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HAL Id: inria-00070317https://hal.inria.fr/inria-00070317
Submitted on 19 May 2006
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
A Polyline Process for Unsupervised Line NetworkExtraction in Remote Sensing
Caroline Lacoste, Xavier Descombes, Josiane Zerubia
To cite this version:Caroline Lacoste, Xavier Descombes, Josiane Zerubia. A Polyline Process for Unsupervised LineNetwork Extraction in Remote Sensing. [Research Report] RR-5698, INRIA. 2006, pp.26. inria-00070317
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ap por t de r ech er ch e
Thème COG
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
A Polyline Process for Unsupervised Line NetworkExtraction in Remote Sensing
Caroline Lacoste — Xavier Descombes — Josiane Zerubia
N° 5698
September 2005
Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
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¬~¥f©h³^pX«ydgztthE©sE²f~|Puwztt«yhE ~|N~gPhmfwztyhEssv~|fx~|, 4) ²ffz¬Y~¥th¨uwdhswgxh¨urX/hzÉsuzXydsvuw~¥ywzPhEvu~hSssudfzPswh7z^Ź°z¬ |tzg ÇhE©fsE²G¦df~©hx~|yz/zuw~|f·svuwz|f[hzPghuw~¥yyzP|su~|Yus³=» |-54 X² 46f²Ô6 ²G©~|fhx|fhu¦z°hYuPy®u~zP|N~shEvªzPwghSN«sw~|f svuw~¥©%fzXyhEssvhSs¦dfzYsvhzPthEyus¨h~|Puhyuw~|f·©~|fhxsvhEgh|YusthSswyw~hSNXr[udfhh|fzg ¬~f©hSs/.#udfh~mg~t/z~|YuE²Gudfh~©h|uwd²¾|Âudfh~mz~hE|Yuu~zP|¾³mc#dfhjd~Pdt¸I/hwªzg|yhzÉudf~¥sgztthE©~|fdPs#hEh|¬hEw~ÇhE z| |X«fghz«s#htgf©hSs³%» |Âwuw~¥y«©E²YuwdhÊË«©~ur i|fr̨fw~z#gztth©Ô~¥s#hEsw/hEy~¥©©r[sv«~ÆuhE[uwzudfhhXuPy®uw~z|zÉzPP[|fhu¦¯zPw°tsE²t©hSt~|fxuwz yz|Yu~|X«fzP«s#©~|fh|hu¦z°Xs¦~uwd¹swg©©y«f¬u«fwh| ªh¦ozg~¥swsw~zP|s|z¬hEthuwhSy®uw~z|s³¼h¬PhwuwdfhE©hSswsE²udfhthuwhEyuw~z|zGsw~|X«fzP«sÉf|ydfhEs ± ydyuwhEw~¥su~y¯z¾swzgh~¬Phs´%~¥sÉ|fzu^h|fzP«fdPyy«uwh·|_udfh[thuwhSy®uw~z|¿z^Ê10Ì~|YuwhswhEy®u~zP|s ± (~|YuwhEswhEyuw~z|sm¦~uwdU|¿y«fuwh[|f©hS´¨hg~|sf~·y«©ÆuS³ÈÁhhYuh|_uwd~smgztth©~|f[uzÂgxzPwhxyzPgx©h¹zthSy®usE²¾sw«ydÁPs¨~|2 4ɪzyhE©©ÉhEyzP|f~Æu~zP|¾²G¦dfhhxzthSy®usmwhj¬ w~¥©hhEswz©«tuw~z|ÁthªzPwgf©hjuwhEgf©uwhSs³ÅNzhxhfyuw©r²¾zP«fm|fh¦ gxztthE©K²9y©©hE¿i¯uwzPfdf~¥yxk~h|YuwhS"º» |fh¼=hu¦z°½ÉXuPy®uw~z| ± i=k¨º»¼=½¯´²Y~¥ssvuw~¥©/fztyhEss8¦dhh=zthSy®us¯wh=~|YuwhEPy®uw~|f/z©rX©~|fhSs¯yzgzYsvhSXr|·«f|f°X|fz¦||X«fgjhEz9swhghE|Pus³c#dfhmyzP|f|fhEyuw~z| hu¦¯hEh| swhPgh|Yus¯~s¯udY«s¯hEg7/hEffhE·~|Yuwzjuwdfh¨zfhEy®uthÇ|f~uw~z|¾²X¦dfhEwhSs¯~| 4SX²/6¾udfhyzP|f|fhSy®uw~z|ÂdPsuwz/h7thÇ|fhE«fuwzswg©©¾yz|su|Yu ± sE²Xuwdhfzf~©~urzhfyuyz|f|hEy®u~zP| ~¥s|X«f©©«f|thÉudfh=hªhEwhE|yh#ghEPsv«fhS´³3'%~|©©r²udfh=zPPj|xw~¬hEG«f|y®u~zP|s8yE|hgxztthE©hSxhEsw~©r7uwdfz«fPd7udfhthÇ|f~uw~z|zÉ~w¦~¥svhm~|Yuwhyuw~z|¾³¯u©PsuS²f/z©rY©~|fhEs=yE|Â/huvuhÇfusw~|X«fzP«s#©~|fh|hu¦z°uwd|ÂgxztthE©s=swhEz|NswhPgxhE|YusudfwzP«fdÂxwhE©hE¬|Yu=yz|suw«yuw~z|z9uwdfhfujuwhEwg³
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* ( &* *,& (3* &&" !(+*
c#dfh·i=k¨º»¼=½ogxztthE©~smfhEswh|YuhEN~|)pthEy®u~zP|Áf³7c#dfhztu~g~ÃEuw~z|¹¸=thSswyw~/hE¹~|¿pXhEyuw~z| "[¸~sfz|fhx¬X~¹sw~g7«f©¥uhE¿||fhE©~|fN«sw~|_¹=ÄYÅ"iÅ"i ©zPw~uwdg,thSt~yEuhEÁuwz¹uwdfhsv~g7«f©¥u~zP|)z=z«7svuw~¥©¯fzXyhEss³Âc#df~¥s©Pz~Æudfg~¥suwhEsvuwhS~|¹pXhSy®uw~z| 6z|hgzuh©r[svhE|swhE ~gPhEs ± h~©¾|suwhE©©~uwhfuP´³
# ; ' *,2 <>&
* $ $ Î ' Î ¡¢7K¡¢ ÎPÑjÒ ¡ Dz~|YufwztyhEssvhSs8fz¬X~th¨7~zz«sɪgh¦z°jswhE·zP|[ghEPsv«whuwdfhEzrjuzthEsy~/hxsyh|hXr·| «f|fzPthEwhS·swhuz¯/z~|Pus¨~|) yzgPy®u
F ⊂ Rd µ 7I³'fz n ∈ N
²¾©huΩn
/hxuwdfhswhuz¯yz|tÇ«fu~zP|s x1, . . . , xnuwduyz|sw~¥su
zn«f|zthhEÁ/z~|Pusz
F³¹ 9^Ò ( ' 9 ¡ Ò £ Î E z|
F~sjNgff~|f
XªwzPg Âfzf~©~ur"swPyhuzNuwdfh svhu7z
yz|tÇP«fuw~z|sΩ =
⋃∞n=0 Ωn
²Ôsw«ydNuduªzm©©9/z«f|thE @¯zPwhE©9swhuA ⊆ F
²/uwdfhx|X«fg7/h¨z8/z~|YusNX(A)
©©~|f~|A~s=jÇ|~Æuh|tzg ¬w~¥f©h³
c#dfhyE|fzP|f~¥y©fÊwyzPgx©huwh©r[|tzgjÌ/z~|Yu=fwztyhEss#~sudfh ' ( Ò ¡, Ò (S Ò ' 9^Ò ( ' 9 ¡ Ò £ Î E ³ /=|thudfh©¥ ¦z¯·z~¥ssvzP|¹zP~|YumfzXyhEssz8~|Yuh|sw~Æur
λ²Guwdhj|X«fgjhE¨z8zP~|Yus
NX(A)ªzP©©z¦=s z~¥swswz|N©¥ ¦¦~Æud"ghS|
λ|A| ²|Ô²G~¬hE|NX(A) = n
²udfhx|_/z~|Puswhj~|th/h|fh|Yuw©r¹|¹«f|f~ªzgx©rNt~¥suw~f«tuhE¹~|A³mc#dhj©¥ ¦z^[%zP~sswz|
fztyhSsws#z9~|Puh|sw~urλzP|
F ⊂ Rd ~sthÇ|fhS Xr[uwdfhmªzP©©z¦~|jfzf~©~urghEPsv«fhmz| (Ω,B) .
µ(B) =
∞∑
N=0
λN e−λ|F |
N !
∫
F N
1B(x1, . . . , xN) dx1 . . . dxN
± 4S´
¦dfhEwhB ∈ B ³
c9zgztthE©Xuwdfh#zsvhEw¬PhE7syhE|fhYrjswhuÉz/zthSy®us²¦hy|x«fPgh|Yu¨/z~|YuwztyhSsws9Xr7t~|hYu~|tªzguw~z|± (zthSy®ughuwhEs´uwzhEydm/z~|YusE³%pX«ydjfzXyhEssÔ~¥sy©©hE ¢t¡ D Î Ó 9^Ò ( ' 9 ¡ Ò £ Î E z9|Ò "! Î £ 9 ¡ Ò £ Î E ³ gw°PhE/z~|Yu^fzXyhEss8zP|
F²P¦~uwd·gw°ts8~|·7swPyh
M²P~¥szP~|YufwztyhEssÉz|
F ×M sv«yduwduN(A×M) <∞©gzPsvusv«fh©rxªz#|Xr:@zh©/zP«f|thS[swhu
A ⊂ F ³» |·udf~s#yz|YuwhYuS²Yuwdfh¨«f|f~ªzg z~¥swswz|·wztyhSsws^~s#7gw°PhE/z~|YufztyhSsws¦dhhzP~|Yus7h·t~svuw~f«tuwhS¿yEyzt~|fuwz¹Â«f|~ƪzPwg z~¥swswz|ÁzP~|Yu7wztyhSswsmz~|YuwhE|sw~Æur
λ²É|"gw°ts
Pswswzty~¥uwhS uwz[hSydÂ/z~|Pumh«f|f~ªzg©rÂt~¥suw~f«tuhEN~|M³=c#dfh©¥ ¦zuwd~sfzXyhEssz|
F ⊂ Rd ~¥sfhÇ|fhSNXr[udfhªzP©©z¦~|xwzPf~©~ÆurghEPsv«wh .
µ(B) =
∞∑
N=0
e−λF
N !
∫
(F×M)N
1B(c1, . . . , cN) dx1dPM (m1) . . . dxndPM (mN )± ´
¦dfhEwhmudfhci = (xi,mi)
hmg°hE/z~|YuszF ×M ²|
PM~suwdfh«|f~ƪzPwgfzf~©~ur·ghEPsv«fhz|udfhg°
swyhM³
©Æudfz«fPd5~| gzPsvu ff©~yEuw~z|s~Æu[~¥s·|zu·hE©~¥su~yuz ssv«fghÂudu[/z~|Yus hÂsyuvuhhE |tzPgx©r²z~¥swswz|fztyhSswswhEsxh «svhª«f©#uwz)f«f~©¥UgzhÂyzgf©h gztthE©sE³U» |fhhE¾²^~|Puhyuw~z|sxy|U/h~|Yuwztt«yhSUXr swhSy~ªrX~|f¿Ó Î ' (ª ÏÁ¦~ÆudNwhSsv/hEyuuzuwdfh7whªhh|yh¨ghSsw«fwh
µ³¯ºhu
h/h|fzP|f|fhYu~¬Phª«f|y®uw~z|NzP|
Ω³^c#dh|¾²uwdfhghEPsv«wh
νd ¬X~|fth|sw~urh¦~uwdÂwhSsv/hEyu#uwz
µ~¥s=thÇ|fhS Xr .
ν(B) =
∫
B
h(C) µ(dC)± "Y´
» 0 < ν(B) < ∞ ²udfh| ν yE|N/h7|fzPwg©~ÃhS[uz·fz¬X~th·fwzP~©~Æur ghEsw«fh π thÇ|hEÂXr ν(B)/ν(Ω)
³=» |Nudfhyz|YuwhYuz¯zfhEy®uhYuPy®u~zP|¹ªzg~ghSs²¾uwdfhthE|sv~ur¹~¥s«sw«©©rÂy®uz~ÃEhE"~|Yuz u¦z uwhgs³'%~svuE²¾PhzPgxhuw~yE©|ÂuwzPzP©zP~¥y©¾yz|su~|Yush~|yz/zuwhS·udfz«fPd¹f~zPth|sv~ur
hp³pXhEyz|¾²/·uxuhg
hd~s«swhEuwz[Çfu
udfhfuf³8c#dfhyzgf©huwhfh|sw~Æur[z%uwdfhswu~©GfwztyhEss#~sudfh|NthÇ|fhEÂPsªz©©z¦=s.
h(C) ∝ hp(C) hd(C)± 6X´
¦dfhEwhC = c1, . . . , cn
~¥syzP|tÇ«uw~z|[z%zPthEyusE³#B $&%"')(+*,%)-/.10&0&*%' σ 23.10&45*687.9%):8;<(')(8.='?>[email protected]&0CBD@57E*0F%)*G')% A ⊆ F '(8*,-/.1H8H8$&I+4 x1, . . . , xn 7→ NX(A) $&%-9*.1%:87.1680&*5J
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" * ! * ! &5(+"
* $ * 6 Ò Ó Î 7 Ò ¡[¢ ' ' D ' ÒXÑ ' ' " Î ¡ Ò 9^Ò 7 Ï 7( ' Î c#dfhi=k¨º¾»¼=½¿wztyhSsws~¥s8¨|tzPg0yzP|tÇP«fuw~z|xzGzP©rX©~|hEsE²©ztyEuwhSx~|yzPgyu^svhu
Fz
R2 yzwhSsv/z|f~|fuz7udfhzPswh¬hEsyhE|fh²X¦dfzYsvh|X«fg7/h
Nz¾zP©rX©~|fhSs~s¯«f|f°X|fz¦|¾³» | zuwdh¦¯zPfsE²udfh7i=k¨º¾»¼½ fztyhEss¯~s|
zPthEyu=fwztyhEss¯zP|Fz
R2 ¦dfhh¨uwdfhzPthEyuswhm/z©rY©~|fhEsE³½^ydÂ/z©rY©~|fh c ~¥sthSswyw~hSYr .
~Æus~|f~Æu~©ÔzP~|Yu p1 ∈ F
~Æus¦~¥Xuwd e ∈ [emin, emax]
|«f|f°X|fz¦||X«fg7/h n ∈ 1, . . . , nmaxzswhPgxhE|Yus
uwdfhswhPgh|Yu©hE|fuds lj ∈ [Lmin, Lmax]²j = 1, . . . , n
uwdfhswhPgh|Yu=t~hEyuw~z|s αj ∈]− π, π]²j = 1, . . . , n
³|hfgf©hªzP
n = 3svhEgh|Yus#~¥s#~¬hE| ~|*'%~³ 4³
p1
l2
l1
α2l3
α1
α3
n=3
e
'%~P«fwh 4 .8 urXf~¥y©¾zfhEy®uzuwdhji=k¨º¾»¼½ gztthE©K³
ÈÁhÇsufhÇ|fhjwhªhh|yhmfwztyhEss¦df~¥yd~¥s«|f~ƪzPwgz~¥swswz|g°hS /z~|YufztyhEssE³ /=|thudfh©¥ ¦§zuwdf~¥sfztyhSswsE²%uwdfh |Y«g7/h
NzzP©rX©~|hEs7ªz©©z¦=s_z~¥swswz|)©¥ ¦¶|Ô²udfh[~|f~uw~¥©^/z~|Yus ± /z~|Yusjzuwdh[fztyhEss´m|
/z©rX©~|fh7ghuhs ± gw°ts´h~|th/h|th|Yuw©r¹|N«f|f~ªzg©rNt~¥suw~f«tuhE_~|Nudfh~¨whSsv/hEyuw~¬h7svuuwhjswPyh³¨c#dfhghEPsv«whjz^udf~¥swztyhSsws~¥sP~¬Ph|"YrNhY«u~zP| ± P´=¦dfhEwhjuwdhx«|f~ƪzPwg wzPf~©~ÆurÂghEPsv«fh7zP|¹udfhgw°Nswyh7~¥sP~¬Ph|Yr .
PM (B) =
nmax∑
n=1
1
nmax
∫
[emin,emax]
∫
V n
1B(e, v1, . . . , vn)dn(v1, . . . , vn) de
|V |n (emax − emin)
± ´
¦dfhEwhB~s"|h©hgh|Yu_zuwdfh¿uw~f«sswzXy~uwhS uz5uwdfhUgw°osvyh
M = [emin, emax] × ⋃nmax
n=1 V n ²¦dfhEwhV = [Lmin, Lmax]×]− π, π]
|V n = V × . . .× V ±
nuw~ghEs´®³
c9z·~|Puwztt«yh7|0" $'&)(+*,&)(zP|NzP©rX©~|hjswd/hEs|N~|YuwhEPy®uw~z|shu¦¯hEh|¹/z©rX©~|fhSs²/¦¯huwdfhE|_sv/hEy~ƪrudfhi#¸k¨º»¼=½ w~z¨fzXyhEssXrN fw~zmthE|sv~ur
hp¦~uwdÁwhSsv/hEyuuz[udfhjhªhEwhE|yhjfzXyhEss©¥ ¦³c#dfhxhtfhEssv~z|Nz
hp~¥s
P~¬Ph|~|¹pXhSy®uw~z|Nt³ 4³ 6 uh=xfhEswh|Yuuw~z|[z9udfh/zPssv~f©h¨~|YuwhEPy®uw~z|s¯/hu¦hhE|ÂzP©rX©~|fhSs#gthm~|NpthEy®u~zP|¹t³ 4³ "X³
* $ 8 Ò 7 Ï 7 ( ' Î ( ' Î ¡¢f£ ( Ò ' ÈÁhyzP|sv~¥thE#u¦¯zxurXhSs#z~|Yuhy®u~zP|¾³
c#dfhÇsuz|fh~sPsvhSz| 7h©¥u~zP|z 9 ¡ Ò (+;( Ï ∼p/hu¦hhE|·zP©rX©~|hEsE³c#d~sh©¥uw~z|[~sthÇ|fhE s^ªz©©z¦=s.
u ∼p v ⇔(
∃j ∈ 1, . . . , nu : d(pju, v) < dmax
|d(pj+1
u , v) < dmax))
zP (∃j ∈ 1, . . . , nv : d(pjv , u) < dmax
|d(pj+1
v , u) < dmax))
± P´
¦dfhEwhdyzPwhEsw/z|fsuzuwdh7½8«y©~¥thS|¹t~¥su|yh²
nuthE|fzuhEs#udfh7|X«fgjhEzÉswhPgh|Yus=yzg/zPsw~|fudfh/z©rX©~|fh
u²
|pj
u
fh|fzuwhEsudfhxydhEy°X~| ± zPmyz|Yuwz© ´#/z~|Yu¨|X«fgjhEjfhEsy~f~|f
u³m» |_zuwdfhE¦zfsE²u¦z[/z©rX©~|fhSs¨whxsw~
y©zYsvh~ƾu¦zjyz|swhEy«tuw~¬hzP~|Yus¯zÔuwdfhÇsu#/z©rY©~|fhwh¨u7t~¥su|yh©z¦¯hE8ud|dmax
zÔudfh¨swhEyz|[zP©rX©~|h³É=|htgf©hz¾u¦z7/z©rX©~|fhSs^dfzP©f~|f7uwdf~¥swhE©uw~z|~¥s~¬h|~| '%~«fh¨t³ (K( ' Î ¡¢f£ ( Ò ' (K Ò ¡ (Ó%Ó Î ' ~| zth^uwz ¬Pz~¥Âz¬h©¥ff~|f·z8zP©rX©~|hEsE³¨È dh|¹udf~¥s~|YuwhEPy®uw~z|_zXyEy«fsªzm·~¬hE|¹/z©rY©~|fhxyz|tÇ«fu~zP|
C²Gudfhth|sw~ur
hp(C)~suwdX«s#h Y«©GuwzÃEhz³
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* ( &* *,& (3* &&" !(+* 7
dmax
u
v
'%~«fh7 .¯8wz t~g~Æur·h©¥u~zP|¾³u ∼p v
hSy«svhmu¦z·yzP|swhEy«fuw~¬h/z~|Puszvwhut~¥su|yh©z¦h#ud|
dmaxz
u³
c#dfhswhEyz|"~|Yuhy®u~zP|ÁyzwhSsv/z|fs=uzÂuwdfh £ Ò 'F' Î £ A( Ò ' z#/z©rY©~|fh¬Y~¥z|fhz~ushXuwhg~Æu~hSs¨uwzN|zuwdh/z©rX©~|fh³» u~¥smPsvhS_zP|_udfh½8«y©~¥thE|Áf~svu|yh/hu¦hhE|_udfhh|fzP~|Yus
p1c
|pn+1
c
z# /z©rX©~|fhcyzPgx/zPswhE
znswhPgh|Yusm|"|fzudfhm/z©rY©~|fhzmz|fhxhEfhz¯uwdfhyzPgyu
F³x» ^udfht~¥su|yh
d(pkc , o)
/hu¦hhE|pk
c
|o± zP©rX©~|fhjzhEtPhS´~¥s©z¦hud|"uwdwhSsvdfzP©
ε²c~s¨s~¥Nyz||fhEyuwhENuwz
oudfz«fPd
pkc
³¨º¾huVC,F (pk
c )h7uwdfhxsvhuz
/z©rX©~|fhEs|ÂhSthSs=zC|
Fsv«ydÂudu
d(pkc , o) < ε
³#ÈÁhjthÇ|fh7uwdfhh7svuuwhSsz^/z©rX©~|fhcPyyzPf~|fuzuwdh
yEt~|©~Æur·zudfhmu¦¯zswhusVC,F (p1
c)|
VC,F (pn+1c )
³8zP©rX©~|hc~¥s=sw~ .
I¡ ÎXÎ/²f~c~¥s|fzu=yz|f|fhSy®uhEXr[|Xr·z9~ushYuwhEg~Æu~hSs²( .
VC,F (c) = VC,F (p1c) ∪ VC,F (pn+1
c ) = ∅
A( ' E'7 Î/²X~c~syz||fhEyuwhE Xr[zP|f©r[z|fhmz%~ÆushXuwhEgx~uw~hEsE²( .
VC,F (c) 6= ∅ , ∃k : VC,F (pkc ) = ∅
Ó Ò 97 βt~c~¥s=yzP|f|fhSy®uwhS[Xr·/zudÂz~ÆushXuwhg~uw~hEsE²( .
VC,F (p1c) 6= ∅ , VC,F (pn+1
c ) 6= ∅c#dfhSsvhmudfhhsuuhEshm~©©«suuwhS[~|*'9~³ "³
F FreeSingleDoubleConnection
'%~«fh " .z©rY©~|fhsuuhEs#¦~uwdÂwhSsv/hEyu#uwzyzP|f|fhEyuw~z|sE³
* $5) + .-:/ %021 9 ¡( Ò ¡·Ó Î ' (ª Ïc#dfhmf~z#th|sw~ur
hpz97/z©rX©~|fhmyzP|tÇ«uw~z|
C = c1, . . . , cNy| h¨¦~Æuwuwh|~| l¨~fsv~¥| ªzgs^ªz©©z¦=s.
hp(C) =
0²t~ ∃ ci ∈ C, cj ∈ C/ci ∼p cj
1
Zexp−
N∑
i=1
[U1(ci) + U2(ci, VC,F (ci))]²~Æ|fzu ± 7´
¦dfhEwhZ~¥s^|«f|f°X|fz¦|·|fzPwg©~Ã~|fyzP|svu|YuS²
U1~s¯/zuwhE|Yuw~¥©/swhEzP|uwdfhzthSy®uswdhP²Y|
U2~s¯zuwhE|Pu~©
PsvhS[zP|uwdfh/z©rX©~|fhyzP|f|fhEyuw~z|sE³
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" * ! * ! &5(+"
c#dfhmhE|fhruhgU1Pswswzty~¥uhEuz·xzP©rX©~|fh
cyzPgzYsvhS[z
nsvhEgh|Yus#~¥s#¦~Æuwuwh|NPsªz©©z¦=s.
U1(c) =
+∞ ²t~cswh© ¸ ~|YuwhEswhEyus
U11(n) +
n∑
j=1
U12(lj) +
n−1∑
j=1
U13(αj , αj+1)²f~Æ|fzu ± P´
¦~uwd .U11(n) =
Mn
(n+ 1)2
U12(lj) = MlLmax − lj
Lmax − LminU13(αj , αj+1) = Mα (0.5− cos(αj+1 − αj))¦dfhEwh
Mn²Ml
|Mα
hx/zPsw~uw~¬h¦h~dYusE³U11
9^Î ' ¢7 ( < Î N ¢7+7n²U12
¢ Ò ¡ 7 Ò ' E Î E' Î ' E ²¾|U13 ¢ Ò ¡ U ¢7+7£ ¡f¢t ¡ Î/³UÅNzPwhEz¬hS² Î 7 Ð ( ' Î ¡ Î £ A( Ò ' ( Ò ¡ (Ó%Ó Î ' Xr¿~|Yuwztt«y~|fÁ_dX¸ yzhzuwhE|Pu~©
± (/~|tÇ|f~uwh/zuh|Yuw~¥© ´®³
c#dfhmhE|fhruhgU2~¥s#¦~Æuwuwh|NPsªz©©z¦=s.
U2(c, VC,F (c)) = U21(c|VC,F (c)) +∑
o∈VC,F
U22(c, o)± Y´
¦dfhEwhU21
~s%PsvhSz|7uwdfh#svuuhEs%zzP©rX©~|fhSs|U22
~¥s%swhE7z|jghEsw«fh^zuwdh Y«©~urz/z©rX©~|fh#yzP|f|fhEyuw~z|sE³
U219^Î ' ¢7 (< Î K¡ ÎXÎ ¢ ' Ó ( ' E 7 Î Î E Î ' S udfz«fPdyz|svu|Pu=|[/zPsw~uw~¬hmzuwhE|Pu~©s
ωfhuωs.
U21(c|VC,F (c)) =
ωf²f~
VC,F (c) = ∅
ωs²f~
VC,F (c) 6= ∅ , ∃k : VC,F (pkc ) = ∅
0²t~
VC,F (p1c) 6= ∅ , VC,F (pn+1
c ) 6= ∅
± 4SP´
U22(c, o) ¢ Ò ¡ uwdfh £ Ò 'F' Î £ A( Ò ' /hu¦hh|)|Áh|tzP~|Yu
pkc
|"/z©rX©~|fhz|ÁhEfhouwdwzP«fd)[|hPuw~¬h
/zuh|Yuw~¥©I²¦df~¥ydÂ~s=swhE z|Nxª«f|yuw~z|ÂghEPsv«f~|fxuwdfh Y«©~ur·z9udfh7yzP|f|fhSy®uw~z|³^c#df~¥s#ª«f|yuw~z|Nth/h|fs=zP| udfht~¥svu|yh·/hu¦hhE|
pkc
|o³_c#dfh·gzPwhudfh[t~¥svu|yh[thSyhEPsvhSs²9uwdfh gzhuwdfhP«©~ÆurÁ¬©«fh·~|yhEPsvhSs³Nº¾hu
c
/z©rX©~|fhyzP|f|fhSy®uwhS·uzoudfz«fPd
pkc
³8c#dfh Y«©~Æur[zudfhyzP|f|fhSy®uw~z|~¥s#~¬h|Xr .
σ(< c, o >pkc) =
1
ε2
(
1 + ε2
1 + d2(pkc , o)
− 1
) ± 4 4S´
¦dfhEwhε~sudfhyzP|f|fhEyuw~z| uwdfhEswdfzP©Ô³Éc#dfh/zuh|Yuw~¥©Gª«|y®u~zP|
U22~¥s#udfh|Â~¬hE|[Xr .
U22(c|VC,F (c)) = −∑
k = 1, n + 1
o : d(pk
c , o) < ε
σ(< c, o >pkc)
± 4 ´
c#dfh ¢7478ÓF(E E¢ ' £ Î Ò £ Ò 'H' Î £ A( Ò ' hudY«s ¢ Ò ¡ Î Ó Xr[|fhEPuw~¬h¨zuwh|Yu~©K³
c9z sw«fg «f¾²Gudfh·i=k¨º¾»¼½ f~zP¨fh|sw~Æur ¬Pzs=©z|f zP©rX©~|fhSs¨¦~Æud"[swg©©Éy«f¬uw«wh7uwdumwh7yz|f|hEy®uhE¦~uwd5udfh¹hEsvu·z¨uwdfh¹|hu¦z° | swg©©=t~svu|yhSszyzP|f|fhSy®uw~z|5¦df~©hNªzf~¥ft~|f)z¬h©¥ff~|f³c#df~s[th|sw~urswhSy~ÇhEsm Ñ7Î 7+7 Ð Ó Î ' Î Ó g°hSNzP~|YumfzXyhEss²ÔPsudfh«fhE©©h smsuf~©~ur¹yzP|t~uw~z|2 PÉ~s¨¬hEw~ÇhSÔ³» |thEhEÔ²Ôudfhsvuwz|fPh%yz|t~Æu~zP|z 7 Ò £¢7Ô S¢tF(47 ( Ï·yE|hSsw~©r¨h¬hEw~ÇhE¨ud|°Xsuwzuwdfh~|Puwztt«y®uw~z|7z=dX¸ yzh ± (~|tÇ|f~uwh ´/zuh|Yuw~¥©9¦~Æud¹whSsv/hEyu=uwz·uwdfhjwhE©uw~z|ÂzÉfz t~g~ur³czwz¬Phmuwdfhj©zty©svu~©~ÆurP²/sw«f/h~zP=/z«f|Nªz=udfhu~zhp(C∪c)hp(C)
²PªzP©©C~|
Ω|
c~|F ×M dPs8uz7/hªzP«f|¦dh|fhE¬h
hp(S) > 0³Éc#dfh«swhzÔuwdfh¨dX¸ yzPwh/zuh|Yuw~¥©
~|t«yhEsgt~g© ¬X~|fz/udfhyzgPy®uFXrsvhu¯zG/z©rX©~|fhSs/.%~Æ7zP©rX©~|h
c~s¯ffhEuz7sv«yd·svhuE²Yuwdh|udfh
fz t~g~urj¦~©©/h¬Ph~ÆÇhS|uwdfh¨th|sw~urx¦~©©/h|X«f©©K³Éc#dfhEwhht~svus^uwdX«smudfwhSsvdz©¥Bsw«yd[s8uwdfh¨ff~Æu~zP|·z
/z©rX©~|fhy| |fzu~|t«yhgzhud|ByzP|f|fhEyuw~z|sE³c#dfhthEywhSswhz
U2~¥suwdfhEwhªzh/z«f|thEÔ³
U1hE~|f/z«f|thE
Xr[/h©z¦²Xuwdhh¨~¥s=sw«fhEw~z#/z«f|[ªzP hp(C∪c)hp(C)
³
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) F ÈÁhf«f~©ÂfujuwhEwg swhE z| uwdfhmªzP©©z¦~|fssv«fgtu~zP|s.
H1.8c#dfhPwhEr©hE¬h©Ô¬w~¥u~zP| /hu¦hhE|[udfh|fhu¦z°·|·udfhy°XPwzP«f|·~¥s#©¥h
H2.8c#dfh©zty©¾ ¬PhPh=z%uwdfhPwhEr©hE¬hE©Ô~|sw~¥thmuwdfh|hu¦z°~s#dfzPgzhE|fhzP«sE³
c9z¬hEw~ªrjuwdu/z©rX©~|fh~¥s¯¦¯hE©©¸KÇuvuwhSxuwz7uwdfh¨fuf²Y¦¯hyz|sw~¥thgsw°jz¾f~ÆthE©s¯yzg/zPswhEzÔuwdfh¨svhu¯zÔf~th©¥sVyzhEswzP|t~|fuwzudfh7/z©rX©~|fhj~|Nudfh7~gPh7|u¦z[yz©©~|fhEhP~zP|s
R1|
R2²/zPsw~uw~z|fhS¹u[t~svu|yh
dªzgV²tyzwhSsv/z|f~|fmuz7udfh|hEXry°Xz«|Ô³½¯yd·f~th©/gsw°x~¥st~¬X~¥thE[~|YuzjswhEy®u~zP|s8z9ÇXhS|X«fg7/hz
f~th©¥s=sswdfz¦| ~|*'%~³ 6f³
R1
1 R1
2
R1
2 R2
2
R3
2
1R4
R2
4
R1
3
V
R2
R1
V V
V
V
1 2
3
4
d
d
pixelpolyline
'%~«fh 6.É8~th©¾gPsv°·Pswswzty~¥uwhS·uzj/z©rX©~|fhP³
i¯zP|sv~¥thEw~|ff~th©%¬ ©«hEsz8hEPyd¹suw~_Ps[sgf©hz^zPf«f©¥u~zP|¾²Ô[pYuw«th|Yu¨uv¸IuwhEsvu~¥s«svhSÂuwzthuhg~|fhj~Æudfhx ¬PhPhEs#zÉudfhu¦z[sgf©hEs¨h7sw~P|f~ÆÇ/y|Yu©rÂt~#/hEwhE|PuS³c#dfh7ªzg7«f©¥ªzPudfhuw¸KuhEsvu¨~¥s·uw~z³c#dhuwzP¹vuzuwdh¨u~zj~sudfht~#/hEwhE|yh¨hu¦¯hEh|[uwdfh¨u¦zxsgf©hghS|sE³Éc#dfhmzuvuwzPg wu#~sjghEsw«fhzuwdh¨¬w~¥f~©~Æurzudfhswgf©hP³=hEwhm~¥suwdfhmuw¸KuhEsvu=htfhEssv~z|[ªz#u¦zsgf©hEs
x|
y.
uv¸IuwhSsu(x, y) =
|x− y|√
σ2x
nx+
σ2y
ny
± 4"P´
¦dfhEwhx²σx|
nx
± whSsv³y²σy|
ny´¨hªhuwzÂuwdh[swgf©hghS|¾²uwdfh[sgf©h·svu|f"fh¬X~uw~z|¾²|_udfh
|X«fgjhE=z%zswh¬uw~z|s¯zx± whSsv³
y´³
'%~svuw©r²tudfhyzP|Yuwsvu#dXrX/zuwdhEsw~s#~¥s=ydfhSy°hE[ªzhSydNswhEy®u~zP|M i = V i, Ri
1, Ri2³^c#dfhmuhEsvu=¬ ©«h
ticsswzXy~uwhS
uzM i ~¥s^uwdfhgx~|f~g7«fg zÔuwdfhu¦¯zuv¸IuwhEsvu¯¬©«fhSs^/hu¦hhE|uwdh~|Puh|©/svhSy®u~zP|
V i |udfh=u¦z7hXuwh|©swhEy®u~zP|s Ri1|
Ri2
.tic = min
l∈1,2
[ uv¸IuwhEsvu(Ri
l , Vi)] ± 4A6Y´
c#dfhE|¾²%¦¯h/hwªzguwdfhEswdfzP©t~|fÂztic/hu¦hhE|
τ1|
τ2ªzP©©z¦¯hS_Xr_©~|fhEmuw|svªzguw~z|_ªwzPg
[τ1, τ2]uwz
[−1, 1]~|zthE#uwzztu~|x/zuh|Yuw~¥©I²
Uc(i)²tswhE z|udfhghEsw«fhzPth Y«Pyrz
H1ªz#udfhswhEy®u~zP|
M i .
Uc(i) =
1~tic < τ1
1− 2tic − τ1τ2 − τ1
~τ1 ≤ tic ≤ τ2
−1~tic > τ2
± 4 ´
pXhSyzP|t©r²ÉudfhÂdfzgzPh|fhE~Æur)dXrXzuwdfhSsv~¥sH2
~¥sydfhEy°PhEUYr yzgf«tu~|_udfh¹pYuw«fh|Yuuv¸IuwhEsvu¬©«fhEstihhu¦¯hEh|
sw«yyhEssv~¬h¨~|Puh|©¾swhEyuw~z|sV i | V i+1 .
tih =uw¸KuhEsvu
(V i, V i+1)± 4SP´
GHG ICJKLM
4S " * ! * ! &5(+"
c#dfhE|¾²8¦h·/hwªzg+¹uwdfhEswdfzP©t~|f¹ztih/hu¦hhE|
1|
τhªzP©©z¦¯hS)Xr)_©~|fhSu|svªzguw~z|Áªzg
[1, τh]uwz
[−1, 1]~|zthE#uwzztu~|x/zuh|Yuw~¥©I²
Uh(i, i+ 1)²tPsvhS[zP|[udfhghEPsv«fhmzthP«yr[z
H2ªz V i, V i+1 .
Uh(i, i+ 1) =
−1~tih < 1
1− 2τh − tihτh − 1
~1 ≤ tih ≤ τh
1~tih > τh
± 47´
'%~|©©rP²Xuwdfhux/zuh|Yuw~¥©Pswswzty~¥uhEuz·xzP©rX©~|fhc~¥suwdfhmªzP©©z¦~|.
Ud(c) = pc
I∑
i=1
Uc(i) + ph
I−1∑
i=1
Uh(i, i+ 1)± 4EY´
¦dfhEwhI~¥sudfh[|X«fgjhE7z=swhEyuw~z|sjyzg/zPsw~|Âuwdfh f~ÆthE©^gsw°ÁPswswzty~¥uhE"uwz
c²pc|
phh/zPsw~Æu~¬Ph·¦¯hE~PdYus
hEswhSy®u~¬Ph©rPswswzty~¥uwhS uwzudfhjyz|Yuwsvu=zuwhE|Pu~©Uc|uwdfhjdfzgzPh|h~ur[/zuwhE|Yuw~¥©
Uh³=c#dhuwzu©%h|hr·ªzP
P~¬Ph|yzP|tÇ«uw~z|C~¥sÉudfhsw«fg0z¾fumzuwh|Yu~©sÉzÔhEyd/z©rX©~|fh/h©z|fP~|fmuz
C³Éc#dfh=fu¨uwhEwg ~sÉuwdX«s8P~¬Ph|
Xr .
hd(C) ∝ exp
(
−∑
c∈C
Ud(c)
)
± 4SP´
¦dfhEwhUd~s=~¬h|Xr·h Y«uw~z| ± 4S´®³
![-9E<>¯6-9
9 F / , c9zNhXuwyumuwdfh·©~|fh|fhu¦z°Âªzg,|)~gPh²¾¦h·~guzÂÇ|)NyzP|tÇP«fuw~z|"z#zP©rX©~|fhSs¦df~¥yd"gX~g~ÃEhEsmudfh«f||fzg©~ÃEhE fwztyhEssfh|sw~Æur
h~¬hE| Xr .
h(C) ∝ hp(C) hd(C)± P´
¦dfhEwhhp|
hdwhxhEswhSy®uw~¬hE©rÂP~¬Ph|_XrNhP«uw~z| ± 7´|¹hY«u~zP| ± 4SP´³jc#df~¥s¨~s |fz|)yzP|X¬hNfz©hEg ªz
¦df~¥yd)f~hEyumzPtuw~g~ÃSu~zP|_~¥sm|fzu/zPssv~f©hxP~¬Ph|_uwdfh©¥hsv~Ãhz¯uwdfhsvuuwhsvyhjuwdu~s ∪∞N=0ΩN¦dfhEwh
ΩN~¥sudfh[swhuz=yz|tÇP«fuw~z|szN/z©rX©~|fhSs³È"hwzPzYsvhuzNhEsvuw~guwhudf~¥sgt~gj«fg YrÁNsv~g7«©uwhE)|f|fhE©~|f
sydfhghP²¾¦df~¥yd)yz|sw~¥susz#sw«yEyhEssw~¬Phxsw~gj«f©¥uw~z|sz¯udfhfzXyhEsst~¥suw~f«tu~zP|πT
sv/hEy~ÆÇhSÁYr¹uwdfh·thE|sw~Æurh1/T ²¦~uwd
Tt«©©r·tzf~|juz
0³8fzXz9zyz|X¬hEwPh|yh~¥s~¬hE| ~|,µ¦dh|uwdfhfhEyhEPsvh¨z%uwhEghEuw«fh
T~¥s
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) 9 , DCc#dfh©zPw~uwdfg ydfzPswh|Áuz_sv~g7«©uwhudfh·«f|f|zg©~ÃhSÁgxhSsw«fh
πT~¥sxNh¬Phsv~f©h·Ä«fgUÅ_°z¬Ái¯d~| Źz|Yuwh
iw©z ± =ÄYÅ"iÅ"i#´8©z~uwdfg 4Ef²C44)I³» u¯yz|sw~svusz¾sv~g7«f©¥u~|t~syhuh=Ź°z¬ji¯d~|zÔ~|X¬~¥|YuÉgxhSsw«fhπT¦df~¥yd7/hwªzgs9swg©©«gxs9/hu¦hhE|udfhswPyhEs
Ωi³%c#df~¥s9~uwhEuw~¬h©z~uwdfgtzXhEs9|fzuÉthEhE|zP|uwdh¯~|f~uw~¥©XsuuwhP³
uÉhEPyd7svuwh²=u|sv~uw~z|ªwzPg§udfhy«fh|Yu%svuuwhSuwzm|h¦Usvuuwh
S′ ~¥s%fz/zPswhEPyyzPf~|f=uwzm=fz/zPsw~Æu~zP|°hEw|h©Q(S → .)
³c#dh^u|sv~uw~z|x~¥s%PyyhtuhEj¦~Æudfzf~©~urα(S, S′)
P~¬Ph|jYruwdfh=l¨hh|juw~z³%c#df~sÉPyyhtu|yh¯u~z~¥syzgf«tuhENswzuwduudfhjfhu~©hE©¥|yh7yzP|t~uw~z|N~¥s=¬hEw~ÇhSÔ²yz|t~uw~z|N«|th¨¦df~¥ydÂuwdf~¥s¨©z~uwdfgyz|X¬PhhEsuzπT³8c#d~s=svuwz|xyz|t~uw~z|~¥s~¬hE| Xr .
∫
A
∫
B
πT (dC) P (C, dC ′) =
∫
B
∫
A
πT (dC ′) P (C ′, dC)± 4S´
¦dfhEwhA|
Bwh=u¦¯z7swhus^zÔuwdfhuw~f«[sswzXy~uwhSxuwz
Ω²t|
P~¥s8udfhu|sw~Æu~zP|°h|fhE©zÔudfhmŹ°z¬xyd~|
Ct³
pX«fzYsv~|fxuwduπT (.)Q(C → .)
dPs#Ç|~Æuhth|sw~ur² D ²t¦~uwdwhSsv/hEyuuzjswrXgxghuw~yE©GghEsw«fh ψ zP| Ω×Ω²Xuwdfh
yz|t~uw~z| ± >4E´~¥sswuw~¥sÇhE ~Æ .α(C,C ′)D(C,C ′) = α(C ′, C)D(C ′, C)
± P´sswdfz¦|N~|- 4E%ªzuwdfh7Ç|f~uwhsuuwhjsvPyh7yEswh²/~Æu¨~szfuw~g©uz[g°Phuwdh7fzf~©~ur
αs©wPh7szYswsw~f©huwz
hEt«yh¨uwdfh«tuwztyzwhE©uw~z|[zudfhŹ°z¬yd~|³8c#dY«s²t¦h¨u°Ph .
α(C,C ′) = min 1, R(C,C ′) ± "P´
465HG7498
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¦dfhEwhR~sudfh7l¨whEh|uw~z~¬h|Xr .
R(C,C ′) =D(C ′, C)
D(C,C ′)
± 6Y´
c#dX«sE²fh Y«©~ur ± ´~¥s#¬hEw~ÇhE¾³k|fhN~|YuwhEwhSsu~|f¿/z~|Yu[zmudfh_ÅNhuwz/z©~sv¸ svuw~|fPsv¸l¨whEh| ©Pz~Æudfg ~¥sudu[uwdfh¹wzPzYsw©°h|fhE©
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F ×M ¸fz/zPswhEÁ¦~uwd Âfzf~©~urpb¸¨|)|¿«|f~ƪzPwg thEuwd
± ~|X¬hEswhmfz/zPs© ´8~|uwdfhswhuzÉswhghE|PusC³8c#dfhm«f|f~ªzg f~wuwdNyzP|sv~¥svusz .
«f|f~ªzg©r[f ¦~|f|Â~|~Æu~©ÔzP~|Yu=~| F
«f|f~ªzg©r[f ¦~|f¦~Xud e ~| [emin, emax]
GHG ICJKLM
4 " * ! * ! &5(+"
«f|f~ªzg©r[ydzYzYsv~|fsvhEgh|Yu|X«fg7/h n hu¦¯hEh| 1|
nmax
«f|f~ªzg©r[f ¦~|fjuwdfhghuwhEs#thSswyw~f~|fhEPyd swhghE|Pu ± ©hE|fudÂ|Ât~whSy®uw~z|/´^~| V n ³» |uwdhyswhmz9uwdfhf~wuwdÂzzP©rX©~|fh
c²fuwdfh7l¨hhE| u~z~¥s#~¬h|Xr .
R (C,C ∪ c) =pd
pb
λ|F |N(C) + 1
h(C ∪ c)h(C)
± P´
¦dfhEwhpb
± whSsv³pd = 1− pb
´^~¥suwdh¨wzPf~©~ÆurzydfzXzPsw~|jjf~wuwd ± hEsw¾³ÉthEuwd´²h~s¯udfhmfwztyhEssfh|sw~ÆurP²t|
λ~¥sudfh7~|YuwhE|sv~urNzÉudfhxwhªhh|yhz~¥swswz|¹wztyhSswsE³» |_uwdfhyPsvh7z8udfhjthSud_z^[zP©rX©~|h
c²/udfhl¨hhE|¹uw~z·~¥s
P~¬Ph|Yr .
R (C,C \ c) =pb
pd
N(C)
λ|F |h(C \ c)h(C)
± P´
c#dfhswhEyz|¹urY/hxz¯~wuwd)yzPwhEswzP|fsuwz uwdfhfz/zPsw~uw~z|_z zP©rX©~|hyzg/zPswhE¹z¯sv~|f©hsvhEgh|YuS³7c#dfhyzwhSsv/z|f~|f ± @À¨´=whE¬hsw~f©h¨gxz¬PhyzPwhEswzP|fsudY«s#uwzudfhfz/zPsw~uw~z|z8sv«ydÂ~wuwdN¦~uwdÂudfhfzf~©~urpb
|muwdh¯fz/zPsw~uw~z|zthEuwdz=/z©rX©~|fh¯yzg/zPswhEmzsv~|fP©hsvhEgh|Yu¦~uwd7uwdfhfzf~©~urpd = 1−pb
³» |Âudfh7yPsvhmzsgf©~|¦~Æudfz«tu¨fu~|tªzPwguw~z|¾²Xudfhf~vudNz%uwdfh7zP©rX©~|fhhEf«yhS[uz·svhEgh|Yu=~¥s«f|f~ªzg .udfh7fz/zPswhEÂswhPgxhE|Yu=~s«f|f~ªzgx©rÂt ¦|Â~|
Z = F × [emin, emax]× [Lmin, Lmax]×]− π, π]³c#dfh7thSuwdN~¥s©swz
«f|~ƪzPwg .8xsvhEgh|Yu~s«f|~ƪzPwg©r[ydfzPswh| ~| uwdhsvhuE1(C)
z9/z©rY©~|fhEs~|CwhSt«yhEuzz|fhswhghE|PuS³» |[udfhyswh
zxf~vud¾²tudfh7l¨whEh|uw~zj~¥sP~¬Ph|Yr .
R (C,C ∪ c) =pd pb
λ|F |nmax (](E1(C)) + 1)
h(C ∪ c)h(C)
± 7´
¦dfhEwh](E1(C))
thE|fzuhEs^uwdfhm|X«fg7/hz9zP©rX©~|fhSs~|CwhSt«yhEuzjzP|fh¨swhPgh|YuE³º¾~°hE¦~swh²X~|·uwdh¨yEswhz%7thSuwd7.
R (C,C \ c) =pb pd
nmax ](E1(C))
λ|F |h(C \ c)h(C)
± P´
=uwdfhEuwd| «|f~ƪzPwg©r fz/zPsw~|ÁÁ|fhE¦swhPgxhE|YuE²^~u·¦z«f©¥U/hÂgzhh©h¬|Puxuwz¿«svhNfuÁ~|fªzguw~z|Uuwzfz/zPswhgzhxz uwhE|)swhPgh|Yus¨udu7wh¦h©©Æ¸ zYsv~uw~z|hEÔ³[pXz²¦hfwzPzYsvhxuzNwhEf©Pyhjuwdfh @À¨ °hEw|fhE©ÉXr_2@#ÀPsvhSzP|7Àu ± @#À¨=À¨´³c#dfh¯Çsvu%svuwhEjyz|sw~svus9z/yzgf«fuw~|f=udfh¯wzPf~©~Æu~hSsz©~|fhSsuw«yuw«fh^whSsvhE|yh¯~|jhEydf~th©zGuwdfhzPw~~|©t~gPh#ªz^~¬h|x|X«fg7/h¯zGz~hE|Yuu~zP|s
Nθ³ 'fz8uwduE²Y¦¯h=svuwzPwhªz¯h¬Phrf~XhE©
pi, i=1,...,Npix|h¬hEwr[z~hE|Puu~zP|θk, k=1,...,Nθ
²fudfh7yz|Yuwsvuzuwh|Yu~©uk
i ∈ [−1, 1]¸^~¬hE|Xr h Y«u~zP| ± 4 ´¯¸¯yzwhSsv/z|f~|f
uz[svhEgh|Yu=zzPw~h|Yuuw~z|θkyhE|PuhhEz|
pi³#ÈÁhuwdfhE|Nztu~|¾²fªzP=h¬Phr[zPw~h|Yuuw~z|
θk²g
BkthÇ|f~|f·|
~|fdfzPgxzPhE|fhzP«s ± (/|fz|«f|f~ªzg·´f~wuwd°Ph|fh© .
Bk(pi) =2− uk
i∑P
j=1(2− ukj )
± P´
c#dfh#¦hE°h9~s%udfh#zuwhE|Pu~©tssvzty~¥uhEmuzf~th©pi²udfhsvuwz|h%~¥sudfh#fzf~©~ur
Bk(pi)zfz/zPsw~|f¨svhEgh|Yu
z%zPw~h|Yuuw~z|θkyh|YuwhEwhS·z|
pi³
c#dfhmfztyhSt«fhªzPfz/zPsw~|fxzÉx|fh¦§zP©rX©~|hcPyyzt~|uwz
B1, . . . , BNθ
~sudfh|uwdhmªz©©z¦~|f.
uwdfh¦~¥Xud~¥s#«f|f~ªzg©rt ¦|[~| [emin, emax]
uwdfh©h|uwd l |·udfht~whSy®uw~z| α z9udfhmÇsu=swhghE|Pu=wh¨«f|f~ªzg©rt ¦| ~| [Lmin, Lmax]×]− π, π]
_~Æth© pi~st ¦| Pyyzt~|¹uwz"uwdhg
Bkα
²#yzhEswzP|t~|fNuwzÁudfhÂz~hE|Puu~zP|θkα
sw«ydUuwdu.kα =
argminj [|α [π]− θj |]²f¦dfhEwh
[π]ghE|s#gxztt«©z
π
uwdfhmswhghE|Pu#yhE|Puh p ~¥s«f|f~ªzgx©rt ¦|·~|·uwdfhmsY«whz F ⊂ R2 yzPwhEsw/z|t~|fmuz7udfh~Æth© pi
Yuwdfh¨~|f~Æu~©zP~|Yu
p1c
zc~¥s#uwdfhE|Nyzgf«fuwhE ªzg
p²α²/|
l³
465HG7498
* ( &* *,& (3* &&" !(+* 4A"
c#dfhm~|X¬hswhgz¬h¨yz|sv~¥susz«f|~ƪzPwg©r·whEgz¬Y~|f7x/z©rX©~|fh¨~|E1(C)
³^c#dfhl¨hh| uw~zPs#sswzXy~uwhSuwzjuwdfhmf~wuwd| uwdfhthSud zx/z©rY©~|fh
c = (p1c , e, l, α)
h¨whSsv/hEyuw~¬h©r·P~¬Ph|[Xr .
R (C,C ∪ c) =pd pb
λ|F |Npix Bkα
(ip) nmax (](E1(C)) + 1)
h(C ∪ c)h(C)
± "PP´
R (C,C \ c) =pb pd
Npix Bkα(ip) nmax ](E1(C))
λ|F |h(C \ c)h(C)
± "C4S´
¦dfhEwhip~¥suwdfh~Æth©9yzhEswzP|t~|f7uzfwzhEyuw~z|[z9udfhyhE|Yuwhz
(p1c , p
2c)z| uwdfhux~ghP³
8 8 * B(4 9 7 Î60Ò Î c#dfh)swhEyzP| °X~|z7gz¬hEs[«sw«©©r wzPzYsvhS uwz5swgx©h"zthSy®uÂfwztyhEssvhSs·~¥s[udfhÁgxztt~ÇyEuw~z| zjU|tzg©rydfzYsvhE|_zPthEyuPyyzt~|·uwzNÂswrYgghuw~¥y©%uw|svªzguw~z|³7º¾hu T = Ta : a ∈ E /h[g~©r¹z#svrXgghuw~¥y©u|sªzPwguw~z|sghuwhEw~ÃhSxYrm¬hSy®uz
a ∈ E ³» Ôuwdh=gzXf~ÆÇyEu~zP|z¾|zPthEyu^~s¯tz|h=Xrjf©rX~|f Ta¦dfhEwh
a~s#«f|~ƪzPwg©r ydfzYsvhE|[~|
E²fuwdfh7l¨hhE| u~z~¥s#whSt«yhS·uzuwdfhfh|sw~Æur[uw~z.
R(C,C ′) =h(C ′)
h(C)
± "Y´
ÈÁhfz/zPswhdfhEwhu¦zswrYgghuw~¥y©sw~gf©h=gz¬hSs/.uwdfh¨t~©uw~z|z¾/z©rX©~|fh|udfh=gz¬h=z¾ydfhEy°X~|f/z~|PuS³c#dfh¨t~©uw~z|·zzP©rX©~|hyzP|sw~svus^z¾«f|f~ªzg©rydfzXzYsv~|f
δe ∈ [−∆e,∆e]|·Pft~|f
deuwz7uwdfh¨zP©rX©~|h¦~¥Xuwd
e²¦~uwdxudfht~Æu~zP|©fyz|f~Æu~zP|juduÉudfh|fhE¦ ¦~¥Xuwd~¥sÉ~|
[emin, emax].e′ = emin +((e−emin +δe) [emax−emin])
²¦dfhEwh
[.]th|fzuwhSsuwdfhgztt«f©zª«f|yuw~z|¾³
c#dfhmgz¬h¨z7/z~|Yu=yz|sv~¥susz9«|f~ƪzPwg©r[ydfzXzPsw~|fxxydfhSy°Y~|fxzP~|Yuz9uwdfhydfzYsvhE|[/z©rX©~|fh| fz/zPsw~|xu|sv©¥u~zP| zuwdf~¥s=ydfhSy°Y~|f/z~|YuE³¯c#d~s#u|sw©uw~z|~ssvrXgghuw~¥y©Gu|svªzguw~z|ghuwh~ÃhE[Xr¬PhEy®uza«f|f~ªzgx©r[t ¦|~|NyzgPy®u=yhE|YuwhhE[uwzxudfhz~P~|³
8 8 8 xÓ%Ó Ð ¢ ' Ó Ð ¡ Î Ò Î¹Ò ¢) Î E' Î ' c#dfhmfX¸ |X¸ =hgz¬h ± ´¯/hwuw«fu~zP|[yzP|sw~svus^z9Pft~|f7zP^hgz¬X~|fj7swhghE|Pu#u¯uwdfh¨h|[z^uwdfh¨hE~|f|f~|fzÂ/z©rX©~|fhP³ '%~svuE²N/z©rX©~|fh
c~¥s«f|f~ªzg©rÁydfzPswh|)~|)uwdfh y«fwhE|Yu7yz|fÇ«fu~zP|¾³pthEyzP|Ô²9uwdfh ydfz~¥yhzudfh
gz¬hxurX/h~sfz|fh·PyyzPf~|f uwz¹fzf~©~ur¹PsvhSÁz|"uwdfh·|X«fgjhEzsvhEgh|Yusyzg/zPsw~|c³·» #uwdfh/z©rX©~|fh
~¥smwhSt«yhENuz zP|fhsvhEgh|YuE²ÔzP|f©rÂuwdfht~Æu~zP|"z¯svhEgh|Yum¦~©©Éhfz/zPswhE¾³º¾~°h¦~¥swh²Ô~^uwdhx/z©rX©~|fhx~syzgx¸/zPswhE z
nmaxsvhEgh|YusE²^zP|f©r¿uwdfhNhgz¬©#zm)swhghE|Pu¦~©©/hNfz/zPswhEÔ³5» | uwdfhNzuwdfhE·yPsvhSs²^uwdfhNhgz¬©
|NudfhxPft~uw~z|¹z¯[swhPgh|Yu¨whwzPzYsvhSN¦~Æud¹udfhjwzPf~©~Æur1/2
³k|yhudfhxgxz¬Ph~¥s¨ydzPswh|¾²G|"hXuwhg~ur~¥s·ydfzYsvhE| ¦~uwd fzf~©~ur
1/2³ » | uwdh_ff~Æu~zP|5yEswh²#Á©hE|fud
l|5¿t~whSy®uw~z|
αwh«f|f~ªzg©r t ¦| ~|
V = [Lmin, Lmax]×] − π, π]³¨ºhu
c = (p, e, l1, α1, . . . , ln, αn)/hudfhydfzPswh|N/z©rX©~|fh³» ÉuwdfhydfzYsvhE|NhXuwhg~ur ~¥s
udfh~|~Æu~©ÉzP~|Yup = (x, y)
²Ôudfh|Áudfh|fh¦ zP©rX©~|fh~¥sm~¬hE|_Xrc′ = (p′, e, l, α, l1, α1, . . . , ln, αn)
¦dfhh7udfh|fh¦~|f~uw~¥©/z~|Yu8~s
p′ = (x− l cos(α), y− l sin(α))³É» /uwdhydfzPswh|xhXuwhEgx~ur~¥sÉudfh#Ç|©fzP~|YuS²uwdh=|fh¦ /z©rY©~|fh~s8~¬h|
Xrc′ = (p, e, l1, α1, . . . , ln, αn, l, α)
³=» |¹uwdfhjwhEgxz¬X~|f·yPsvhP²~ÉuwdfhxydfzPswh|NhXuwhg~Æur~¥suwdh7~|f~Æu~©9/z~|YuE²/uwdh|Nudfh|fhE¦zP©rX©~|h~s~¬hE|ÂYr
c′ = (p′, e, l2, α2, . . . , ln, αn)¦dhh
p′ = (x + l1 cos(α1), y + l1 sin(α1))|Ô²zuwdfhEw¦~¥swh²
c′ = (p, e, l1, α1, . . . , ln−1, αn−1)³
» |yswhEs¦dfhEwhuwdfhm|X«fgjhEz9swhghE|Pusz¾udfhydfzPswh|·/z©rX©~|fh¨~s#/hu¦hhE|2|
nmax − 1²Pudfhl¨whEh| uw~z7~¥s
P~¬Ph|Yr·uwdfhu~zz9uwdfhfh|sw~Æu~hSs/.
R (C,C ′) =h(C ′)
h(C)
± " "P´
¦dfhEwhC ′ ~¥suwdh7yzP|tÇ«uw~z|ztu~|fhEXr[udfh/hwuw«fuw~z|Âz%uwdfhxy«fh|Yu=yz|fÇ«fu~zP| C ³#» |ÂudfhzudfhyswhEsE²udfh7l¨whEh|uw~z~s#P~¬Ph|Yr .
R (C,C ′) =h(C ′)
2 h(C)
± "6Y´
c#dfh yz|su|Yu1/2
~|Yuwh¬hE|fhEs7smudfh[t~Æu~zP|)z=NsvhEgh|Yuuz_ÂzP©rX©~|h[yzPgx/zPswhEÁz#zP|fh·swhghE|Pu ± whSsv³[udfhhgz¬©GzjswhPgh|Yuz%x/z©rY©~|fh¨yz|Yu~|~|f
nmaxsvhEgh|Yus´^~sfz|fh¨¦~ÆudÂfwzP~©~Æur
1|·udfhm~|X¬hEswhgz¬Ph²
(%uwdh·whEgz¬ ©Éz=NswhghE|Puz=ÂzP©rX©~|fh yzg/zPswhEÁz#u¦z¹svhEgh|Yus ± hEsw¾³[udfh[Pft~uw~z|¿z=NswhPgxhE|Yuuwz_
GHG ICJKLM
4A6 " * ! * ! &5(+"
/z©rX©~|fhyzPgzYsvhS[znmax − 1
swhPgh|Yus´8~¥s=ydfzYsvhE| ¦~uwdÂfzf~©~ur1/2
³
c#dfhyzg7~|uw~z|jzudf~¥sÉsv«ff¸I°Ph|fh©X¦~uwdxz|fh#zuwdfhu¦zsv«ff¸I°Ph|fh©¥szf~wuwdt¸ |t¸ thSud7z/zP©rX©~|hEshEt«yhEuwzzP|fhswhPgh|Yu%©©z¦=sÔuzP«|YuwhEhÉuwdh¯~whEf«y®u~f~©~Æurmzuwdfh#Ź°z¬¨yd~|Ph|fhEuwhEYr¨udfhÅNz|Yuwhi©z©z~uwdfg³c#df~¥syzgjf~|uw~z|N©©z¦=suwz· ¬Pz~¥·udfh / @#À0°h|fhE©¾¦df~¥yd¾²¦~uwdf~|¹|ÂzPtuw~g~ÃSuw~z|ªghE¦¯zPw°G²X~s|fzu=h©h¬|Pu.¯f~wuwdÂzzP©rX©~|hmyzg/zPswhE zÉsvhE¬h©GswhghE|Pus#~sudfh|NgzYsu©r·hhSy®uhEÔ³
8 85) 9 7 ( Ð ¢ ' Ó Ð Î ¡E Î Ò = Î E' Î ' EsªzP7udfh·u|sv©¥u~zP|¾²%uwdfhff~Æu~zP|U|¿uwdfh whEgxz¬©^z_ydfhEy°X~|f¹/z~|Pu©©z¦ uz_/hwuw«f¿~| Nh©h¬|Yu7¦# rÁ/z©rY©~|fhffz t~gu~¬Ph©r)¦¯hE©©¸I/zPsw~uw~z|fhSÔ³Uc#dfhSsvhgz¬hEsthÇ|fhN"whE¬hsw~f©h[gz¬PhyE©©hS pXf©~uv¸ |X¸ ÅNhEwPh[zpXhEgh|Yus ± ptÅ_p´#¦df~¥yd ~¥s~©©«svuwuwhS[XrÇ«fhf³
z ~ U( )
Split Merge
h
bl
'%~P«fwhm .^ptf©~uv¸ |X¸IghEwPhzswhPgxhE|YusE³
¢ 9 7(ª ºhu
s = (pj , pj+1)hmxsvhEgh|Yu#z9jzP©rX©~|h
c³Éc#dfhswf©~uvu~|jz
s~¥s¯zP|f©rxwzPzYsvhS·~¾uwdfhm©h|fuwd z
s²l²X~¥s©wPh
ud|2Lmin
²f|[~c~s#|zu=yzPgx/zPswhE[z
nmaxswhPgxhE|YusE³%ÈÁhm«swhm| «tt~©~wr¬w~¥f©h
Z~|zthuwzxzfu~| u¦¯z
|fhE¦ swhPgxhE|Yuss′1 = (pj , p′)
|s′2 = (p′, pj+1)
sv«ydxPsp′~s©ztyuwhEj~|7udfh#whSy®u|f©hz©h|uwd
l−2Lmin|j¦~¥Xuwd
2Lmin| ¦dzPswh¨g~|Nt~¥s=yzPwhEsw/z|fs¯uzuwdfhswhPgxhE|Yu
s³^» | udf~¥s=¦# r²Xuwdh©hE|fudsz%uwdh|fh¦swhghE|Pushm~|
[Lmin, Lmax]³c#dfh7¬ w~¥©h
ZyzhEswzP|fsuwz «f|f~ªzg t ¦~|fzÉ/z~|Pus=~|ÂuwdfhjwhSy®u|fP©hz©h|uwd
l − 2Lmin|¦~Xud2Lmin
.
Z =
[
H ∼ U([−Lmin, Lmin])B ∼ U([Lmin, l − Lmin])
] ± "Y´
'wzPg uwdfhPh|fhEuwhE_¬hSy®uwzPz = (h, b)
²¾¦hztu~|_udfhghuwhsv1 = (l1, α1)
thSswyw~f~|fuwdfhswhghE|Pus′1«sw~|fuwdfht~#GhzgzPwfd~swg thÇ|fhEXr .
v1 = ηv(h, b) =
[ √h2 + b2
α+ arctan(hb )
] ± "PP´
¦dfhEwh·uwdfh ghuhsv = (l, α)
zswh·ÇfthEÔ³¹ÈÁh[zPtu~|)udfh| uwdfh ghuhs
v2 = (l2, α2)fhEsy~f~|fNudfh
swhPgxhE|Yus′2ªzg
v1huv = (l, α)
sªzP©©z¦=s.
v2 = T (v, v1) =
√
(l sin(α)− l1 sin(α1))2 + (l cos(α)− l1 cos(α1))2
arctan(l sin(α)− l1 sin(α1)
l cos(α)− l1 cos(α1))
± " 7´
6 Î ¡ E Î
ºhusj = (pj , pj+1)
|sj+1 = (pj+1, pj+2)
uwdhu¦zjyz|swhEy«tuw~¬h¨svhEgh|Yus^z9/z©rX©~|fhc³Å¹h~|fudfhEswhu¦z
swhPgxhE|YusyzP|sw~svuszhf©¥y~|fsj|
sj+1Yr
s′j = (pj , pj+2)³8c#dh¨ghEwP~|fzu¦zswhPgh|Yus~¥s#z|f©r·fz/zPswhE ~Æ
465HG7498
* ( &* *,& (3* &&" !(+* 4S
udfhzP~|Yupj+1 ~s¨©ztyuhEN~|¹udfhwhSy®u|fP©h7Pswswzty~¥uhEÂuz sl,l+1
ªzP¨ svf©~umwzPzYsv~uw~z|¾³mc#dfhyzP|t~uw~z|suwz¬hEw~ªrwh¨uwdX«suwdfh¨ªz©©z¦~|f .
d(pj , pj+2) ∈]2Lmin, Lmax]
uwdfhf~svu|yhmz pj+1 uwzuwdhmswhPgh|Yu s′j = (pj , pj+2)~¥s#©z¦huwd|
Lmin
uwdfhzPvudfzPz|©XfwzhEyuw~z|xz pj+1 z| s′j~¥sÉ©zXyEuhExu¯mt~¥su|yh
bzpj uwdu^~s8hu¦¯hEh| 2Lmin
|l−Lmin
³
£ m¡ Ò 9^Ò E¢7 D Î ¡ ' Î 7
ºhuCh uwdfhÂy«wh|Yuxyz|tÇ«fu~zP|¾³"c#dfhEwhh
NS(C)swhghE|Pusuduy| hÂswf©~u|
NM (C)yz«f©h z
swhPgxhE|Yus%uwdu¯yE|xh=ghEwPhEÔ³%c#dfhh=huwdX«sNT (C) = NS(C)+NM (C)
/zPssv~f©hgxz¬PhEszurX/h=svf©~uv¸ |X¸ ghhzsvhEgh|Yus³Ác#dfh·Çsvuxsvuwh yzP|sv~¥svusz=«f|f~ªzg©r¿t ¦~|_¹gz¬h gz|f_uwdfh
NT (C)/zPssw~f©h[gz¬PhEsE³_» =udfh
ydfzYsvhE|_gz¬hx~¥s swf©~Æu ± whSsv¾³jghhS´²¾¦hjfztyhEhE_Psmhtf©¥~|fhE_~|)P´ ± whSsv¾³x´w´®³c#dfhfwzPzYsw©9°Ph|fh©Q
~¥sudY« P~¬Ph|Xr .
Q (C → A) =∑
si∈S(C)
qSi (C,A)
NT (C)+
∑
(si,si+1)∈M(C)
qMi (C,A)
NT (C)
± "PP´
¦dfhEwhS(C)
fh|fzuwhEs=uwdfhjsvhuz^swhPgxhE|Yus#uduyE|N/hjswf©~ÆuS²M(C)
th|zuwhSs#uwdhjswhuzÉyz|swhEy«tuw~¬hjsvhEgh|YusuwduyE|[/h¨gxhEwPhEÔ²
qSi (C,A)
yzPwhEswzP|fs8uwzjuwdfhmsv©~uvu~|fz¾uwdfhmswhghE|Pusi²f|
qMi (C,A)
yzPwhEsw/z|fs8uwdfh¨gh~|fz9udfhyzP«ff©h
(si, si+1)³^c#dfhswf©~Æu=vu
qSi (C,A)
~¥s~¬hE| Xr .
qSi (C,A) =
∫
Σi
1A(Si(C, z))dz
2Lmin (li − 2Lmin)
± "PP´
¦dfhEwhΣi = [−Lmin, Lmin] × [Lmin, li − Lmin]
~¥sudfhjyzgPy®u=~|¹¦df~¥ydÂuwdfh¬w~¥f©hZ~¥sf ¦|N|
Si(C)~¥s
yz|tÇP«fuw~z|z/z©rY©~|fhEs#¦dfhEwhmudfhgxhuwhEsvi = (li, αi)
z%uwdfhswhPgxhE|Yusiwhmhf©¥yhE Xr
(l1, α1) = ηvi(z)|
(l2, α2) = T (vi, ηvi(z))
³8c#dhgxhEwPhmwuqMi (C,A)
z%uwdfh°Ph|fh©Ô~¥s#~¬hE| Xr .
qMi (C,A) = 1A(Mi(C))
± 6YP´
Mi(C)~¥sxyzP|tÇP«fuw~z|·z/z©rY©~|fhEs¦dfhEwhuwdfhmghuwhEs^z¾udfhyzP«ff©h
(si, si+1)wh¨hf©¥yhEXruwdh¨gh¸
uhs#z%uwdfhswhPgxhE|Yu#zPtu~|hE Yr·uwdfhgh~|fzsi|
si+1³
Ó ¡ ÎXÎ ' ¡¢t A( Ò £ Ò 9 9 E¢f A( Ò '
c#dfhswrXgxghuw~yE©ÔghEPsv«fhψz|
Ω× ΩydzPswh|·uz·th~¬ uwh¨uwdhgxhSsw«fh
πQ ~s#udfhmªz©©z¦~|f .
ψ(A,B) =
∫
A
∑
si∈S(C)
∫
Σi
1B(Si(C, z))dz
|V | dµ(C)
+
∫
A
∑
(si,si+1)∈M(C)
1B(Mi(C)) |Jφ−1(vi, vi+1)| dµ(C)
± 6 4S´
¦dfhEwhV = [Lmin, Lmax]× [−π, π]
²f|φ~¥s¯uwdfhmt~#/hEzgzfdf~¥swgyzhEswzP|t~|fuwzjuwdhªzP©©z¦~|f7¬~¥f©hyd|fh .
(v1, v2)φ←−− (z, v)
¦dfhEwh(v1, v2)
yzwhSsv/z|xuzudfhghuhsÉzÔudfhu¦zjswhPgh|Yus^ztu~|fhE· uwhEswf©~Æuwuw~|f7udfhmsvhEgh|Yu^zgx¸huwhs
v = (l, θ)«sw~|fxudfh«tt~©~¥r¬~f©h
z³Éc#df~¥st~#GhzPgzfdf~¥svg~¥sP~¬Ph|Yr .
φ(z, v) = (ηv(z), T (v, ηv(z))± 6X´
¦dfhEwhηv|
TwhhEswhSy®u~¬Ph©r_~¬hE|"Xr"h Y«u~zP|s ± "´| ± " 7´®³
Jφ−1
thE|fzuhEsmudfh[ÄYyzPf~¥|_zφ−1 ¦dfzYsvhthuwhEwg~||Yu=~¥s#~¬h|Xr .
|Jφ−1(v1, v2)| =l1 l2l
± 6 "P´
GHG ICJKLM
4S " * ! * ! &5(+"
'%~|©©rP²udfh¨l¨hhE|·uw~zyzPwhEsw/z|t~|fmuzuwdhswf©~u¯zswhghE|Pu¯z¾©h|fuwdl~|u¦z7swhghE|Pus8z¾©hE|fuds
l1|
l2~¥s#~¬hE| Xr .
R (C,C ′) =NT (C)
NT (C ′)
Lmin(l − 2Lmin)
π (Lmax − Lmin)
l
l1 l2
h(C ′)
h(C)
± 6 6Y´
º~°Ph¦~¥svhP²t~|uwdhgxhEwP~|xyEswh²Xuwdh7l¨whEh|uw~zx~¥s#~¬hE| Xr .
R (C,C ′) =NT (C)
NT (C ′)
π (Lmax − Lmin)
Lmin(l − 2Lmin)
l1 l2l
h(C ′)
h(C)
± 6X´
8 8 3 9 7 ( Ð ¢ ' Ó Ð Î ¡E Î Ò 9^Ò 7 Ï 7( ' Î ÈÁh7fz/zPswh¨u¦¯z/hwuw«wuw~z|s=zurXh7swf©~Æuw¸ |X¸ ghhzÉ/z©rX©~|fhEsE³#c#dfhEswhu¦zwhE¬hsw~f©hmgz¬hEsh¨~©©«svuwuhE~|ÁÇ«fh 7Y³·c#dfhxÇsu/hwuw«fuw~z| ± ptÅN 4S´myzP|sw~svusmzt~|ÂÂswhPgxhE|Yum©~|f°X~|fu¦zÂzP©rX©~|fhSs|¾²¾whE¬hEswh©r²hgz¬X~|jxsvhEgh|Yu¯z%7zP©rX©~|fhP³8c#dfhmsvhSyz|·/hwuw«fu~zP| ± pfÅN¯´#yz|sw~¥sus^zwhEgz¬Y~|fxjyz|f|fhSy®u~zP|·hu¦¯hEh|u¦zsvhEgh|Yus|Ô²th¬PhsvhE©rP²XywhSu~|fyzP|f|fhEyuw~z|Xr[gz¬X~|fxzÉ|h|tzP~|Yu=z%/z©rY©~|fh³
'%~«fh7 .hwuw«fu~zP|sz%urXhswf©~Æuw¸ |X¸ gxhEwPhz9/z©rX©~|fhEsE³
¢ 6 $ Ò Îºhu
Sj,δe
h[uwdfhhEvu«fu~zP| ¦df~¥ydUsv©~usN/z©rX©~|fhYr)hgz¬X~|¹_swhghE|Pujz_zP©rX©~|fh z=¦~¥Xuwd
e|
yzgzYsvhS·Xr[u#©hEPsuuwdwhEhmswhPgh|Yus#|·¦df~¥yd©©z¦=s^uwzthÇ|fh¨u¦¯zxzP©rX©~|hEs¦dfzPswhm¦~tuwdswh¨h Y«©uze+ δe|
e− δe.
Sj,δe(p, e, v1, . . . , vn) = (p1, e+ δe, v1, . . . , vj−1), (p
j+1, e− δe, vj+1, . . . , vn) ± 6YP´c#dfh¬w~¥u~zP|[z¦~¥Xuwd
δe~¥s«f|f~ªzg©r[f ¦|~|ÂyzPgyusw«ydÂsudfhmu¦¯zx|fhE¦§¦~¥Xuwds=hm~|
[emin, emax].
δe ∼ U([−B(e), B(e)])²f¦dhh
B(e) = mine− emin, emax − e± 6>7´
c#dfh~|X¬hEswhj/hwuw«wuw~z|)yzP|yhEw|s¨udfh·yzP«ff©hz#zP©rX©~|hEs(ci, cj)
¦dfzPswhÇ|©ÉzP~|Yupni+1
i
zci|Á~|f~Æu~©
/z~|Yup1
j
zcj¬h~ªr .
Lmin ≤ d(pni+1i , p1
j ) ≤ Lmax
± 6YP´ÅNzPwhEz¬hEE²Puwdfhsw«fg zudfhswhghE|Pu|X«fgjhE
(ni + nj)dsuzh©z¦huwd|
nmax.
ni + nj < nmax
± 6YP´c#dfhghEwPh¨gz¬Ph
M(ci, cj)ªz
ci = (p1i , ei, (v
ki )k=1..ni
)hucj = (p1
j , ej , (vkj )k=1..nj
)~sthÇ|hEÂsªzP©©z¦=s/.
M(ci, cj) = (p1i ,ei + ej
2, (vk
i )k=1..ni, vij , (v
kj )k=1..nj
)± P´
¦dfhEwhvij
yzhEswzP|fs¯z%uwdfhghuwhEszuwdhsvhEgh|Yu(pni+1
i , p1j)³
c#df~¥s8thÇ|f~uw~z|z/uwdh=svf©~uv¸ |X¸ ghhz|f©r7~|y©«fhEsudfh=yzP«ff©hzGzP©rX©~|fhSs¦dfzPswhÇ|©fzP~|Yu^|x~|f~uw~¥©zP~|Yu¬Ph~ƪrxuwdhfz t~g~urxyz|t~uw~z| ± 6PP´®³» u¦z«©/hh©h¬|Pu^uwz7ghEwPh©¥svzjyz«f©hSs^¦dzPswh=~|~Æu~©hYuwhEg~Æu~hSs^zP8Ç|©hYuwhEg~Æu~hSs¬Ph~ƪrÂudf~¥s¨yz|f~Æu~zP|¾³70ÇsvusvzP©«tu~zP|_¦z«f©¥¹/hjuwzNf"·|fhE¦0fz/zPsw~uw~z|¹°Ph|fh©%~|"uwdhx=ÄYÅ"iÅ"i©Pz~Æudfg uwduyzPwhEsw/z|uzuwdfh~|X¬hEsw~zP|[gz¬hP² (gz¬h¨uwdu=~|Y¬PhsvhSs¯/z©rX©~|fhsªzP©©z¦=s.
(p1, e, v1, . . . , vn)← (pn+1, e, v′n, . . . , v′1)
± 4S´¦dfhEwh
v′j = (lj , αj − π)~αj > 0
²f|v′j = (lj , αj + π)
zuwdh¦~swh³
465HG7498
* ( &* *,& (3* &&" !(+* 4 7
hh²t¦htz|fzu=«swhsv«yd°Ph|fh©Gf«tu¦¯ht~whSy®u©r·~|yzzPuwhuwdf~¥sgz¬h¨~|Yuzuwdfh°Ph|fh©Q
³8» |thEhEÔ²¦¯hPfÂsuhuwduyzP|sw~svus#z%|fzg©r[fwzPzYsv~|fxhEwgj«tuuw~z|Âz%uwdh7svhE|swh¨z9udfh/z©rX©~|fh ± s´³ 'z=swf©~u=gz¬hSj,δe
²tudfhmwhSsv«©Æu~|f~z%/z©rX©~|fhSs#~s#udX«s#«f|f~ªzg©r[ydzPswh|Âgz|fxudfh¨ªz©©z¦~|fxªz«f~s.
S1j,δe
(p1, e, v1, . . . , vn) = (p1, e+ δe, v1, . . . , vj−1), (pj+1, e− δe, vj+1, . . . , vn)
S2j,δe
(p1, e, v1, . . . , vn) = (pj , e+ δe, v′j−1, . . . , v
′1), (p
j+1, e− δe, vj+1, . . . , vn)S3
j,δe(p1, e, v1, . . . , vn) = (pj , e+ δe, v
′j−1, . . . , v
′1), (p
n+1, e− δe, v′n, . . . , v′j+1)S4
j,δe(p1, e, v1, . . . , vn) = (p1, e+ δe, v1, . . . , vj−1), (p
n+1, e− δe, v′n, . . . , v′j+1)
± P´
ÈÁh|fz¦yz|sv~¥th#uduuwdfhgh~|f/hu¦hh|u¦zzP©rX©~|hEsci|
cjy|/htzP|fhXr·ªz«=~s#zh|t/z~|Yus.
pi, pj1 = pni+1i , p1
j pi, pj2 = p1
i , p1j pi, pj3 = p1
i , p1nj+1
pi, pjm = pi, pj4 = pni+1i , p1
nj+1³%ºhu
pi, pjk/h[Â~7zhXuwhg~uw~hEsmuduxy|)/h·©~|f°PhEÁuzNgxhEwPh
ci|
cj³Nc#dfhwhSsv«©Æu~|fN/z©rX©~|fh·~¥s«f|~ƪzPwg©r
ydfzYsvhE|gz|fjuwdfhmªzP©©z¦~|fxu¦¯z/z©rY©~|fhEs.
M1k (ci, cj) =
(p1i ,
ei+ej
2 , (vil)l=1..ni
, vij , (vjl)l=1..nj
)~Æk = 1
²( pi, pjk = pni+1i , p1
j(pni+1
i ,ei+ej
2 , (v′il)l=ni..1, vij , (vj
l)l=1..nj)~Æk = 2
²( pi, pjk = p1i , p
1j
(pni+1i ,
ei+ej
2 , (v′il)l=ni..1, vij , (v′j
l)l=nj ..1)
~Æk = 3
²( pi, pjk = p1i , p
1nj+1
(p1i ,
ei+ej
2 , (vil)l=1..ni
, vij , (v′jl)l=nj ..1)
~Æk = 4
²( pi, pjk = pni+1i , p1
nj+1
M2k (ci, cj) =
(pnj+1i ,
ei+ej
2 , (v′jl)l=nj ..1, vji, (v′i
l)l=ni..1)
~Æk = 1
²( pi, pjk = pni+1i , p1
j(p
nj+1j ,
ei+ej
2 , (v′jl)l=nj ..1, vji, (vi
l)l=1..ni)~Æk = 2
²( pi, pjk = p1i , p
1j
(p1j ,
ei+ej
2 , (vjl)l=1..nj
, vji, (vil)l=1..ni
)~Æk = 3
²( pi, pjk = p1i , p
1nj+1
(p1j ,
ei+ej
2 , (vjl)l=1..nj
, vji, (v′il)l=ni..1)
~Æk = 4
²( pi, pjk = pni+1i , p1
nj+1± "P´ºhu
NS/huwdfh#|X«fg7/hÉz/swhPgxhE|Yus%¦dfzPswhwhEgxz¬©X©©z¦=suwdfhPh|fhEuw~z|zu¦z/z©rX©~|fhEsÉ|
NMuwdh|X«fgjhE
zG~szGhXuwhg~Æu~hSsɦdfzYsvh©~|°Y~|f©©z¦=s%uwdfhh|huw~z|xzÔm|fhE¦ /z©rX©~|fhP³% gz¬Ph~¥s8«|f~ƪzPwg©rjydfzYsvhE|xgz|fudfh
NT = NS +NMzYswsw~©hgz¬hSs³c#dfh°Ph|fh©
Q ~¥suwdh|Â~¬hE| Xr .
Q (C → A) =∑
sji∈S(C)
1
NT (C)
4∑
k=1
1
4
∫ B(ei)
−B(ei)
1A((C \ ci) ∪ Skj,δe
(ci))dδe
2B(ei)
+∑
pi,pjk∈M(C)
1
NT (C)
2∑
m=1
1
21A((C \ ci, cj) ∪Mm
k (ci, cj))
± 6Y´
¦dfhEwhsj
i
th|fzuwhSs=udfhxswhPgh|YujzÉudfh7/z©rX©~|fh
ciz^¦~¥Xuwd
ei²pk
i
thE|fzuhEsuwdfhjzP~|Yukzci² S(C)
~¥s=udfhxswhu¨zswhPgxhE|Yus¦dfzYsvh#hgz¬©t©©z¦=s9udfhhE|fhu~zP|7zu¦¯zm/z©rX©~|fhEsE²| M(C)
uwdfh=swhuÉzGswhPgxhE|Yus%uwdu8y|xPh|fhEuwh/z©rX©~|fhmXr[ff~|fswhPgxhE|Yu#©~|f°X~|fxudfhg³8c#dfh°Ph|fh©Ô~¥suwdX«sP~¬Ph| Yr .
Q (C → A) =1
8NT (C)
∑
ci∈C
1
B(ei)
ni−1∑
j=2
4∑
k=1
∫ B(ei)
−B(ei)
1A((C \ ci) ∪ Skj,δe
(ci)) dδe
+1
2NT (C)
∑
ci,cj∈C
4∑
k=1
2∑
m=1
1A((C \ ci, cj) ∪Mmk (ci, cj))
± P´
c#dfhghEsw«fhψz|
Ω×ΩydzPswh|uwzxth~¬ uwhuwdfhgxhSsw«fh
πQ ~¥s¯jsvrXgghuw~¥y©ghEsw«fhyz|yh|YuwuhExz|
⋃∞n=0ΩN ×ΩN+1 ∪ ΩN+1×ΩN
³Éº¾huA|
Bhswhus#z¾udfh¨uw~f«z
Ωsw«yd Ps
A ⊆ EN|
B ⊆ EN+1³
c#dfhghSsw«fwhφ~sudfh|Â~¬hE| Xr .
ψ(A,B) =
∫
A
N∑
i=1
ni−1∑
j=2
4∑
k=1
∫ B(ei)
−B(ei)
1B((C \ ci) ∪ Skj,δe
(ci)) dδe dµ(C)
ψ(B,A) =
∫
B
N∑
i=1
N∑
j = 1j 6= i
4∑
k=1
nmax(emax − emin)
2dk(ci, cj)λ|V |1A((C \ ci, cj) ∪M1
k (ci, cj)dµ(C)
=
∫
B
∑
ci,cj
2∑
m=1
4∑
k=1
nmax(emax − emin)
2dk(ci, cj)λ|V |1A(C \ ci, cj) ∪Mm
k (ci, cj)dµ(C)
± P´
GHG ICJKLM
4S " * ! * ! &5(+"
¦dfhEwhΣth|zuwhSs=udfhjsvuuwhxsvPyh7Pswswzty~¥uhEuwz[uwdfhjzP©rX©~|h7¦~¥Xuwd¾²Ô|
dk(ci, cj)thE|fzuhEsuwdfht~¥su|yhhu¦¯hEh|
udfhmu¦¯z/z~|Yusz p1, p2k³8c#dfh¨yuwz 1
2dk(ci,cj)
~¥st«fhmuzuwdfh¬w~¥f©hmyd|fPh .
(e1, e2, p′)←− (e, δe, lj , αj)
¦dfhEwhe1|
e2h=uwdh¨¦~¥Xud[z
c1|
c2zPtu~|fhS·Xrhgz¬X~|f7z¾udfh¨swhPgh|Yu
jz97/z©rX©~|fh¨z¦~¥Xud
e²X|
p′yzPwhEswzP|fs¯uwzxudfhhE|fhuhE·~|f~uw~¥©¾/z~|YuE³
'%~|©©rP²Xuwdfh7l¨hhE|[u~zyzPwhEswzP|t~|f7zudfhsv©~uz%x/z©rX©~|fhmz%¦~XudeXr·whEgz¬Y~|fxxswhPgh|Yuz9©h|fuwd
l~¥s~¬hE| Xr .
R (C,C ′) =NT (C)
NT (C ′)
8 l λ |V | min(e− emin, emax − e)nmax(emax − emin)
h(C ′)
h(C)
± 7´
¦dfhEwh |V | = 2π(Lmax − Lmin)³É» |uwdfhh¬Phsvh¨yPsvhP²Xuwdfh7l¨hhE| u~z~¥s#~¬h|Xr .
R (C,C ′) =NT (C)
NT (C ′)
nmax(emax − emin)
8 l λ |V |min(e− emin, emax − e)h(C ′)
h(C)
± P´
6 * Ò Îc#dfh·swhEyz|)ptÅN/hwuw«fuw~z|Á¦Psmf«f~©ÆuuzNgxhEwPhxy©zYsvh/z©rX©~|fhEs¨udu7fzN|fzu¬hEw~ªrNudfhfz X~g~Æur¹yz|f~Æu~zP|¾³hh²9uwdfhghh~¥s7tzP|fhXr_hgz¬X~|fudfhÇsuzPuwdfh©¥svu7swhghE|Puz#Â/z©rX©~|fhªzP©©z¦¯hS_XrÁft~|f¹©~|f°X~|fswhPgxhE|YuE³8c#dfh|fhE¦ ¦~¥XuwdN~¥s#h Y«©GuwzudfhghE|z%u¦zx¦~¥Xudsz9uwdfhghEwPhE zP©rX©~|fhSs³$'z=yzP«ff©hz%zP©rX©~|fhSs(ci, cj)
¬hEw~ªrX~|ni + nj ≤ nmax
²¦¯hmd ¬Ph¨dh~dYuzYswsw~~©~Æu~hSs#z|fh¦o«f|fzPfhhE·/z©rX©~|fhSssswdfz¦| ~|Ç«wht³gxzP|fudfhEswh7dfhE~PdYuzYswsw~f~©~uw~hEsE²¦hz|©r°hEhÂuwdh7fz/zPsw~Æu~zP|szÉswhPgh|Yus¦dfzPswh©h|fuwdN~¥s/hu¦hhE|
Lmin|Lmax
²¾¦df~¥yd"©©z¦=suwzÂhE|fhuh7¦h©©Æ¸ thÇ|fhE_zP©rX©~|fhSs³'fzPm ~¬hE|_ghhP²¾[/z©rX©~|fhx~sm«|f~ƪzPwg©r¹ydfzPswh|gxzP|fxu¦¯z/z©rX©~|fhEsyzg/zPswhE Yr·uwdfhsghsvhEgh|Yus#f«tu=~|Ât~#Ghh|Yuzthc#dfh=swf©~Æuyz|yh|s%udfhzP©rX©~|hEs8yzgzYsvhSxYrxu8©hEPsuÉu¦zsvhEgh|YusE³ ydfhEy°X~|fmzP~|Yu
pji
~¥sÉ«f|f~ªzg©rxydfzPswh|gxzP|fNudfh /z~|Yus
(p2i , . . . , p
ni
i )³¿c#dfh[/z©rX©~|fh ± z¦~Xud
e´~¥sudfh| y«tuuxuwdf~¥sjzP~|YuE³)c#dfh¦~tuwdsjzudfh[u¦z
hEsw«f©uw~|f)zP©rX©~|fhSsh hY«©#uze + δe
|e − δe
²¦dfhhδe~s«f|~ƪzPwg©rUt ¦|U~|
[−B(e), B(e)]thÇ|fhE ~|
hP«uw~z| ± 6>7´®³ 'zxz|fh z=udfh[u¦z_/z©rX©~|fhSs²^ydzPswh| ¦~uwd fwzP~©~Æur1/2
²udfhydhEy°X~|¹zP~|Yu~sxhPh|fhEuwhEPyyzPf~|fuz[«f|~ƪzPwg t ¦~|fz8©hE|fud
l|N·t~whSy®uw~z|
α³ 'fzP«fzPfhhEzP©rX©~|hEsyE|Âudfh|¹/hPh|fhEuwhE
Xr[h¬Phsv~|fjz%uwdfhswhghE|PuzPthEE³Éc#dfhEwhwhuwdX«sdh~dYuzYswsw~©hswf©~ÆusªwzPg ydfhEy°X~|fxzP~|Yupj
i
³ºhu
NM (C)hudfh|X«fgjhE=zÉ/z~|Pus#uwdu©©z¦=suwdfh7gxhEwP~|xzu¦¯z/z©rY©~|fhEsz
C|
NS(C)uwdfh|X«fgjhE
zj/z~|Pus uwdu©©z¦=s·udfh¿sv©~uvu~|f5zjU/z©rX©~|fhÁzC³ gxz¬Ph_~¥s«f|f~ªzgx©rydfzYsvhE|§gzP|f udfh
NT (C) =NF (C) +ND(C)
zYswsw~©hmgz¬hSs³Éc#dfhm°hEw|h©Ô~¥suwdfhE|P~¬Ph|Xr .
Q (C → A) =∑
ci∈C
ni∑
j=2
1
NT (C)
8∑
k=1
1
8
∫ B(ei)
−B(ei)
∫
V
1A((C \ ci) ∪ Skj,δe
(ci))dv dδe|V | 2B(ei)
+∑
ci,cj∈C
8∑
k=1
1
NT (C)
2∑
m=1
1
21A((C \ ci, cj) ∪Mm
k (ci, cj))
± P´
c#dfhl¨hhE|[uw~zyzhEswzP|t~|fuzxswf©~uz7/z©rX©~|fhmz¦~¥XudeXr·h©Py~|fjswhghE|Pu#z%©hE|fud
l²X~sudfh|
P~¬Ph|Yr .
R(C,C ′) =NT (C)
NT (C ′)
16 l λ |V |min(e− emin, emax − e)nmax(emax − emin)
h(C ′)
h(C)
± PP´
º~°Ph¦~¥svhP²Pudfh7l¨hh|uw~zyzhEswzP|t~|f7uzjghEwPhz9u¦zj/z©rX©~|fhSs#whEf©Py~|f|~|f~uw~¥©z#Ç|©¾svhEgh|Yu#Xr|fhE¦§swhPgxhE|Yu#z%©h|uwd
l²t~¥sP~¬Ph|Yr .
R(C,C ′) =NT (C)
NT (C ′)
nmax(emax − emin)
16 l λ |V |min(e− emin, emax − e)h(C ′)
h(C)
± C4S´
/ / / » |)udf~s7svhSy®uw~z|²9¦hydfhEy°_uwdfh/hd ¬X~zz¯udfh[Å"iÅ"i0©zPw~uwdg ªz7sgf©~|f¹«f|f~ªzg,z~¥ssvzP|"fztyhSswsm¦~uwdnmax = 10
|λ|F | = 100
³ 'fzPuduS²¦hjyz|sw~¥thuwdfhjhgf~w~¥y©% ¬PhhSszN²/uwdhuwzu©%|X«fg7/hz8/z©rX©~|fhEsE²
| Nii=1..10²uwdfh|X«fgjhEÉzGzP©rX©~|hEsÉyzPgzYsvhS7z
iswhPgh|YusE³9c#dfh#ghE|sÉhyzPgx«tuwhS7ªzg ÇXhS7|X«fgjhE
z~uwhEuw~z|sI0± dfhh²
I0 = 30000´®³ 'fzg
I0²/¦¯hjyz|sv~¥thswgx©hhE¬hEwr
P~uwhEuw~z|s ± dfhEwhP²
P = 10000´®³¯c#d~s
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3
2
4
5
e = 4ij
e = 4ij
e = 4ij
e = 4ije = 4
ij
3214
5
pj1
e = 5
e = 3
i
j
pi1
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Q = 1/2Q + 1/2Q ²/¦dfhEwh
Q ~¥sudfh@#À¨°hEw|fhE©¾¦~uwdfzP«tut~zghuw~yE©Ô~|tªzguw~z|²|Q
~s=uwdfhx°Ph|fh©I³c#dfh7©Pz~Æudfg yz|X¬hEwPhE ~|©hEssuwd| "6swhEyz|fs¨¦~uwd)[wztyhSswswzm l à ±
3.5 × 106 ~uwhEuw~z|s´®³7c#dhxhEgf~~¥y© ¬hEhSsyzPgf«tuwhS"f«fw~|fudfh·©zPw~uwdfgwhswdfz¦|_~|_Ç«fhx³xc#dhxhEgf~~¥y©ÉhEwzmtzP|fhjzP|_udfhzP~|Yu|X«fgjhE¨~¥sm¦hE° .m/z«tu0.2
ªzudfhzPwhuw~¥y©¬©«fhz
E(N) = λ|F | = 100³7Źzhz¬hEE²uwdhxhEgf~~¥y©¬©«fhSsmz8udfhht/hEy®uu~zP|_z¯uwdfh|X«fgjhE¨z
/z©rX©~|fhz#sv~Ãhi±i = 1, . . . , nmax
´¨hxy©zYsvhjuwz uwdfhudfhzPwhuw~¥y©%¬©«fhxzfu~|fhS_XrÂuwdhdYrX/zudfhEsw~s¨z «f|f~ªzg©¥ ¦ zP|[udfh|X«fg7/h=z%swhPgxhE|YusyzPgzYsv~|fzP©rX©~|fh .
P (i) =1
10∀i = 1..10 ⇒ E[Ni] =
E[N ]
10= 10 ∀i = 1..10
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x 106
0
20
40
60
80
100
120
Iterations
E[N]E[N
1]
E[N2]
E[N3]
E[N4]
E[N5]
E[N6]
E[N7]
E[N8]
E[N9]
E[N10
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Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 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 - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399