´ ´ Qf±Oe gHjl}_kackgijl`QT @ V ©RTkxe~R}_VdrckiVF} o¿rngiU acRTVe(gijl`bacr~wykj5an`Okj£¯ - ± 6 − − λ∗i .[yi (→ w ∗ .→ xi + b∗ ) − 1] = 0. DÂo ∗ k`9jlgij !(Vdrcg|¨HanRTVe gHrcrnVd`c{;gijl}_kjTh → − kVd`9V Twie¬an^gijacRTV*;giTj|}Twyrn^L¯pwyavwW}_k`banwijle V*VF¢H|wyIacg R|wyλoi anRTVUwyrnhikjo¿rcgHU anRTVWY*\ 2°±¬²YvacRTxVdirc©kx`cVi¨ λ∗ kx`fjBT wijl}acRlV"e gHrcrnVd`c{;gijl}_kjTh → kVd`fgHj!acRTV − xi i hHgBgB}`ck}_V*gio anRTV |gHTjl}Twirc^H. P>PRQ STUVU.
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(30) @. $. vgyacVfacRlwya → kx`&w*kjTVFwyr&e(giU"Tkj|wackgijWgio;anRTVfHVde¬angir~` → o¿gHr©RTkxe~RWacRTV 2wyhHrnwijThiV9U"Tmank{lkV(r − − w9∗acgKT¤Q9RlVd`cV?HVde¬angir~`WwyrnV?e(wiVd} x i " ¯1\B«*±"wijl}wyrnVanRTVAe gH`cVd`ba. " . k x W ` T j y g W a d V 5 ¢ l i w
(31)
(32) λ∗i VdVdUWV(j5an`¥ang?anRTV?Y*\2+2Q9RTVWgi{_ankUwi§{lwyr~wyUWV anV(r~` (→ → − }_g!jTgia+e~RlwyjThHV"ko9jTgij_ `cT{T{;girca − w ∗ , b∗ , λ ∗ ) HVde acgir~`9wyrnV°rnV(UWgHVd}o¿rngiU acRTV*anrnwikjTkjTh`cV ad Q9RTV}TVde kx`ckgHjo¿Tj|e¬ackgij f k`hHkHV(j?5^ 6 − − − f (→ x)=→ w ∗ .→ x + b∗ =. N X. − − λ∗i yi → xi .→ x + b∗. i=1. 9Q RlV+e xwi`n`bk¡|edwackgij!gio&wjTV(©gH_stVde a → rnVd¢5TkrnVd`9anRTV"e(giUW{T_a~wackgijAgioacRTV}_gya¥{Trcg_}_|e¬av;V at©OV(Vdj − x i w l j 7 } n a T R V B \ + « T ® i g n r ( V g i d V d r i ¨ c a T R. V _ } F V e x k b ` k i g j o¿Tjle ackgij f k`}_V(¡ljTVF}e gHUW{TV(acVd^7B^WanRTV\B« wyjl}?}_gBVd` → − jlxgyaf}_V({;V(j|}!gij?acRTV jTgHj_ t\B«+ 7 87. = =A?/ AE<?/?/P ?/ . D jLacRTVW{TrnV(Bkgil`¥`cTl`cVde¬ankgHj³¨I©ªVWwi`n`cTUWV+anRlwa¥acRTVanrnwikjTkjTh`bV(a¥©9wi`°kjTVFwyrn^®`cV({lwirnwiTV+wyjl}L©ªV `cRTg©RTg©ang7¡ljl}®acRTVWe xwi`n`ckm¡lVdrv©kacRLacRTV|VF`ta*Uwyrnhikj³ D jLacRTkx`*`bll`bVFe¬ankgHj³¨I©ªV"hiV(jlV(r~wyk1!dV acRTV kjTVdwiredwi`cV&ang°Rlwijl}_Vªjlgij_ ÂkjTVFwyrn^*`bVd{lwyr~wylV&acr~wykjTkjlh°`cV an`d:D jl}_V(VF}²¨anRTVªkjTVdwirRB^B{;gyacRlVd`ck` k` ang5g. rnVd`bacrnkxe¬ackiV+o¿gir*rnVdwy Âko¿Vwi{T{TkedwankgHjl`*wyjl}®acRTVdrcVe(wijL;VWjTgikx`bVH¨³UWVdwH`bTrnV(UWVdjHa¥V(rnrcgHrn`°gir*gH_ackVdrn` kjacRTV*anrnwikjlkjThW`cV ad H T ¹ Q9RlV{TrcVdBkgHl` wyjlwi^_`ck`*kx`V BacV(j|}_Vd}B^Lkj5acrngB}Tle kjTh . =" $ {ξi }i∈<1,N > kj£gHrn}TV(r*acg wig© `cgiUWV e xwi`n`bk¡|edwackgij?UWk`banw AiVd`9kjacRTV*anrnwikjlkjThW`cV a+¯p`cV(V . khiTrnVi±¬ Q9RTVF`bV°wyrnkwiTVF`¢5lwyj5ankmo¿^"acRTV°UWk`banw AiVd`&Uwi}_V°5^"acRlVfacr~wykjTkjThV _wiUW{TVF`(:D jl}_VdVd}²¨BQ9RTV¥`bxwieA wirckxwyTV ξi e gHrcrnVd`c{;gijl}_kjTh*angacRlVfacr~wykjTkjTh+HVde acgir xi k`2!(Vdrcg+kmo²anRTV°HVde acgir&k`ªgijanRTV¥higBg_}`bkx}_Vvgio anRTVfanT;V¥}_V(¡ljTVF}B^ {→ ¨igiacRTVdrc©kx`cVfkmaOk`ªVd¢5lwiTangacRlV°}_kx`ta~wyjle(Vf;V at©OV(V(j7acRTVvacT;V°;giljl}Twyrn^ − w , b} y w | j } n a T R. V ; { i g k jHa xi → − − w .→ x + b = ±1 kmo yi.(→ − − w .→ xi + b) ≥ 1 0 → − m k o ξi ( w , b) = → − → − → − → − 1 − yi .( w . xi + b) yi .( w . xi + b) ≤ 1 Q9RTV¥`bgio aOUwirchHkj\B«¥ wykU waOUWw _kUWk%!(kjThanRTVvUwyrnhikj©RTkV¥UWkjlkUWk%!(kjThanRTV°Vdrcrngir~`d¨y©RTkxe~R kUW{TkVF`9acRTV*jTVd © °q&q 6 5 5$&. 6 ½ <_ * ). p¯ H± `bl_stVde aOang 6 yi(→ ¯Ây± − − w .→ xi + b) ≥ 1 − ξi , ξi ≥ 0, ∀i ∈ h1, N i ©RlV(rnV kx`edwyVF}<anRTV
(33) +"#$"
(34) " u wirchHV(r?X wi`n`bkhij|`wKRTkhHRTV(r{|Vdjlwyat^<gHj anRTVacr~wyCkjTkjTh?VdrcrngirF¨Iw`cUwyVdr XacVdjl}T`¥acgkjle rnVdwH`bV+acRTVWUwyrnhikjwijl}AanRTVacr~wykjTkjlh?Vdrcrngir~`(Q9Rlk` N 2 − X k→ wk +C ξi 2 (w,b,ξ) i=1. min. Y[Z:PAY\.
(35) 3.
(36) !"#$". w.x + b = −1. Optimal Separating Hyperplane. Optimal Separating Hyperplane. xl. w.x + b = −1 Support Vectors. Support Vectors. ξl yi = −1. yi = −1. yj = 1. yj = 1 xk. xi. . ξk. 2 = kwk. xi. . .. w.x + b = 1. khilrcV -,6 2kjlVdwyredwi`cVi. 2 = kwk. .. w.x + b = 1. khiTrnV 6 \Bgyo afUwirchHkj!e(wH`bVH. {|wyr~wyUWV anV(rv}_V(¡ljTVF`°w7acr~wi}_V( /gi¶L|V(at©ªVdV(jLwwirchHV(rfUwirchHkjAgHjAgijTV"Rlwyjl}Lwyjl}Aw?`bUwy2jBTU"|Vdr°gio Vdrcrngir~`ªgij?acRTV giacRTVdrRlwyjl}³ u°`+o¿gHracRTV{TrnV(BkgHl`"{TrckUwyO{TrngiTV(U ¯1`bVdV L&¢;À¯ - ±b± ¨&©OVkj5acrngB}Tle V 2wihir~wyjThHVU+Tack{TkVdrn` wyj|} {µi}i∈<1,N > wH`c`cg_e kxwacVF}®rnVd`c{|VFe¬ankHV(^©kacRacRlV7e gij|`tanrnwikj5an` yi(→ − − {λi }i∈<1,N > w .→ xi + b) ≥ i w l j } 6 1 − ξi ξi ≥ 0 N N N 2 − X X X → − → − − ||→ w || − − − +C ξi − λi .[yi (→ w .→ xi + b) − 1 + ξi ] − µi .ξi L(→ w , b, ξ , λ , → µ) = 2 i=1 i=1 i=1. Q gUWkjlkUWk%!(V§anRTV 2wihir~wyjThHkwijv©kacR rnVd`c{|VFe¬a³ang ∂L − ∂→ w. → − − {→ w , b, ξ }. − = → w−. N X. ¨FV a2l` e gHU{l_acVanRTV{lwirbankwii}_V(rnk·wyackiVF` 6. − λi y i → xi = 0. i=1. N X. ∂L ∂b. = −. ∂L ∂ξi. = C − λ i − µi = 0. λi y i = 0. i=1. 0O^`cTl`backacTackjThWacRTVF`bV VF¢5lwankgHjl`Okj5acg Lª¢|f¯#@H± ¨T©ªV*gH_anwikj 6 max W (λ) = − λ. `cT_stVde aacg 6. N X i=1. P>PRQ STUVU. ¯$@H±. N N N X 1 XX − − λi λj y i y j → xi .→ xj + λi 2 i=1 j=1 i=1. λi yi = 0 , 0 ≤ λi ≤ C , ∀i ∈ h1, N i. ¯83H± ¯ - H± ¯ -C- ±.
(37) -. . "
(38) . Rm. $. F. Φ. .. khilrcV 6 zgHV*gio anRTV Uwy{T{TkjTh Φ . Q9Rlk`gi{TackUk%!dwyackgij {TrcgHTV(U kx`acRTV£`nwyUWVwH`|V(o¿girnVi¨vV Te V({TaanRlwaAjTg© ¨ Rlwi`?ang;VKT{T{;V(r ;giljl}_Vd}³ Q9RBl`(¨9wo anV(rRlw·BkjTh£`bgHHVd}acRTkx`{TrngiTV(U!¨O©ªVArnVde(giVdr+anRTV®`bgHTλackgij → o¿rngiU acRTV − w∗ `nwyUWV*Vd¢5lwyackgijK¯1¦y±9wH`9|V(o¿girnV vgyacV*anRlwaanRTV´ ´ QÀe(gijl}_kackgij|`9©RTke~RAwig©Áacg7e(giUW{T_anV b wircV 6 − − λi .[yi (→ w .→ xi + b) − 1 + ξi ] = 0 µi .ξi = 0. © Rlke~RUWVdwyj|`OacRlwya 6 "kmo λi = 0 ¨OacRlV(j µi = C ¨ ξi = 0 wyjl} [yi(→ anRTV®iVFe¬angir xi kx`7gHj acRTV − − w .→ xi + b) − 1] > 0 hHgBgB}`ck}_V*gio anRTV*acT;Vi¨ 9ko ¨;acRTVdj ¨ wijl} → acRlV+iVFe¬angir xi kVF`vgHj − → − anRTV 0;gi<Tjlλ}li wy<rnkVFC`ªgio anRTV*ac0l<|VHµ¨liwyjl<}Ckx`wξi\B=«+¨ 0 [yi( w .xi + b) − 1] = 0 kx`jTgiagijKacRlV7higBg_} °kmo λi = C ¨acRlV(j µi = C ¨ ξi > 0 wijl} [yi(→ − → − `ckx}_V+gioacRTV"acl|V!¯¿VdkmanRTV(r*©OV(e(wH`c`ck¡lVd}!kjl`ck}_V"w ac.RlxV+i +anTb)|V−wijl1}A+kj®ξiac]Rl=V"rn0khHR5xa¥i `ck}_V"gyoanRTV7Y*\2¨Igir |wi}_^7e xwi`n`bk¡lVF}gijacRTV*gyanRTV(r`ck}TV°gio³acRTV+Y*\ 2¥±¬¨_TTa9ka9k``back²wW\B« `bkjle(V°kafe gij5anrckT_anVd`&acgacRTV }TV ¡ljTkackgij!gio → wijl}?wi`cgacgacRlV {|g5`bkackgijgyoacRTV"Y*\ 2 − w ½ ) » #/ * ) ¹ T 6 ; . D jUWgH`bafe(wi`cVd`d¨5acRTV¥anrnwikjTkjThW`bV(ajTV(VF}T`Oacg|V`cV({|wyr~wacVF}B^wUWgirnV¥e(giUW{TkedwacVF}`cRlwy{;V¯pwWe kr~e V o¿gHrvV _wiUW{TV·±¬vQ g7Rlwijl}_V"wjlgij_ ÂkjTVFwyr¥Y*\ 2+¨lanRTV+kjTVdwir¥\B«¥ wyrnVhiVdjTV(r~wyk%!(Vd}B^!kj5anrcg_}_le(kjlh wo¿Tjle ackgij Φ acR|wa Uwy{l`°acRlVW}lwanw?gyo&acRTVWkjT{T_a`c{lwie(V Rm kj5angw?RTkhiRTVdr*}TkUWV(j|`bkgijlwi2o¿VFwanTrcV `c{lwHe V F ©RTVdrcV¥anRTV}Twa~wWwyrnV¥kjTVdwirc^7`bVd{lwyr~wylVW¯p`cV(V . khiTrnV i± 6 5 5 5. . Φ : Rm → − x. −→ F − 7−→ Φ(→ x). Y[Z:PAY\.
(39)
(40) !"#$". -C-. DÂo9©ªV"¡ljl}wUwy{T{TkjTh `cle~RacRlwya¥anRTVWjTV(©acr~wykjTkjTh`cV a kx`*kjTVFwyrn^ → − `cV({|wyr~wyTVi¨ª5^¤wyjlwigHhi^©kΦacRacRlVA{TrcVdBkgHl`wyjlwi^_`ck`d¨anRTVY*\2 {(Φ( {|wyxr~wyi UW), V yaniV()}r~`i∈<1,N wircV!hH>kHV(j5^<acRTV `cgi_ankgHj?gio2acRlV*o¿gig©kjTh °qq 6 max W (λ) = − λ. `cT_stVFe¬aang 6 ©Rlke~RhikiVF`. N N N X 1 XX − − λi λj yi yj Φ(→ xi ).Φ(→ xj ) + λi 2 i=1 j=1 i=1. N X. λi yi = 0 , 0 ≤ λi ≤ C , ∀i ∈ h1, N i. ¯ - i± ¯ - H±. i=1. − λ∗i yi Φ(→ xi ). ¯ - 5±. − − λ∗i yi Φ(→ xi ).Φ(→ x ) + b∗. ¯ - ¦i±. → − w∗ =. N X i=1. − f (→ x) =. N X i=1. 5^?kma~`fwyTV kj!acRTV*o¿VFwanTrcV Q9RlVd`cV*o¿girnU+Txwi`wircV*giTanwykjTVF}?B^`bT|`tankman_ackjThanRTVkjT{T_avHVde acgir → − x `c{lwHe V Φ(→ − vgyacV*anxRl)waF¨lkj L&¢;°¯ - i±9wijl}K¯ - ¦i± ¨ Φ e gHj5acrnkTTacVd`fgijT^7angWacRTV}_giaf{TrngB}Tle¬af;V at©OV(Vdj?o¿VFwanTrcV V TwyUW{TVd`d°uf©ªV"jTV(VF}Aang ABjTg© kx`fanRTV"wiTV"gyo§anRTV}Tgya°{lrcg_}_le a°o¿gHr¥wijB^e(giT{TV+gio&{;gikjHa~`vgio anRTV kjT{T_av`c{lwie(Vi¨_©RTkxe~R!e(wyj!;Ve(wiVd } 6 − − − − K(→ x,→ x 0 ) = Φ(→ x ).Φ(→ x 0). XªgHjl`cVd¢5TV(j5an^H¨lko&©ªV¡ljl}!anRTk`°V
(41) _{TrnVd`n`ckgHj³¨TanRTV(jAanRTV+V _{Tkxe kaIABjTg©Vd}ThiVgio Φ kx`°jTgya¥jTVde(Vd`n`cwirc^H Q9Rlk`9kx`9acRlV ! $ T - V u l`bV(o¿T|e(rckacVdrckgijang ABjTg©¤ko2w*o¿Tj|e¬ackgij7edwyj;V°w*wykx} AiV(rnjTVdTk`&hikiVdj5^AV(r~e Vdr§acRlV(girnV(U 6 w`b^BUWUWV anrckxeo¿Tj|e¬ackgij´ k`Ww AiV(rnjTVdªko°wyj|}£gijT^Kkmo K(x1, x2 ) kx`w®{;gH`ckmankHV?}_V(¡ljTkacVUWwyacrnk ² Q9Rlk`vacRTVdgirnV(U wyg©f`l`°acge giUW{TTacV"}TkrnVde ac^?anRTVW}_gya¥{Trcg_}_|e¬a −→ −→ ©kacRTgH_a¥e(giUW{T_ankjTh anRTVvkUwyhHVd` Φ(−x→1 ) wijl} Φ(−x→2 ) Q9RBl`anRTVvV _{|Vdjl`bkiV°e giUW{TTanwankgHj7K(gyo xΦ1,kxx`ª2w·)Hgikx}_Vd}³¨Hwi`ª©OV(|wi`ªkan` V _{Tkxe kav}_V(¡ljTkackgij³ vgyacVvacRlwyaacRlV AiVdrcjlV(le(girnrcVF`b{;gijl}l`acg+w+}_gya&{lrcg_}_le a°¯p`c^BUWUV(acrnkev}_V ¡ljlkmanVv{;gH`ckmankHVo¿Tjle ackgij|± kjWacRlVfo¿VdwyacTrnVv`c{lwie(ViQ9RTVfgH{_ackUWk1!Fwackgijkx`&`back;we(gijBiV °qq<©kmanRkjTVdwir&VF¢H|wykmat^e gij|`tanrnwikj5an` `ckjle V¥acRlV AiV(rnjTVd²k`9{;gH`ckmankHV }_V ¡ljlkmanVi. ¹ |¹ ) ¹ /1 ´*VdrcjlV(x`"e~RlwirnwHe¬anV(rnk1!dVWacRTVA`ckUWkwirckat^£}_V(hHrcVdVgir"acRlVe(girnrcVdwyackgij£;V at©OV(V(j<at©ªgK`nwyUW{TVd`dKQ9RlV(^ U"l`baWR|w·iVRTkhHR ·wilVd`o¿gire(g5`bVdr7`cwiU{lVF`wijl}¤g©#·wilVd`o¿girWo¿lrbanRTV(r7`nwyUW{TVd`dÁQ9RTV!Ug5`ta e(wH`c`ckxe(wy AHV(rnjTV(x`*¯¿©RTkxe~RAwyrnV¥©kx}_Vd^7|`bVF}?kjrcVdUgiacV `cV(jl`ckjlh5±OwyrnV 6 5 5 E. 6 ½ . P>PRQ STUVU.
(42) -. . "
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