HAL Id: inria-00072105
https://hal.inria.fr/inria-00072105
Submitted on 23 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
Image segmentation using active contours: calculus of variations or shape gradients?
Gilles Aubert, Michel Barlaud, Olivier Faugeras, Stéphanie Jehan-Besson
To cite this version:
Gilles Aubert, Michel Barlaud, Olivier Faugeras, Stéphanie Jehan-Besson. Image segmentation using active contours: calculus of variations or shape gradients?. RR-4483, INRIA. 2002. �inria-00072105�
ISRN INRIA/RR--4483--FR+ENG
a p p o r t
d e r e c h e r c h e
TH `EME 3
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Image segmentation using active contours: calculus of variations or shape gradients?
Gilles Aubert — Michel Barlaud — Olivier Faugeras — St´ephanie Jehan-Besson
N° 4483
June 2002
$/0* 1$1 2)345$67*0 /8
9,:<;=;=>@?BADC0EF>@G*HJILK/MN:=OJP0>@;Q#RSG;TRSCUVWK/X;=:=YZ:=>@GW[/RSC0\]>^G_RS?a`WK/bZHc>@d0P0RSe0:=>
f >@P0RSegQ>@?a?h]e V
i"jlk
manmoprqts@uvmawyx^zuy{}|@s
j
|@nnm*~n#xz
j
{sm^
{n#x ma*|^ssm*mz|@ssx{yx^szma
wy|tmu"m*m
x|^wyu+mwymz
j
mawyz
j
ms@op^sm)^^po@x m
W¡¢£¤_¥¦£§©¨
mz|@sy{mawu
j
m)wy|^ª«}man¬|0vm ^nms@uv{s #xsW{n#x m®u
j
wy|^
j u j
m)n{s{n{¯xuy{}|@s°|
x^s4msmawv ^Lz*wy{}uymwy{}|@sL{s@±|^«}±{s Dwym* ^{}|@sx^s²ª³|@sx^wvL=sz*uv{|^sx«´
¨
m°
j
|Jµ¶u
j
xu|^smBz*xs4 ^|
=wy|^n·|^sm¸z«xuv|u
j
m¸|u
j
mwª@Ly|«±@{s
|{yy|^s¹|^wºm«n
j
|^«}uy¯¹m»xuv{|^sµ"{}u
j
µ¼m«}«}~z
j
|^yms
ª|^³sxwyz*|^s{}uy{}|@s*´D½y{s ,u
j
{)ma»{}±Jx«}maszm^µmBvu4u
j
mDz*x^vm#|+x«xwy mBz*«x^y|&wym* ^{}|@s
=³szuy{}|@sx«+ª@°yuyxs³xwnmu
j
|+|0u
j
m)zx«z*«"|±Jxwy{xuv{|^s³x^smwy{±m®u
j
mz|^wwym³|@s{s
¾
«mwv~T¿x ^wxs m&m»xuv{|^s´
¨
m®wvm±@{y{}u¼u
j
{&wv|@ª«mnÀy{s u
j
ms|^uv{|^s¸|
j
xmmawv{±Jxuy{}±^mxs³
j
|JµÁu
j
xu/u
j
myx^nm"m»xuy{}|@s/zxs#ª³mm*«m* ^x^s@uv«mawv{±maµ"{u
j
|^u/ ^|{s ®u
j
wy|^
j u j
mssxuywx«
yuvma®|z|@s@±mwyuv{s "u
j
mFwym* {|^s{s@uvm* @wyx«{s@uv|ª³|@sx^wv{s@uym* ^wx«´
¨
mx«v|+m*Âsmx&«x^wv ^mwz*«x^yl|
wym* ^{}|@s)=³szuy{}|@sx«Fªx^vma#|^s#u
j
m+myuv{n#xuy{}|@s¸xs¸z*|^n#xwy{v|@suy|xwy|uv|^uóm+|lu
j
m³wv|@ªxª{«{}uÃ
msy{}uÃ{vuwv{ªuy{}|@s,|/{n#x ^m®Ämxuwymx^s7
j
|Jµ
j
|JµÅu
j
m
j
xmmwy{±_xuv{±m)uy||«x«}«|Jµ++uv|
maxy{}«7z|@n#uvm¸u
j
m¸z|@wywym³|@s{s 7Æ#ÇxuvmaxÈLmwy{}±Jxuy{}±^mx^s
¾
«mwv~T¿x ^wxs mm»xuv{|^s´
¨ m
ÂSsx««})x^«u
j
{0sm*µÉ=sz*uv{|^sx«@uy|u
j
m"wy|^ª«mnÅ|wvm {|^sym* ^nmas^uxuy{}|@s{sym»ms³zm|Sz|«|^w
{n#x ^m´
¨
m)ªwy{}m*ʳ°mayzwv{ªm|@w"snmwy{zx«z
j
mnmxs³W
j
|JµNy|^nm®mÈmwy{nmas^ux«wym«uy´
ËÌ@Í]ÎÏФ_Ñ¢§ ÒÓ#ÔÕ^ÖW×lÖTÕÓ¸ÖØSÙTÔÙ=Ú=ÛØÜÝ+ÖÕÚ=ÛJØÞ×]ÖÕÓ¸Ö*ØÙÔÙ<Ú=ÛJØÜßà*Ù<ÚâáJÖLã/ÛØÙÛJäåçæçÜßàÙ=ÚâáJÖBÝ+Ö*è
ÕÚ=ÛJسæçÜÒ Ó¸Ô*Õ@Ö×]ÙTÔÙ=ÚæÙ<Ú=à æçÜÝÖTÕÚ=ÛJØéläØ]à*Ù<Ú=ÛJØ]ÔêæçÜëÛJäØSì^ÔJå íéäØSàÙ<Ú=ÛJØSÔêæçÜ+ã/Ôêîà*äêïäæÛtð,ñSÔJå Ú=ÔÙ<Ú=ÛJسæçÜ
×òÔ óÖ#ôó]Ù=ÚâÓÚ}õaÔÙ<Ú=ÛJØÜ0×òÔçóÖ#ö&åyÔ^ìJÚ=Ö*ØÙ=Ü÷äê}Öå è<øÔ*ÕåyÔJØÕ^ÖÖù*ä³ÔÙ<Ú=ÛJسæçÜöúÔÙÖçÔJäûBì^Öå ÚâáJÔÙ<ÚâáJÖ Üü"ÔåyõaÖØ
ýDÚâØSì^ÛJþÁ÷æ Ù<ÚâÓ¸ÔÙ<Ú=ÛJØÜøZÖ*á_Öê]×lÖ*Ù/ÿ7ÖÙ=òÛaì_æ
!"# %$'&()!*+, -/. 01112. 034#1456&/789-:<;
/ ="!!>?"@A !B C%-:,EDGFIH >JKLDG!MFN2 O@2*!<;
P%QGR
-D
S
,UT#-
V ,RGW ,EDX-:YDZ>[<5\0114Y ]O_^J*!OO5a`cbd3Te1f31#0:5C-:2.EDZ>[_&/!+gJ5\h >*!#;
Ki7! +jG ;> ;Kk
$/ $67 6, .$/ 8
Ì¢ Ì §
|^m*u{|^s«mwy|^ª«
k
manm)m«x°ym* @nms@uyxuv{|^s,l¹sm{n#x mmas7n{s{n{yx^s^u®sm
masmwy {m=x{yx^s@u{s@uvmawv±^ms{wma®Ä|^sz*uv{|^ssm*««}mamBwm* ^{}|@s7mum#=wy|^s@uv{
k
mwym´
|^³)n|^s@uywy|^s»m
z*mmÈ uÃm#mWÄ|^s³zuv{|^s³sm*««}ma#v|@s^u«}{*mmxw«}ma m»xuv{|^sm
|^{y|^s²|^ÞmDºm*«n
j
|«uv¯
x_±^mz®maz|@s{uv{|^sxÈD«}{n{}uymª{}mas,z
j
|{v{m´&Æw_Çxz*m
k
x¸zmuvuvm²m»{±Jx«msz*m³s|@Dmuy{}|@s+sm
«xwy m®z«xymm®Ä|@szuy{}|@ssm«}«m"m)wm {|^sBx^w"m"nÉmu
j
|ma"z«xv{»m&m)zx«z*«m&±Jxwy{xuv{|^s
mu/ms°¼m³{y|^s³«}ma#ma»xuy{}|@sl¹
¾
«mwv~T¿x @wyx^s m&z|@wywym³|@sx^s^uym´
|^¼wvm*Ä|^wn«|^s0«mnDÇmanm
³wv|@ª«
k
mnm"ms¸uymwnmFmmawv{±Fm*mam+Ä|@wynmm*u¼n|@s^uwv|@s/»m«}ma¼nDÇmanmama»@³xuv{|^s³/³ma±mas@uÇmuywym
muyx^ª«}{mm#n#xs{
k
mwym²m«m* ^x^s@uvmxsx^yymwxw«<¹vmuyxm{suy{}«mmz*|^s@±mawyy{}|@s,m{s^um* @wyx«}mam
wm {|^s®ms{s^um* @wyx«}mal=wy|^s@uv{
k
mawvm^´
|@0¼mÂSs{y|^sxy{³sm"z«xvmmsz*|^wym¼«0«xwy m&m¼Ä|^sz*uv{|^s~
sm*««}ma»@{F*¹x³mas@u®w«T¹myuv{nxuv{|^s4m¸«x«}|^{/mBwy|^ªx^ª{}«{um#mBzmawvux{smzxwxz*umawv{vuy{»ma
{n#x ^mWmuBn|@s@uywy|^s#»m,«T¹|@uv{«mawv{±Fm*mWmÄ|^wnmmwnm*uBm,zx«z*«mw¸x{mnmas^u«m¸mawv{±Fm*ma
mDÆ#ÇxuymxȲz*|^wwy³|@sx^s^uymx{sy{¼»m°«}ma²ma»@³xuv{|^s³l¹
¾
«mwv~T¿x ^wxs m´
|@)uymwn{s|@s)mas
x^«{»x^s@u¼zmuvuyms|@±m«}«mÄ|^sz*uv{|^ssm«}«m®xBwy|^ª«
k
manmm«xvm ^nms@uxuv{|^s°mwm {|^s/³xs&sm
ma»@msz*m®¹{n#x ma+z|@«mw´
|^+mzwv{±|@s&ªwv{
k
m±mnmas^u"s|^uywym{n#«*manmas@uyxuy{}|@ss@³n mwy{»m®m*u
³wmavmas@uv|^s³¼»m«»ma"wmay«}uxuym*Èmwy{nmas^uxÈl´
'г£a¢Îç¦Ì¢D§ ×]ÖÕÓ#Ö*ØÙÔÙ<Ú=ÛJزìÚâÓ¸Ô*Õ@Ö æçÜ"àÛØÙÛJäåçæÔàÙ<Úðæ#å ÖyÕÚ=ÛJسæÔàÙ=ÚâáJÖ æçÜ/æÙÔÙ<ÚæÙ=Ú=ù*ä³Ö æÚâÓ¸Ô*Õ@Ö Ü
ðÛØSàÙ=Ú=ÛسØ]Ö*êâêîÖæå ÖyÕÚ=ÛJسÜðÛJØ]à*Ù<Ú=ÛJØØSÖêâêîÖæFðçåyÛJØSÙ=ÚÖ*åyÖܼàÔêîà*äêìÖæ®áJÔJå Ú=ÔÙ=Ú=ÛØæçÜÛ óSÙ<ÚâÓÚæ*ÔÙ<Ú=ÛJØ ìÖFðÛJå Ó¸Ö Ü
ÕåyÔ^ìJÚ=Ö*ØÙìÖðÛJå Ó¸ÖÜÖaùä³ÔÙ=Ú=ÛØæWì÷äêîÖ*å è<øZÔÕåyÔØ@Õ@Ö Ü)ìÖ*å ÚâáÖÖ,ì^ÖLöúÔÙTÖçÔJäûÜÓÖÙÄòÛì^ÖìÖæ Ù<ÚâÓ¸ÔÙ<Ú=ÛJØ
ì^Öü+ÔJåyõaÖسÜFÓÖ*Ù=òÛaìÖæìÖ*Øæ*ÖÓ*êîÖ æì^ÖسÚâáJÖÔäL
o
00
£a¤_ÐÑ ¦£ ÄÐ
¤_ÐS¡ ÄÌ l£a¥£Ì ²Ì
£
´&|@sx^wvBx^swvm {|^s¸=³szuy{}|@sx« ´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´
´ ¾
Èxn#«m&|0y³z
j
|@uv{n{}¯axuy{}|@sWwy|^ª«}man&{sD{n#x m
³wv|zmayy{s Å´´´´®´´®´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´
´o sDmÈxn#«m®{s@±|^«}±{s #u
j
m)nmaxsN´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´
´ sDmÈxn#«m®{s@±|^«}±{s #u
j
m±Jx^wv{xsz*m©´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´
¤_Ì¢a¢ ÄÐ Ð! +¤_Ì#"! ÄÐ ¦£ ÄÐ ¥ Ä¢°¥¢¸¡/Рѥ³¤Í$ ¦£ ÄÐ ¥ =¢B¥ Ñ ¦Ð !% ̤_¢Ì âÍ &
o³´
i
wxsÄ|@wyn#xuy{}|@sB|0wym* ^{}|@s¸=sz*uv{|^sx«+{s@uv|¸ª|^s³xwy¸=sz*uv{|^sx«´´´®´´®´´´´'a
o³´ i
wxsÄ|@wyn#xuy{}|@sB|0ª|^s³xwy¸=sz*uv{|^sx«&{s@uv|¸wym* {|^s°=sz*uv{|^sx«´´´®´´®´´´´'(
) * Ð Z£¥£ ÄÐ Ð+ "£-,ZÌLÌ % Ð £ ÄÐ Ì#.¥£ ÄÐ Z¢ " ¥¡/Рѥ¤aÍÞ¥ / ¤aÐS¥¦0, +
´
m* {|^s~T{sma³masms@u&Ämaxuy³wvma´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´'_
´^´
i
wxsvÄ|^wn#xuy{}|@sB|0wvm {|^sB=szuy{}|@sx«"{s@uv|¸ª³|@sx^wvB=szuy{}|@sx«´´´´'_
´^´ 1¼|^n#uxuv{|^sW|0u
j
mÆ#ÇxuvmaxÈDmwy{}±Jxuv{±m´®´´´´´®´´®´´´®´´®´´´´'ao
´^´o 1¼|^syuywzuy{}|@s |xs |@uv{n#x«±m«}|z{uà ±mazuv|@wBÄ|@w°u
j
m²m*±^|«uv{|^s |xs
xzuy{}±^mz|^s@uy|^w´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´'
´ Æmasmwx«zxym´´´®´´®´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´'
´´
i
wxsvÄ|^wn#xuy{}|@sB|u
j
m)wym* ^{}|@s¸=s³zuv{|^s³x«{s@uv|Bx#ª³|@sx^wvB=szuy{}|@sx«´´'
´´ 1¼|^n#uxuv{|^sW|0u
j
mÆ#ÇxuvmaxÈDmwy{}±Jxuv{±m´®´´´´´®´´®´´´®´´®´´´´'2
´´o 1¼|^syuywzuy{}|@s |xs |@uv{n#x«±m«}|z{uà ±mazuv|@wBÄ|@w°u
j
m²m*±^|«uv{|^s |xs
xzuy{}±^mz|^s@uy|^w´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´'a
* Ð Z£¥£ ÄÐ Ð! £,ÌÞÑ̤- % ¥£ % Ì ¢ " ¢-,Z¥ ÌÞÑ̤ % ¥£ ÄÐ £ÐZРĢФ43(5ÐÏ £Ð
6 Ì@Ì ¥²¤_Ì"+ ÄРФ Ä¥£ ÄÐ 87 9
:
´ qts@uwv|z*uv{|^sW|0uywxsvÄ|^wn#xuy{}|@s4´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´'
: ´ 1¼|@n#uyxuv{|^sD|u
j
m)mwy{}±Jxuy{}±^mv{s ¸
j
xm)mwy{±_xuv{|^s°uv||« ´®´´´®´´®´´´´'
: ´´
m*«xuy{}|@sWªm*uõm*masWu
j
mÆ#Çxuymx^ÈWmawv{±Jxuv{±m®xs³Wu
j
m)
j
xmmwy{}±Jxuv{±m´´·
: ´o 1¼|@syuywzuy{}|@s|u
j
m¸±m«}|z*{}uñmz*uv|@w|"u
j
mBx^zuv{±mBz*|^s@uv|@w=wv|@n·u
j
mDÆ#Çxuymx^È
mwy{}±Jxuy{}±^m ´´´´®´´®´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´·^
: ´o³´
m* ^{}|@s~T{sma³masmas^uÄmaxuy³wvma¼´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´·^
: ´o³´ Æmsmawyx«z*xym ´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´·o
: ´ «{zxuy{}|@s´´´®´´®´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´·
¥£¦0,8 ";,< Ä¢£Ð!"¤_¥ ²¢ +
= * Ð ÄФ>,8 Ä¢£Ð!"S¤a¥ ²¢§¢aÌ#" ²Ì £¥£ ÄÐ Ð+ +¤_Ì"! ÄÐ ¢? $% ÄÑÌÐ ¢Ì.Ì ¦^Ì@¢ @9
´ qtn#«mnmas^uxuy{}|@sN´®´´®´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´1o^
´ ¾
ȳmawv{nms@uyx«wvmay«uyN´´®´´´®´´´´®´´´®´´®´´®´´´´´®´´®´´´®´´®´´´´1o@
9 * Ð ¦ Z¢ ÄÐ
:
$°670
xs@Lwy|^ª«mn#®{sL{n#x ^m#³wv|zmayy{s ³Zz
j
xym* @nmas@uyxuy{}|@sluwyx^z^{s l|^wz«xy{îÂzxuy{}|@slz*x^s
ªm¸z*x^vu{s4u
j
m=wyx^nmµ¼|@wD|¼|^uv{n{¯xuv{|^sLu
j
m|^wylm´ ³´#xu
j
m¸n{s{n{}¯axuv{|^s|&v|@nmmasmwy
nmaxwvm^´
i"j
mWmsmawv ^{|Äuyms y|^nmWz*|^nª{s³xuv{|^s |wym* {|^s²|@wª|^s³xwy4=szuy{}|@sx«´
i"j
m
n{s{n{¯xuv{|^sÁ{Bx««}Ás|u¸uwv{±{x«xs³Án#x^s^Ánm*u
j
|³
j
x_±m7ªm*mas m*±^m*«|^mÁuv| wymx^z
j
x^s
|@uv{n³n µ j {z
j
n#x_°ªm|^s«}B«|z*x«<´
¨
mx^wym
j
mawvmu
j
m®wy|^ª«mn |u
j
m|^uy{n{¯xuv{|^s¸|wvm {|^s#|@wFª|^s³xwy=³szuy{}|@sx«Fµ"{u
j
u j mnmu
j
| |x^zuy{}±^mLz|@s^uy|^w´ zuy{}±^mz|@s@uv|^³wy
j
x_±m7ªm*ms {s@uywy|z*m ª@)x*
¨
{u@{s
x^s
i
mwy¯*|@³|@«|^ FxsLµ¼mawvm|^wy{ {sx««},ª|^s³xwy7nm*u
j
|³*´sxm ªx««|@|@sa /|^w
^m*|mav{z"xz*uv{±mz*|^s@uv|@wy ]x^wvm+wy{±ms#uy|Jµx^wy³u
j
mm ^mF|xs#{n#x mu
j
wy|^
j u j
mn{s{n{}¯axJ~
uy{}|@s²|+x7ª|^s³xwy{s@uvm ^wx«/|=³szuy{}|@s|&Ämxuwymmms{s |^s²m m´ z*uv{±m°z*|^s@uv|@wy
³wv{±masª@#wvm {|^s=sz*uv{|^sx«F{s¸x^{uv{|^s#uy|ª³|@sx^wv=sz*uv{|^sx«
j
x_±m+x^mxwym«xuymw´qts@uywy|~
³zmaDª@ xso@ u
j
m*
j
x_±mª³mmsW=wyu
j
maw+m±m*«|^mW{s : :
@³o^o^o³o³o
:
:
³
x^s :
@@ ´qtsmSmazu*@u
j
mym"|xz*uv{±m+z|@s@uv|^³wyÄ|@wu
j
m|^uy{n{¯xuy{}|@s|lxz*wy{}uymwy{}|@s{sz*«{s
ª|u
j
wvm {|^s°x^sª³|@sx^wv#=³szuy{}|@sx«+x³³maxw¼uy|Bª³m)wymx«}«B|Jµ¼mawt=«T´
qts masmwx«T]Ämxuywym|u
j
m#{n#x m#wym* {|^s7uv|,ª³m¸ym* @nmas@uvmuywxzml]m*uz´´*´yx^wvmmnª³mam
{s wvm {|^sÞ=sz*uv{|^sx«Bµ
j
{}«m,u
j
mLª³|@sx^wvÉ=sz*uv{|^sx«x««|Jµ+¸yn||u
j
sm¸xs³Áwym* @«xwy{}uà |
u j
mLwym* ^{}|@sÁª|^³sxwy´
i"j
mLª³xy{z,wy{sz{«m7{°uv|Éz*|^syuywz*uBx xwxª|«{zD³xwyuv{x«{]mwyms@uv{x«
ma»xuy{}|@s
¾
=wy|^n u j
m#msmawv ^,z*wy{}uymwy{}|@sllm´ ³´ª@7z*|^n#uy{s ,y|^nmy|^wyu|
¾
«}mawt~«x ^wxs m
ma»xuy{}|@s@u
j
{
¾
z j xs ^m/u
j
m®
j
x^³m|u
j
m®z*wwvmas@u¼z*³wv±^mxzz|@wy{s uy|y|^nm±^m*«|z{uÃÂm*«
µ j {z
j
z*x^sªmBu
j
|^
j
u|"xx,{wvmazuy{}|@s|&mzms@u®|&u
j
mBmsmawv ^Lz*wy{uvmwy{|^sl´WÆ{±mas4x,z*«}|@vma
zwy±mmsz*«}|@v{s #xsW{s{uv{x«wvm {|^sB|@sm®u
j
masz|^n#uvm&u
j
m)y|«uv{|^sW|u
j
{
¾
Ä|^w&u
j
{"{s{uv{x«
z*|^s{}uy{}|@sl´
i"j
m)z|@wywym³|@s{s )=x^n{}«B|0z*wy±ma&mzwvmaxymu
j
m®masmwy Bzwv{uvmawv{|^sDxsDz|@s@±mwy ma
uy|Jµx^wyÁx <«}|z*x«
n{s{nn|u
j
mzwv{uvmawv{|^s
j
|@³m*=«}« z|^wwym³|@s{s ²uv| u j
m7|^ªtmzuBuy|ɪ³m
ym* @nms@uvmal´
i
|¸z|@n#uvm)z j x
¾
³ym*±^mwx«]nmu
j
|
j
x_±mªm*mswy|^|^ym´
@|@nmx^u
j
|@wy|Ds|u®z|@n³uvmu
j
mu
j
m*|^wym*uy{zx«mÈwymv{|^s7|/u
j
m±^m*«|z{uÃWÂm*«<ª³xy{zx««}
u j
m ^wx{ms@uZ|³u
j
m¼masmwy z*wy{uvmwy{|^s
ªuz
j
||^ym¼xmÄ|@wyn#xuy{}|@s)|u
j
m"z*³wv±^mFu
j
xuµ"{}««nx!mu j m
zwv{uvmawv{|^smaz*wymxym"
:
(# $=u
j
m*¸z|@n#uvm+x{wymz*uv{|^s|mayz*ms@u
´u
j
mw¼xu
j
|^w% :
@oo@o
:
^
z*|^n#uym¼u
j
m&u
j
m|^wym*uy{zx«mÈwymv{|^s)|³u
j
m&±m*«|z{uñmazuv|@w=wy|^nÅu j m ¾
«mwv~¿x @wyx^s mma»xuy{}|@s*´
i"j
m¼z|@n#uyxuv{|^s®{mwvÄ|^wnm®{su
j
wvmm¼n#x{s)yuvm³*´'&{wvu_wym* ^{}|@s®{s@uym* ^wx«wymwymyms@uy{s +wym* ^{}|@s
=³szuy{}|@sx«0xwym¼uwyx^stÄ|@wynma{s@uy|®ª|^³sxwy{s^uym* @wyx«y{s u
j
m+Æwym*mas~
{}man#xssu
j
m|^wymnD´(@maz ~
|@sl@u
j
m®z|@wywym|^s{s
¾
«}mawt~¿x ^wxs ^m¼ma»xuy{}|@sFx^wvmmwy{}±^mlxs°vma¸uv|mÂsmx)s³xn{z
z
j
manmuv|m*±^|«±mu
j
m{s{uv{x«wym* ^{}|@sl´ s|^u
j
mawx«}uymwsxuy{}±^m+{¼uy|)m*maBu
j
mwym* ^{}|@s#Ä|^wn«xuy{}|@s¸uv|
z*|^n#uymu
j
m @wyx^{ms@uF|u
j
mmsmawv ^¸z*wy{}uymwy{}|@s#µ"{}u
j
wym³mazu¼uv|u
j
m®wvm {|^sBª|^s³xwy#{syuvmax¸|
wym³z{s )wym* {|^s{s@uvm* @wyx«0uy|ª|^s³xwy{s^uym* @wyx«´Zqts* : @u
j
mx^u
j
|^w/wy|^|^ym"uv|z*|^n#uym+u j m
mwy{}±Jxuy{}±^m)|¼u
j
m#z*wy{}uymwy{}|@sµ
j
{«}muyx!^{s B{s@uv|xzz|^³s^uu
j
m#{yz*|^s@uv{s{}uy{}max^z*wy|^"u
j
mz*|^s@uv|@w*´
qts :
]@ u
j
mz*|^n#uxuv{|^s|u
j
mm*±|^«uv{|^sDm»xuv{|^s{xz
j
{}m±mDu
j
wv|@
j j
x^³mmwy{}±Jxuy{}|@s
³wv{sz{«m´
i"j
{Lz*|^n#uxuy{}|@s ªmz*|^nmLn|@wvm {s@±|^«}±^mNµ
j
msÅ ^«}|@ªx«{sÄ|^wn#xuy{}|@s x^ª³|@u7wym* {|^s³{
³wvmavmas^u¼{s°u
j
mmsmwy #z*wy{}uymwy{}|@sl^u
j
m®v|~zx««}maBwvm {|^s~Ãma³masms@uFzxym´0qÃu
j
xms³/Ä|@wFm*Èx^n#«}m
µ j
ms#vuxuy{yuv{z*x«Ämxuywym|]xwym* ^{}|@sz
j
xÄ|@wmÈx^n«}m^u
j
m+nmaxs#|@w0u
j
m+±Jx^wv{xsz*m&|]u
j
m+{s~
uymsy{uÃxwym"{s@±|^«}±^m{s#u
j
mn{s{n{}¯axuv{|^s´
i"j
{z*xym
j
x^Fªm*ms¸yuy³{}ma¸{s*2³@!*+
:
:
^+ T´
2
qts :
@^ ³u
j
m+xu
j
|@wy
j
|JµÉu
j
xuu
j
m+n{s{n{¯xuy{}|@s|S=szuy{}|@sx«{s@±|«±{s wym* ^{}|@s~mmsmas@u
Ämaxuwvma{s³zmax{uv{|^s³x«uvmawyn#0{su
j
m&m*±|^«uv{|^s)ma»xuy{}|@s)|Su
j
m"x^zuv{±m&z|@s@uv|^³wu
j
xux^wvm{n~
|^wyuyx^s@uF{sWwxz*uv{zm^´
i"j
mavm®x³{}uy{}|@sx«]uymwnxwymmaxy{}«Bz|@n#uvmaBu
j
xs uv|#
j
xmmwy{}±Jxuy{}|@s
uy||«*´
qtsu
j
{xwyuv{z«mµm)z«xwy{îÄWu
j
mwvm«xuv{|^s
j
{"ª³muõ¼mmsu
j
mª|^s³xwyWx^swvm {|^sW=sz*uv{|^sx«
u j
xuxwy{ymsxuywx««{sBym*±^mwx«{n#x mwy|z*mv{s )ux *´
¨
m
j
|Jµ {sBymz*uv{|^sBou
j
xuF|@smzxs¸ ^|
=wy|^n¶|^smuv|u
j
m+|u
j
mw¼ª^v|^«}±{s
|{v|@sl¹ma»@³xuv{|^sµ"{u j
{wv{z
j
«m*uFz*|^s{uv{|^s³|^wºm*«n
j
|«uv¯^¹
ma»xuy{}|@s°µ"{u j
man#xssz*|^s{uv{|^s³*´
¨
mBu
j
ms²z|@szmas^uwyxuvm|^s4u
j
mBwv|@ª«mn·|FÂs³{s ,«}|z*x«Fn{s{n#x|"xD«x^wv ^m¸z«x|&wym* ^{}|@s
=³szuy{}|@sx«*´¬qtsNvmazuy{}|@s ɵmÂwyyuDuywxsvÄ|^wn!u
j
man {s@uv| ª|^s³xwyÞ=³szuy{}|@sx«xsNx«
nmu
j
|&=wy|^n u j
mz*x«z*«+|0±_x^wv{xuy{}|@s&uv|Bz*|^n#uymu
j
mz|^wwym³|@s{s ¸Æ#ÇxuvmaxÈmwy{}±Jxuv{±m
x^s z|@syuywzuBx ±^m*«|z{uÃɳm« |@s u j
mwvm {|^s ª³|@sx^wv^´
i"j
{°Â³m« m*Âsma°x
¾
µ j
|^ym
y|«uy{}|@sllÄ|^wxD ^{}±^ms{s{uv{x«¼ª³|@sx^wvLz*|^s{}uy{}|@slZm*ÂsmxW|@sm~Ãx^wyx^nmuvmaw+=x^n{«L|&wvm {|^s
µ j {z
j
@{sBwxzuy{z*mz|@s^±^mwy ma0uv|Jµ&xwx«}|zx«n{s{nnÀ|]u
j
m+=³szuy{}|@sx«T´
i"j
mwy|^ª«}man|u j m
m*È{yuvmaszmxs³D³s{»masm&|0x¸v|^«uv{|^sWuv|#u
j
{
¾
{+s|u"x^wymvma°{s°u
j
{+xwyuv{z«m´
¨
msmÈuz
j
x^s mF|^w³|^{s@uZ|±{}mµ_x^s)wym~Ãmwy{±mu
j
mxnm/ma»@³xuv{|^s³{sx+y{n#«mwxs³)n|^wym
s³xuy³wyx«@µ&x__{T´m^´µ"{u
j
|^u0 |^{s u
j
wv|@
j u j
m&uywy|^³ª«}m|Suyws{s ®wvm {|^s{s@uvm* @wyx«{s^uy|ª³|@sx^wv
{s@uvm ^wx«u
j
{+{x^z
j
{}m±ma{s,ymz*uv{|^s
:
ª@x«{s °
j
xmmwy{±_xuv{|^s,nm*u
j
| @<2 T´
i"j
{
«{sm&|lxwy|^x^z j)j
x^x«wvmax)ªm*mas)Ä|^«}«|Jµ¼ma{s o( ³{s
j
{µ|^w ®|^su
j
m&mavuy{n#xuy{}|@s|Su
j
m&|^uy{zx«
Ê|Jµ´
¨
m°u
j
ms²uwys²{s ymz*uv{|^s 2,|^³w)xuyuvms@uy{}|@s²uv|Lx,smµ z«x|+wym* ^{}|@s~ª³xym,=³szuy{}|@sx«ª@
z*|^sy{mwy{s
j
{vuy| ^wxn#l|³{nx mÄmaxuwvma*´
i"j
m¼
j
x^³mmwy{}±Jxuv{|^suy|@|^«x««|Jµ²³Zuy|mx^v{«®mwy{}±^m
u j
m±m«}|z*{}uÃ#³m«Du
j
xu+m*Âsma"u
j
mm±|^«uy{}|@s°|u
j
m)wvm {|^sWª³|@sx^wv^´
@mazuy{}|@s,{"m*±^|uymWuv|¸xsx³«}{z*xuv{|^s°|0u
j
m)wym*±{|^&nm*u
j
|"uy|u
j
m)wy|^ª«mn |wym* ^{}|@s
ym* @nms@uyxuv{|^sµ"{u
j
x ^{}±^ms)z*|«|^w
j
{vuy| ^wxn {sxvma»msz*m¼|³{nx ma*´wmȳmawv{nms@uyx«@wym«uy
j
|Jµ u j xu"u
j
muvmaz
j
s{»m
j
x&{smmy|^nm®{s@uvmawvmavuy{s #|uyms@uv{x«´
$,(
qts²n#xs@7{n#x ^mwy|zmayy{s Dwy|^ª«}manZu
j
m¸{ym#{®uy|Âs²xvmu®|&{n#x ^m¸wvm {|^su
j
xu)n{s{î~
n{¯*m#xB ^{}±^ms,mawywy|^wz*wy{}uymwy{}|@sl´
i"j
{®z*wy{uvmwy{|^s{|ÄuvmasLx°z*|^nª{s³xuv{|^s7|¼wym* ^{}|@s,xsª³|@sx^wv
=³szuy{}|@sx«*´
N«}|zx«]n{s{n{}¯mw¼Ä|^wy³z
j
xzwv{uvmawv{|^s#{sz*«{s #ª|u
j
wvm {|^s¸x^s°ª|^s³xwy=szuy{}|@sx«¼{
³yx«}«
j
xwuv|Dz|@n³uvm^´
i"j
{{n|^yuv«,muv|°u
j
m)=x^zuu
j
xuu
j
mym*u|F{n#x ^mwvm {|^sS{<´m´
u j
mvmu&|wym* @«xw|^msD|^n#x{s{s =µ
j
|^ymª|^s³xwyB{"xz*«}|@vmal n#xs{}Ä|«
|@ma"s|u
j
x_±mx¸vuwy³zuy³wvm®|±^mzuy|^w"xz*mwym*±^ms@uv{s ¸uv|Bym{sx¸yuywx{
j
utÄ|@wvµ&xw#=x^
j
{|^sD @wyx^{}mas@u
mzms@unmu
j
|´ qtsÉ|^wmwuv|²z{wyzn)±^ms@uu
j
{#{Bz*«uÃz*x«z«³|±Jxwy{xuv{|^sxsÞ
j
x^³m
|@uv{n{}¯axuy{}|@sLuymz
j
s{»mzxs4ª³mBª³wv|@
j
u®uv|,ªmxw®|@sLu
j
mBwy|^ª«mnD´
i"j
m¸ªx^v{z{maxD{uy|,vm
u j
mnÅ{s|^wmwZuv|®mwy{}±^m¼x
xwyuv{x«
{Smawvmas@uv{x«
¾
»xuv{|^s
¾%
u j
xuµ"{««³wv{±m¼u
j
m"ª|^s³xwy|
x^s#{s{}uy{x«Swym* ^{}|@suy|Jµx^wy#x«|z*x«n{s{nn |lu
j
mmwwv|@wz*wy{}uymwy{}|@sl´
i"j
mm*#³|^{s@u/{Fuy|z|@n#uvm
u j
mB±^m*«|z*{}uñmazuv|@wxumaxz
j
³|^{s@u)|"u
j
m°ª|^³sxwyxumx^z
j
uy{nmB{syuyx^s^u´Wqts²u
j
{x^³mawµ¼m
³wv|@³|@vm®x=wyx^nmµ¼|@wÄ|@w"xz
j
{m*±{s u
j
mavm ^|^x«{sx#snªmw&|0³wyx^zuv{z*x«}«¸{n|^wyuyx^s@u+z*xym´
i
|ÂÈ{maxZ{s²u
j
m¸uõ|~Ã{nmasy{}|@sx«/zxymZu
j
m¸m*±^|«±{s ,ª|^s³xwy|^wxz*uv{±mBz*|^s@uv|@w*Z{
n|m«}ma7ª@DxB³xwxnm*uwv{zz*wy±m #
]µ
j
mwymn#x_Dªm{uyxwz ~«}mas u
j
x^s{x^s)m±|«uy{}|@sx^wyx^nmuvmaw*_u
j
muy{nm^´
i"j
mx^zuv{±m&z|@s@uv|^³wZ{0u
j
mas³wv{±masª@u
j
mÄ|^«}«|Jµ"{s
¾
%
µ"{u
j
µ j
mwym{®xsL{s{uv{x«/zwy±m#mÂsm4ª^,u
j
m¸ymw®xs
% u j
m#±m«}|z*{}uñmazuy|^w|
´ i"j
{
±^m*«|z{uÃW{u
j
m#s s|Jµ+su
j
xu®nyu®ª³m#mwy{±mD=wy|^n-u
j
mmawywy|^wz*wy{}uymwy{}|@s,v|Wu
j
xuu
j
my|«uy{}|@s
z*|^s@±mawv ^muv|Jµ&xwxzwv±^mx^z
j
{m*±{s DxW«}|zx«/n{s{nn x^sLu
j
*
j
|^m=««Zuy|Jµx^wy³
u j
m)ª|^s³xwyB|u
j
m|@ªtmz*uuy|Bvm ^nms@ux!#"%$ ´
&(') *,+.-0/01024365728/0193;:=< >+ /@?-/0ACB;>+ /028DE
¿m*u³s|Jµ m*Âsmn|^wymwymz*{ym*«°u
j
mwym* ^{}|@sWx^s,ª|^s³xwy°=s³zuv{|^s³x«*´"¿m*u.F ªmxBz*«x^y"|
|^n#x{s =|@³maslwym* ^«x^wª|^s³mvmuy{T´m^´
|
x^sHG xsm*«mnms@u0|6F |ª³|@sx^wv
G
µ j {z
j
µ¼m#v|@nm*uv{nm®s|^uvmIF´ Àª³|@sx^wvD=s³zuv{|^s³x«TKJL*nx_7ªmmÈwymvma7x®xDª³|@sx^wv
{s@uvm ^wx«]|v|@nm®z*x«x^w¼=szuy{}|@sNM¸|0{n#x ^mÄmaxuwvma
JL
G PO
QSR
M T@¥
µ j mwym
G {u
j
mDª|^s³xwy²|+u
j
mDwym* ^{}|@s xs
T@¥
{uy#x^wvmax,m*«mnms@u´
i"j
mDmwy{±_xuv{|^s |u
j
{
ª|^³sxwy¸=s³zuv{|^s³x«l{+z«xv{z*x« ³ x^sD«}max&uy|#u
j
m®Ä|^«}«|Jµ"{s ±^m*«|z{uø±mz*uv|@w
% L M VUXW,Y
M [Z]\ \
µ j mwym
\
{0u
j
m&{s@µ&xws{u/s|@wyn#x«±^mzuy|^w|Cx^s
U
{}unmaxsz*wy±Jxuwvm^´
i"j
m&{mx{0uv|ymx
«|z*x«Sx^wyx^nm*uywy{¯xuy{}|@s|[uv|#wvmazm
uv|#xyuyx^sx^wy¸wy|^ª«}man{sBu
j
m®z*x«z*«¼|±Jx^wv{xuy{}|@s*´
Àwym* {|^sD=³szuy{}|@sx«T8J_^n#x_,ªmm*ȳwvmayym7x^x^s7{s@uvm* @wyx«<]{sLxW|^n#x{s`G |Fl|¼v|@nm
=³szuy{}|@sba7|v|@nm®wym* ^{}|@s¸Ämaxuy³wvma
J_^G
4 OR a
cG T
T
qtsu
j
xuz*x^vm^Su
j
m¸z|@n³uyxuv{|^s|Fu
j
m#±m*«|z{uÃ,±mazuv|@w+Ä|@w
{s|u®xmaxy´
¨
m¸wy|^|^ymuv|
z*|^n#x^wvm)uõ|Wn#x{sxwy|^x^z
j
m´
i"j
mÂSwyyux^wy|^xz
j
{®ªxym|^s7u
j
m{max°|/uywxsvÄ|^wn{s
x«}«=sz*uv{|^sx«¼{s^uy|ª|^³sxwy=szuy{}|@sx«u
j
mawvmaª@#wym³z{s <u
j
wy|^
j
x«}|z*x«S³xwxnm*uwv{¯xuv{|^s
|u
j
m7ª³|@sx^wv
u j
m7wv|@ª«mn5|®n{s{n{¯xuv{|^sÞuv| x²vuxs³xwÉwy|^ª«mn {sÁu
j
m7z*x«z*«#|
±Jx^wv{xuy{}|@s¸=wy|^nµ
j
{z j uj
m²z|@n#uyxuv{|^s |)u
j m Æ#ÇxuymxÈ mwy{}±Jxuy{}±^mBÄ|««}|Jµ+´
i"j
m²vmaz|@s
x^wy|^x^z
j
{ªxymB³³|@s¸u
j
m®ym|Z
j
x^³mmwy{±_xuv{|^s#uy|@|^«´0qtsBxyms³vm{}u¼{n|^wymsxuywx«Sv{szm
{u%mm&u
j
m)wym* {|^sWwvmawymyms@uyxuv{|^sl´
|uvmu
j
xuu
j
mz*x«x^w"=sz*uv{|^s a{sT
{ ^msmawyx«}«Wwym* ^{}|@s~Ãmmsms@u*´
i"j
{{{n#|^wyuyx^s^u
y{szmu
j
{ma³masms³zD|^sDu
j
mwvm {|^snyuªm)uyx!masW{s@uv|°x^z*z*|^s@u"µ
j
mas,vmaxwz
j
{s Ä|@w+x¸«|z*x«
n{s{nn |0u
j
m®=s³zuv{|^s³x«Tx"{yzvma°{sD«xuvmw"ymz*uv{|^s´
«y|s|uvmu
j
xu&µ¼m®z|@«
j
x_±^mx³mWxma³masmaszB|[M#|^s
G{T´m^´µ+wv{uvm M G {s
uy| mmDu
j
mvn#nm*uywy°µ"{u j u j
m)wym* {|^sW=s³zuv{|^s³x«T´
i"j
{"{s|^u+smz*myx^wv°y{szmu
j
mwym«}u"{s
ymz*uv{|^sW³´ï{sxwyuv{z*«xw i"j
m|^wymn |#{sW=x^zu+wy|J±{m®xsx^svµmw¼Ä|@w&u
j
{+z*x^vm^´
&('V& 2D:E +K?#E -0A + B;> > 2B6>+ /3;+D: E >/ > 24<K:
3;+ A:[E E >/ <
sD{n#x ^m®{"wymwymyms@uymDª^°{}u"{s@uymsy{}uÃ
mÂSsmD|^sDy|^nmwvm {|^sB| ´
zuv{±m+z*|^s@uv|@wµmwym&|^wy{} ^{s³x««}{s@uywy|³zmauy|)vmaxwz
j
Ä|^wn{s{nx®|]ª|^³sxwy=s³zuv{|^s³x«*´
qts ³0 0u
j
m°Ä|^sz*uv{|^s,M{xD=szuy{}|@s |"u
j
mWn#x @s{uym°|"u
j
mW{n#x m¸ ^wx{}mas^uu
j
wy|^
j x
yuywy{zuv« mzwvmaxy{s ,=sz*uv{|^s
$ " ®KM
Y
z j u j
xu
" Lx^ " $ ´
ºmsz*m M
Y ´ i"j
mn{s{n{¯xuv{|^s¸xn|@s@uyuy|u
j
mn{s{n{}¯axuv{|^s#|Su
j
m«ms u
j
|]x
zwy±m{sx
{mn#xss{x^sDx^zm´0¿|zx«n{s{n#x¸x^wvm|^ªux{smD±{xu
j
m)yuvm*ma³mavuv~mzms@u"nm*u
j
|l´
m {|^s®=³szuy{}|@sx«
j
x_±mx«v|ªm*mas{s@uywy|zmal´
i"j
mwym* ^{}|@s®{sÄ|@wyn#xuv{|^s){mnªm³m{su j m
=³szuy{}|@s a4|%
´ i"j
mym=s³zuv{|^s³x«
j
x_±^mªm*msLvmaÄ|^wn#xs@x«{z*xuy{}|@sz
j
x^n|J±{s
|@ªtmz*uy+m*uvmazuy{}|@so^o³J] 2 ³{n#x ^m®ym* @nms@uyxuv{|^s
:
8
:
S³o^oo^ T|^w"z*«x^yy{îÂSz*xuy{}|@s
:
oo
:
T´ &³|@wmÈx^n«}m^m*|@«}m
j
x_±mBym z
j
yuyxuv{vuy{zx«Ämaxuy³wvma|+x7wym* ^{}|@s G xu j m
nmaxsW|^w&u
j
m±_x^wv{xs³zm !
"
!#%$
R
&R &('
R VT µ"{u
j G)
'R T
*
R
&R &('
R
+ W
$ R T
¨
m)ymu
j
mavm®uõ|m*Èxn#«m&uv|Bn|^uv{±_xuvm®u
j
m{s@uywy|z*uv{|^sW|x ^msmawyx«lwvm {|^s°=szuy{}|@sx«
J_^G
8 O R a
,#G
,
G
,.-)VG T
<o
µ j mwymu
j
m®=sz*uv{|^sx«/,.0Fxwym ^{}±^msWª^
, 0 VG
4 O
R)1 0 G
T
2
3 =
"
j
|Jµ+s°{sDu
j
{"m»xuv{|^slu
j
m®=sz*uv{|^s
1 0 {&{}uvm«îwvm {|^s~Ãma³masms@un|@wvmwymz*{ym*«
1 0 cG
1
0
450VG
40
G
407698 G
:
µ j mwym
450;:VG
O
R)<
09:
.T
=
;>?0 2
_ 3 2
|uym"u
j
xu/µ¼m
j
x_±m"yuv|^³³ma#u
j
m³wv|zmayxu/u
j
mvmaz|@s«}m±m*«ªu/{uFz|@«¸z|^s³zm*{±Jxª«z*|^s@uv{sm´
¨ m j
x_±^mz
j
|@vmas7u
j
{ymz*{x«/z*xym#|&mmsmsz*ªmz*x^ym{u|Äuyms²xwy{vma{s²x«}{z*xuv{|^sZx