A Polyline Process for Unsupervised Line Network …...ISSN 0249-6399 ISRN INRIA/RR--5698--FR+ENG...

HAL Id: inria-00070317https://hal.inria.fr/inria-00070317

Submitted on 19 May 2006

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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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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 ⊂ Rd µ 7I³'fz n ∈ N

²¾©huΩn

/hxuwdfhswhuz¯yz|tÇ«fu~zP|s x1, . . . , xnuwduyz|sw~¥su

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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

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F²P¦~uwd·gw°ts8~|·7swPyh

M²P~¥szP~|YufwztyhEssÉz|

F ×M sv«yduwduN(A×M) <∞©gzPsvusv«fh©rxªz#|Xr:@zh©/zP«f|thS[swhu

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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

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Fz

R2 yzwhSsv/z|f~|fuz7udfhzPswh¬hEsyhE|fh²X¦dfzYsvh|X«fg7/h

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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

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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

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hp¦~uwdÁwhSsv/hEyuuz[udfhjhªhEwhE|yhjfzXyhEss©¥ ¦³c#dfhxh­tfhEssv~z|Nz

hp~¥s

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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

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u³m» |_zuwdfhE¦zfsE²u¦z[/z©rX©~|fhSs¨whxsw~

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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|

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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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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

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o∈VC,F

U22(c, o)± Y´

¦dfhEwhU21

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ωfhuωs.

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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´

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pkc

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c

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pkc

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1

ε2

(

1 + ε2

1 + d2(pkc , o)

− 1

) ± 4 4S´

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U22~¥s#udfh|Â~¬hE|[Xr .

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k = 1, n + 1

o : d(pk

c , o) < ε

σ(< c, o >pkc)

± 4 ´

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x|

y.

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σ2x

nx+

σ2y

ny

± 4"P´

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nx

± whSsv³y²σy|

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Ri2

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l∈1,2

[ uv¸IuwhEsvu(Ri

l , Vi)] ± 4A6Y´

c#dfhE|¾²%¦¯h/hwªzguwdfhEswdfzP©t~|fÂztic/hu¦hhE|

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1~tic < τ1

1− 2tic − τ1τ2 − τ1

~τ1 ≤ tic ≤ τ2

−1~tic > τ2

± 4 ´

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H2ªz V i, V i+1 .

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−1~tih < 1

1− 2τh − tihτh − 1

~1 ≤ tih ≤ τh

1~tih > τh

± 47´

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i=1

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i=1

Uh(i, i+ 1)± 4EY´

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−∑

c∈C

Ud(c)

)

± 4SP´

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∫

A

∫

B

πT (dC) P (C, dC ′) =

∫

B

∫

A

πT (dC ′) P (C ′, dC)± 4S´

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± 6Y´

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/ @À ~s#udfhsv~gf©hEsvu=fwzPzYsw©G°h|fhE©Ô¦df~yd©©z¦=suwzg°hswg©©X«fgs/hu¦hh|NswyhEszÉt~#/hEwhE|Yusw~ÃhEsE³É» uyz|sw~svusz=«f|f~ªzgf~wuwd¿z=Â/z©rX©~|fh·~|

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

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ni∑

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1

NT (C)

8∑

k=1

1

8

∫ B(ei)

−B(ei)

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± P´

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NT (C ′)

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h(C ′)

h(C)

± PP´

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NT (C ′)

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16 l λ |V |min(e− emin, emax − e)h(C ′)

h(C)

± C4S´

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10∀i = 1..10 ⇒ E[Ni] =

E[N ]

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80

100

120

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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