Détection de feux de forêt par analyse statistique de la radiométrie d'images satellitaires
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Détection de feux de forêt par analyse statistique de la radiométrie d’images satellitaires Florent Lafarge — Xavier Descombes — Josiane Zerubia. N° 5369 Novembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5369--FR. Thème COG. apport de recherche.
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(43) .C"$ )C (X(t + τ ), ..., X(t + τ ))1 k (X(t ), ..., X(t )). q¯n*]&|M`§~|M¥|Mm oKsty b]_oK`coKp^aq_bbl6m|M_l*l6s =stm l6m*|Mm6soKq'q&|Msty br¨_Æq°st`¤|Mpbay6sb"KÑ,=¥;|,l6m*|A« m6soKq'q&|My6stm Xª 'qn*]&|M`§~_l|M¥|Mm oKsty bOl6stprq'sv±&bO³ ª 'q baq_l ba`"'¥b'b~'sv¡'ba¥lXª 'q_bst`¤|Mpbl bay*| =sl m6y6st'_l ba¥oKq´¥;|,`c¢a`cb¥oKsV~&|Myfm6y*|Mq_l6¥;|Mm6soKql6~&|Mm6s;|M¥br¨ ´ 4 5YË65ÏÉ 4. 1. k. 1. k. Y,- /'
(44) H '-V:"SW9X<SWY /
(45) Y t∗ E F N A , L K ,G $
(46) J ' ' 'W
(47) +
(48)
(49) +C + R(s,X(t) ' t) = E[(X(s) − E[X(s)])(X(t) − E[X(t)])]. # %$. Z[Z]\^.
(50) . . . " '
(51) , ' . M . . .
(52) J C/ !
(53) J t = s = t∗ '
(54) J CC
(55) !
(56) J ' G $
(57) ' I 'N
(58) J C &. R(s, t) #´%$ 4 Y5 Ë6Ï5 É 4 7 & ∂ R(s,t) Y' ' C !
(59) J G E F 2N G ∂s ∂t $
(60) ' X (t) = lim X(t+hδ )−X(t) ' 'i $>
(61) ' * Ch→0 $> $> ,'Nh 7 C( ! +$Y$ ,- -'./ I ) -E ! * !
(62) X (t) 2. i. i. i. &. A. i. ti $Y$ " ' !
(63)
(64) ! E F N !) * !
(65) A LK *.C t G $>
(66) ' I G !
(67) (' '
(68) " #),"' i E 7C(
(69)
(70)
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(74) J A C . ´ 4 YË ÏÉ. $> $J$> I ') *E' ' C
(75) #
(76) * ,W$ ,- ) ' / ' $J$ # ' / µ = E[X] C G # C # σ2 = E #
(77) +
(78) Y' ! +
(79) *
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(82)
(83) ! α = (α * /' n G $
(84) 6E ' $> J , ..., Pn ' E #. UC + $ $
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(86) J$ J C( I Y,- /'
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(88) C ! # %$. . . . .
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(90)
(91)
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(94) Y *
(95) $ ' Y' ! +
(96) /' '
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(110) #EF
(111) ('. ´ YË ÏÉ. . . / * E .C,- . . s − tG. $>
(112) '. ! 2 %>@ .
(113) +
(114) ∀u E F S ⊂ E F N G
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(118) . # %$. . E #
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(128) . . # %$. χk (Au ). ' $>
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(130) J J' . • F (t) = u ∂F (t) = 0 G ∀i ∈ [1, N − 1] • ∂x i ∂F • ∂x (t) > 0 N 2 F • D(t) = ∂x∂i ∂x (t) j. i,j∈[1,N −1]. 8 2
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(132) ' * 7C( ' . . . ! 2 %>V! T . * /$J$ ",
(133) 6E$
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(136) w : h → w(h) =. B. t∈S. sup |X(t) − X(s)| kt−sk<h. . EA$JÉ $> ; ' 5Ï Ë&. 8 &
(137) I G t ∈ E F N WNK ,G $
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(142) %* C # #W,- ' G /', $>
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(144) 2C( N *
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(147) (' -.C I 'W C(E F ' ! +$Y$ 'G '∂S .CW'
(148) ,- G
(149) * / ' '. '
(150)
(151) G +C( E ' C( + ' C ' /' E C
(152) Y' J C & G ./ $ ' !
(153)
(154) Y I ' ' C ' ! $Y$ ' ')
(155) # #
(156) φ(x, x , ..., x , z) G $> E /'
(157) J X, X , ..., X , Z
(158) Z ' $>DC( E . . . . . . . . 1. . N. 1. N. MZ. ')(. *'+.
(159) .
(160)
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(162) ,- ./. $>
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(167) C # ' #
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(169) $> '"E"
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(172)
(173) ./ i h ↓ij0 ∀ > 0 P (max [w (h), w (h)] > ) = o(hN ) . . . . $>
(174) ' G. . i,j. i. . ij. • E(χk (Au )) = λ(S)(−1)k. R. R. xN (det D)φ(u, 0, ..., 0, xN , z)dxN dz .
(175) $>"'
(176) Y $>" ' x C >0 !
(177)
(178) ! #
(179) # '" # * 7C( ' . N. . EF. z∈. N (N −1) 2. !
(180) E $ '.C$&'. D. C($>E. k. '. . • E(χ(Au )) = λ(S)(−1)N −1. EA$JÉ $>; 5Ï & Ë A & /
(181) ' , $>I . B. R. xN >0. R. z∈. EF. N (N −1) 2. xN (det D)φ(u, 0, ..., 0, xN , z)dxN dz . E NK ,G $>
(182) J *E' ' + BK
(183) ,W
(184)
(185) #G E F N E ,W
(186) % C N ' Y'Y Y' $ '"
(187)
(188) C')E)$>)
(189) + $>
(190) ' G $ ' / # W"$>W ( E F * Y' +.C-E"$ /' , $> 0 E ' +
(191) A = A (X, S) u '
(192) C # / . . . . . . Eχ(A) = λ(S)(2π).
(193) . N +1 2. σ 2 = E(X 2 (t)) G Λ. 1. 2. u |Λ| 2 σ −(2N −1) exp− 2σ 2. P[ N2−1 ] j=0. '$>-, * + "
(194) +C + W. 2j N −1−2j (−1)j aj σ 2j CN −1 u. Xj (t) G. . aj =. (2j)! j!2j. . xoK'y ~'¥t_l'b'am*|Mst¥l l6'yf¥bl 'a`coKq_l m6y*|Mm6soKq_l'bnbl 'ba=¡1~'y oK~'y6sam l_¥b®¥bnam ba'y!bl6m stqFFstm O¬,noKq_l6'¥tm bay ²{f=¥bay = @!= 2¨ ÆqÁ~&|My6m6sna'¥tsbayX~+oK'y®'qÁn ]&|M`§~Å|M¥|Mm oKsty b§p|M_l l6sbaqÁ'b,=st`cbaq_l6soKq½=+¥ª²bl6~ ay*|Mq_nb§'b,¥;| Ñ®\ n|My*|Knam ay6sl6m6s³F_b"bl m 'oKq'q_~&|My"Ä · =º¹ u Eχ(A) = λ(S)(2π) |Λ| σ exp− 2σ !oK_lf~ oK'oKq_l|K_|M~'m baynb®m6]_oKy ^a`cbO_|Mq_lf¥b'oK`¤|Mstq_b'b®¥ª st`¤|Mpbay6sbr¨xoK'ynba¥;|=_¥bl ]FkF~ oKm6]_^l bl 'b¥;|~'y oK~'y6sam O,'oKstbaqm¢am6y bay6sv±&bl"Ä ©|noKqm6stqF'stm Ã'bl¼'ay6stbl¼~&|My6m6sba¥t¥blzÇ_l ³F ª¿|M l bnoKq_ÐoKy =y b=Ðn*]&|M`§~ |Mstq_l s!³F_b • ¥;|°q_apr¥tstpb|M'st¥tstm °¬°¥ª²oKy =y b N 'bl,`¤|A¡Sst`¤|'bU¥ba'y`coS='¥b1'b¼noKqFm6stq'stm · nb1³F's9bl m ~'¥t_l u oKy6m5¹fl6stprq'sv±&baqm+baq°m bay6`cbOXª st`¤|Mpbay6sbr³F_b¥ª st`¤|Mpb"'oKstm®|oKsty'q_bnbay6m*|Mstq_bO]_oK`coM« paq_astm ®_|Mq_l9¥ª stqm baq_l stm 'bl bl9~'sv¡'ba¥l·Ïs¨²br¨S'q1nbay6m*|MstqU¥tsl*l*|Mpbº¹bam9q_oKm*|M`§`cbaqm ³F ª²ba¥t¥b!q_b 'oKstm~&|Kl ¢am6y bm6y oK~¯'y6'stm br¨ . . 3 2. Z[Z]\^. 1 2. −3. 2. 2.
(195) =. . . " '
(196) , ' . M . . . xoK'y"³ ª 'q_b,st`¤|Mpb§l*oKstm`coF'a¥tsl*|M'¥b,~&|My'q_by |M¥tsl*|Mm6soKq½Xª 'qn ]&|M`§~p|M_l l sbaq ]_oM« `coKp^aq_br_st¥u|M'm ³F_b!¥;|,=sl m6y6st''m6soKq°'bl(stqm baq_l6stm l!'bl(~'sv¡=ba¥lf'bnbam6m b®st`¤|Mpb"l6'stb'q_b ¥oKsXq_oKy6`¤|M¥bU· nba¥;|,y baSsbaqm!¬c|oKsty 'q_b=sl6m6y6st''m6soKqq_oKy6`¤|M¥b'b¥ª ]'sl6m oKpry*|M`§`cbº¹?¨ Æq m bay6`cb°Xª st`¤|Mpbay6sbr9nbam6m b¯~'y oK~'y6sam q_oK_lc~ bay6`cbam¤'b°³F&|Mqm6sv±&bay¼¥b¯q_oK`O'y b¯`cokbaq 'bna¥t_l m bay l!|M¯l ba'st¥ ´~ oK'y 'q_b®st`¤|Mpb`coS'a¥tsl b~&|My'q°n ]&|M`§~¯p|M_l*l6sbaq ¨ + 6-¸% ©ª²am6_'b¤y |M¥tsl*b¤_|Mq_lnbam6m bc~&|My6m6sbUlª¿|M~'~''sb¤l6'yO¥;|´m6]_oKy6sb¼'blOn*]&|M`§~_l,|M¥|Mm oKsty bl p|M_l*l6sbaq_l¦'oKqFm¦q_oK_lbaq_oKq_l£'bfoKsty£nbay6m*|Mstq_bl£~'y oK~'y6sam l¨FxoK'y9³_bfq_oK_l¦~''sl l6soKq_l¦'m6sv« ¥tsl*bay(nblf'bay6q's^ay blº=¥bl9'oKq'q_blº'nrª²bl6mÇ«¶¬A«2=sty b®¥bl(st`¤|Mpbl \fµ } 'bl9u ba=¡XF'oKstbaqFm ~ oK'oKsty ¢am6y b`coS'a¥tsl blf~&|Myf'blfn ]&|M`§~_l p|M_l l6sbaq_l¨'oK_lf|M¥t¥oKq_l 'oKq_n|M~'~'¥ts³_bayf~'¥t_l sba'y l(~'y ?« m6y*|Mstm ba`cbaqFm l|A±_q³_b¥bl st`¤|Mpbll6'y ¥bl*³_ba¥t¥blfq_oK_lm6y*|r|Mst¥t¥oKq_l|MsbaqFm 'q_b=sl6m6y6st''m6soKq p|M_l*l6sbaq'q_br¨. •. . . "V T . . .
(197) !. q_b"st`¤|MpbObl6m|Kl l6st`§st¥;|M'¥b,¬§'qn*]&|M`§~p|M_l*l6sbaq l s ¥bl~'sv¡'ba¥l'b"nbam6m bst`¤|Mpbl oKqFm = sl m6y6st'_ll ba¥oKq'q_b,¥oKsq_oKy6`¤|M¥br¨µ2¥lª¿|Mprstm'oKq_n¤'b,y |M¥tsl bay'qÁm bl6m'b,q_oKy6`¤|M¥tstm ¼l6'y ¥;| =sl6m6y6st''m6soKq 'bl,~'sv¡'ba¥l§'bU¥ª st`¤|Mpb¯|A±_q Xª¿|oKsty§'q»'bapry ´'b1nbay6m6stm6_'b¯³F&|Mqm¤¬¥;| `coS'a¥tsl*|Mm6soKqU'b ¥ª st`¤|Mpb ~&|My£'q¼n*]&|M`§~¼p|M_l l sbaq ¨µ2¥b?¡Ssl6m b~'¥t_l6sba'y l£m bl6m l¦'bfq_oKy6`¤|M¥tstm iF~'y baqFm = r r 2rnoK`§`cb(¥b(m bl m'b FoK¥t`coKpoKy oF« iF`§sty6q_o S¥b(m bl6m'bf©Xstq¤bam '¥t]_oK¥ r|MyoK 'sbaq,baq_noKy b£¥b¦m bl6m'b(iF]&|M~'sty oM«
(198) 4st¥ =l¨K!oK_l'm6st¥tsl bay oKq_l¥bm bl6m'Nb FoK¥t`coKpoKy oF« iF`§sty6q_o m bl m"~&|My6m6sna'¥ts^ay ba`cbaqm¤~'y naslO~ oK'y¥blOn*]&|Mqm6st¥t¥oKq_lc'b¼m*|Mst¥t¥b¤st`§~ oKy6m*|Mqm brnrª²bl6mÇ«¶¬A«2=sty br _|Mq_l q_oKm6y bn|Klf~ oK'y ¥blfpry orl l bl st`¤|Mpbl¨ FË ! ¼ÉÎ É T É MÉ X Ì X 5 4 É X © b"m bl m'b FoK¥t`coKpoKy oº« iF`§sty6q_oºbl6muwoKq_'l6'y!¥;|¼noK`§~&|My*|Msl oKqÁ'b¥;|§uwoKq_nam6soKqÃna=« `O'¥;|Mm6stb'b u y ³F_baq_nb N (x) ~+oK'y9¥ª²n ]&|MqFm6st¥t¥oKq½· nrª²bl6mÇ«¶¬A«2=sty b®¥b!q_oK`"'y bXª²oK_l bay6r|Mm6soKq_l stq=uway6sba'y bl¬ x l 'y!¥bq_oK`"'y b§Xª²oK_l*bay6K|Mm6soKq_l®m oKm*|M¥bl5¹|bnO¥;|cuwoKq_nam6soKqÃ'by a~&|My6m6stm6soKq Xª 'q_b9¥oKsSq_oKy6`¤|M¥b(~+oK'y¥;|!~+oK~''¥;|Mm6soKq¯· nrª²bl6mÇ«¶¬A«2=sty b(¥;|!~'y oK&|M'st¥tstm f³F ª 'q_b£oK_l*bay6K|A« F (x) m6soKq¼l oKstmstq=uway6sba'y b!¬ x¹?¨Ñb `¤|Mq's^ay b ~'¥t_l~'y nasl*brrq_oK_l¦'am bay6`§stq_oKq_l£¥ª²n|My6m¦`¤|A¡=st`"'` baq´K|M¥ba'y!|M_l*oK¥t_bbaqm6y b N (x) bam F (x) ¨ . . . !. . MZ. ')(. *'+.
(199)
(200)
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(202) ,- ./. ==. µ}. = max |N (x) − F (x)| x∈. ! oK_l(noK`§~&|My oKq_l |M¥oKy l ¬"'bl9r|M¥ba'y l9nay6stm6s³F_bl(m*|M''¥bl(l ba¥oKq1¥;|m*|Mst¥t¥b®'b!¥ª²n ]&|Mq=« m6st¥t¥oKqUbam¥bfq'stb|M1'b l6stprq'sv±&n|Mm6soKqUoK'¥t¤=¤m bl m¨Æq¤~&|My6m6sna'¥tsbayF~ oK'y¦'bl¦n*]&|Mqm6st¥t¥oKq_l 'b,m*|Mst¥t¥bcql6'~+ay6sba'y bc¬´r1bam~ oK'y'qq'stb|M'b§l stprq'sv±&n|Mm6soKqap|M¥£¬´=¨¿!=r ¥;|Ur|M¥ba'y nay6stm6s³F_bbl6m|M~'~'y o¡Sst`cb"~&|My ¨ Ñ|Mq_l!q_oKm6y bOn|Klº_¥;|§m*|Mst¥t¥b'bO¥ª²n ]&|Mqm6st¥t¥oKqnoKy6y bl ~+oKq_Ã|M°q_oK`O'y b'b"~'sv¡=ba¥l'bO¥ª st`¤|Mpb bam®bl6m®'oKq_n"q_bam6m ba`cbaqFm"l6'~ ay6sba'y®¬Ur=¨+!oK_l'm6st¥tsl bay oKq_l'oKq_nO¥;|¼K|M¥ba'y ~ oK'y|M~=« ~'y o¡Sst`cbay¥;|,K|M¥ba'y!nay6stm6s³_b~ oK'yf'q¯q'stb|M°'bl stprq'sv±&n|Mm6soKqap|M¥¬§=¨¿! =r¨ oK_ly b zÇbam6m bay oKq_l'oKq_n¥ª ]FkF~ oKm6]_^l b,³_b¥bn ]&|M`§~l oKstm®p|M_l l sbaq |bn'q_bO~'y oK&|M'st¥tstm 'b=¨¿! ='b®l bm6y oK`§~ bay l6s¥ª²n|My6mf`¤|A¡=st`"'` oK_l bay6 bl6m(l6'~+ay6sba'y oK´ap|M¥V¬O¥;|OK|M¥ba'y nay6stm6s³F_b ¨ x|Myc|Mst¥t¥ba'y lq_oK_lOy bl m6y bastprq_oKq_lO¥bUn|M¥na'¥f'bc¥ª²n|My6m,`¤|A¡=st`"'`oK_l bay6 ¬ x ∈ [µ − +o M bam l*oKqm¦y bl ~+bnam6stba`cbaqFm9¥;|"`cokbaq'q_bbam£¥;|K|My6s;|Mq_nb'b ¥;|¥oKs&q_oKy6`¤|M¥b 3σ, µ + 3σ] ¬!¥;|K³F_ba¥t¥b9q_oK_lµnoK`§σ~&|My oKq_lq_oKm6y b(n ]&|MqFm6st¥t¥oKq ¨Sd9ba¥;|q_b(n ]&|Mq'pb(~&|Kl¥;|!~ bay6m6stq_baq_nbf=§m bl6m n|My ~+oK'y'q_b§¥oKs¦q_oKy6`¤|M¥brX~'¥t_l"'b 'blOoK_l bay6K|Mm6soKq_lO'b,¥ª²n ]&|MqFm6st¥t¥oKq l*b§l6stm6_baqm _|Mq_l£¥ª stqm bay6r|M¥t¥b [µ − 3σ, µ + 3σ] ¨'d9bam699% m b y bl6m6y6snam6soKqU~ bay6`cbam£'b q_b~&|Kl¦m baq'sty9noK`§~'m b!'bl oK_l*bay6K|Mm6soKq_l9noKy6y bl6~ oKq__|Mqm l |M=¡§uwba=¡¤'bfuwoKy ¢am l£_|Mq_l9¥b m bl m£'b FoK¥t`coKpoKy oF« iF`§sty6q_o oK_l*bay6K|Mm6soKq_l|k|MqFm'blK|M¥ba'y lb?¡=m6y ^a`cblbam³F_b¦q_oK_lnoKq_l6s'ay oKq_lam6y*|Mq'p^ay bl|M,n ]&|M`§~ p|M_l*l6sbaq ¨ 1.63 √ n. 1.63 √ n. 1.63 √ n. Ecart maximun Marge d’acceptation du test Fonction cumulative de frequence de l’echantillon. )+*,+. /¼\Vbl m'b F oK¥t`coKpoKy oº« iF`§sty6q_oº. fonction de repartition d’une loi normale. Z[Z]\^.
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(206) ʨ ÆqXª¿|M'm6y b,m bay6`cbr q_oK_ly bn*]_bay n ]_oKq_l"¥b§noK'~'¥b m ba¥ ³F_b§Ä (π∗, g∗) . . . . . . . c. 1. 1. K. K. i. i. (π∗, g∗) = arg min W (π, g) (π,g). !oK_l'm6st¥tsl oKq_lsnas'¥ª¿|M¥tpoKy6stm6]'`cb= F«`cb|Mq_l (_|Mq_l¥bfn|Klo+M,¥blq_ok|M=¡coKqFmnoK`§`cb `coS'bU'b¤y ba~'y l baqFm*|Mm6soKq ¥bUnbaqm6y b1Xª stq_bay6m6sb1'b¼¥;|na¥;|Kl l b C ¨© bUnay6stm ^ay b1Xª¿|K'³&|Mm6soKq lºª²nay6stm|M¥oKy l"Ä . . i. MZ. ')(. *'+.
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(235) =.
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