Least-Squares, Sentinels and Substractive Optimally Localized Average
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(4) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Least-Squares, Sentinels and Substractive Optimally Localized Average Guy Chavent. N ˚ 3332 January 1998 ` THEME 4b. ISSN 0249-6399. apport de recherche.
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(53) A K J Y g I Y} I g W F I U g P U [kb I Y W F PZ] I ] W P JUXW I§ g ] W I RkU^S P W `§PV[ n s aPV] [TY W F PV[ u 7 RkW W F I bY} U g P U [kb I #7 I P bu YQF [ WaI g _ Y [TY _ g J 7Q` A F Y W F I Y] I [g W PV[ IeS g PcI b W Y PcYQg [H F PZb F U ] I F U I ] II [pPZ[\ g YC\ YC]aP W PcYQ[ K 6 b U^J I I ] F UXW SZS ] SZIeScI I PZ[ W F W I [IIeScd?I W Ie\ S U g UXu g U \IuCF RTW SVF U U^W=UXW F W IU 7 Y} I0JU b F PV[ I g `PV] I g `§b S YQ] I U [l_'PV[ J YC] W b U ] I ] beY^PV[kbPV_ I ] P W F W F UXW PZ[ W g YT_ R b I _ Y g W F I ] IU g b F Y S YTb UXS Pc I _ U I g U^u PV[ u/XI g [ IeS ] 7Q#` 2 U b QR ] U [l_ + P S 7 I g WU [k_ W F I P g Y ScS Y I g ] 8. ./"010 0%5'5$e E0F PZ] U \l\ g Y U b F U ]yPV[ W g YT_ R b I _-7Q`*2 U b ?R ] U [k_ + P S 7 I g W Y g W F I I ] W P JU^W PZYC[Y]\ UXW P U^ScS `_TPV] W g P 7 R Wa\ I g _Y 7 \ U 7Tg P U^PZJ] IPVW bI bg ] ] PZ[ Y W F I ] \ b g IPZYC] I[ [k6b I Y2 [Tb Y^PV] ] I PV[ [k_ W F + I P Y&7 7k] g I g ]LUXPVW _ PZYC[l]e=NQ] Y Y I U [kg _ I PZ[ PZW [ F PZ]0g ] I bb Y W PZYC7T[/PV[ PV[ W PcYQF [ I U S W U I I W U ?R U S I W IU #U W U/S IU J UXW A Y W F I _ UXWaU F I g I A "! PV] U I b W Y g Y$PZ[Q I g ]aPZYC[bY I /bP I [ W ] W Y 7 I _ IW I g J PV[ I _H] R b F W F Pc UXW [ A ]a \ u P PZ I ]Y?b U ] PcWaYQUX[HW PZ§] W PV[ b U^2 ScS ` b ]aY R ] [k_ [kI _ ] W + P JP UX7 W PZg YC[Y g=] Y PZ[TJ'Y IY S YT`Cb U^S PZ I _ U $ I g PZUX]uCb I8 _ =[ U g Y g R [l_ PV[ U u g I [ U^W UXSS UXW U ?R U S I WWaI J S u 8 UXSZScI U U I U^u u ^N?IYl IePcS [ ]\ P YTb PZYC[ g Y [l_ F PVb F YQ[ [ Y ] P u I U UXW U^S0S UXW U R I#U WW I W JUXWaIW F I!R [ [TY [\ U g U^JIW I g W F I PZ[Q I g ] PZYC[bY I /beP I [ W I b W Y g A $ U [k_ W F I§U I g U^u PV[ u^I g [ IeS 8 !F U IW Y 7 I b F YC] I [ U^S YQ[ uW F I Y ScS Y PV[ u'S PV[ I ] A PZ] U [ I ] W P JU^W I Y 8 =; P I K 6 7 9 A ;
(54) 8 = Y} 9 A ; PV] U ] JU^ScSHI [TY RTu F ]aY W F U^W K oC A PV][TY W _TY J PV[ UXWaI _ 7Q` [TY^PV] I W F g YIU [k _ I g U^u PVPZ[ u'[^I ]g \ [ IeS P 8 PV] YTSb YTb UXPZYCS Pc[ I _ U P W FR SZI ]U]\ u YC]I ] P 7 SZIUX IW ]U^\ SHI SbP UXU^SZS W ` [ IeuQUXW PZ I ] PV_ I K nQ S &Y 7 I ] U [k_ P W F UW Y WUXS JU ]] !? 8 ? b S YC] I W YYC[ DI YQ[k_TP PZYC[k] oC [k_ nQ g YC[ ` P Pc ]aY b [ \ b Y g F [ YQ[ `Y beYC[k] W g W R b W PV[ u KS Y?b UXS U Pc I _ U K I g UXU u PVI[ u0XI S g [ IeS Rk]eUX S 2 WRTUXWW UXSZI S ] F Y RT#S _ I F U U I PVI[d beY I JW J J YC[ W I F UXW WW U F I ` ] UXIpW PZ] U ` W F I 898;:=</>>>?.
(55) o 6. , '. RkU [ W P WaUXW Pc I beYC[k_TP W PZYC[ K 6 ) g Y J W F I _ I [kP W PcYQ[ nk 6 Y I ] IeIW F UXW7 9 A ; ]aY W F U^W K 6 g I g P W I ] A = 8 =; F PZb F U ] I _kY[kY W0 [TY lb U [ 7 I ] UXW PZ] I _ PV[ UXSZS PZ[l] WU [kb I ]YC[ S `/P . A. . =. A. K^K E0F I bYQ[k_TP W PZYC[ PV]§[ I b I ]] U g ` U [k_] R /bP I [ WW Y I [l] R g I/W F UXW/ K 6 F Y S _k] U k[ _ #I ] F XU SZS ] R \k\lYQ] I g Y J [TY YQ[ W F UXW P W PZ]] U^W PV] I _H E0F I [ Y} 9 A ; 7 9C A W A ; A 8W A ;. 7 9C A ; ; A K > . A 7 9 ; A ; Y} 9 A A , I [kb I ] UXW PV] `QPV[ u K oQ U^J Y R [ W ] W Y bYC[ W g Y S W F I bY} U g P U [kb I#I P u F WaI _[TY g J A Y W F I PZ[Q I g ]aPZYC[bY I /bP I [ W I b W Y g A 2 I Y g I#I PZ[l_TPZb UXWaI F Y W Y ] U^W PV:] `C 7lY W F K oC U [k_ K n?[kY W PVb IW F UXWW F I `bY gag I ]\lYQ[k_ W Y U _? I g ] IS `bY J \ IW PV[ u Y 7?h I b W PZ I ]e W F I P g \ F `?]aPVb UXS PV[ WR P W PZYC[ W Y S _ 2 U b ?R ] U [l_ + P S 7 I g WW F UXW U F P u F [kY^PV] I SZI IS YC[ W F II ] W P JU^W Y g Y 8 = #U ] W F I \ g PZb IW Y\ U ` Y g 8 W Y U b F P I IU F P u F ]a\ UXW P UXS g I ]aY SVRTW PcYQ[HNQYbYQ[k_TP W PZYC[k] K oQ U [l_ K n? Id \ g I ]] W F UXW YC[ I F U ] W Y'b F YQ] IU beY J \ g Y J PV] I 7 IeW#IeI [ [kY^PV] I SZI IS U [k_]\ UXW P U^S g I ] Y SZRTW PcYQ[ Y W F II ] W P JUXW Y g j UXW F I g W F U [_ I ]b g P 7TPZ[ uW F I Y g P u PV[ UXS ]aY SVRTW PZYC[ W Y K 6 K oQ K n? \ g YQ\lYQ] I _ 7Q` 2 U b ?R ] U [k_ + P S 7 I g W #I ] F UXSZS \ g I ] I [ W'U U g P U [ W PZ[ W g YT_ R b I _ 7Q` LS _ I [ 7 I g u U [k_ uCI [ I g U^S PZ I _ 7Q` f P h \ I g ] U [l_ E0F Y \k]aYC[H F N 7l] g b PZ \ P ` YTb PZ _ g N \l\ g Y b F F PVb F PZ]L[TY PV[ J PZ_ I R ] I W I R W U W I' W JU^ScS UXS I I UXuCI U U [ W F PV] U \k\ g Y U b F lYQ[ I b F YC] I ]0 g ] WU WaU g uCIeWU I g UXu PV[ u/^I g [ IeS 8 S YTb U^S PZ I _ UXWLW F I ]\ UXW P U^SS YTb U WE0PcYQF [ W]aY\ 7 I P PV[Qg I ]a] Y W P uQUXPcW YQI [ _H Y P W F PV]0 g !? ] 8 ? ]] 6 _H U 7 [l_H R P [ SZI P]] YQ7 [ I PZ] U§g J_ U h ] YT] b F PV_ ] W 7Q ` P W F g [TP Y] PV[k_ _ IS Y 7gg I Y ]eg \ g YTI b I ]a] U^WU ] UXS I I ] F SVUXRTSZW S ] II 7 I8 S Y E0W0F uQI RT[ I W F I I PV[C RkI W g ] PZWYC[ bSZSY I /IbeSZU^P I W [ I W U I b WR Y gWaI A PV]b UF YCW ] I UX[pSUPZ[ ] R I b F U [kU _ ` W F UXW ] Y A [TPZY ][TY WY W Y?Y SZU g uCI PV]F PV[Tb Y F 97 YQI Y b U^R ] I Y K >PZb 7 P ScS b I [k]] R g Y I W F UXW K PoQPV][k] ] UXW PV] I _ U n?=W PVF]U^] W PZ] J'I _ g J 8* W A W W SVU g u^I F F I UCR I K^K ScS I R g IW F UXW K UXW I ¢ T 2 0z 2 ¦ T '
(56) 5& '/5 5& 8 # !=_ $ % '+() *+ , ./% 0 8. . 1 ! 34' 5 9( 93 .. . ) ) ./&%$ ! " # ' . %. % ' % .. "! '/5 5&" . . . 8 # "!. #U . * +_+-) .! ' %! - .
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(63) b b \ A 7 P W F U ] S P W WaScIU Y&]$7 \lYQ]aPV]a[ P 7 _ ScI ]aPZ_ I[S Y&7 I ] U]aP \ W beF Y \ ? 8 PcYQ[S Y ILA I Rku PcF W ] 6 [ U ] P U A Y W F U =;I WaU P ScF I ] A [k_ gU _g I _ ?WP U PZYCI [ A E0F I W [lF _ I J ScI J RTWUXW U §u I U I W JUXW I 8 W WU U I U^W W F I ] F U \ I Y 8 ?PV] RkU^S PZ I ] W F I ]\ U^W P UXS g I ]aY SVRTW PZYC[ U b F P I I _ UXWW F II gag Y g SZI IeS A PZ] g g 7 F F Y F F N \k\ g Y b F ]=_ YQ\k\ _bY \ `PZ[k_ \ [? _ U [ W W S ` Y IWJF UI j U N ScI0U W \k\ UXg W Y U U^b ScFW U [lRT_ u W F W I I ] I [ W PZ[ IeSU U \k\ g Y U U b F °UU [k_ I J'IR S b F IIU g S P I g J W F UScI[ W IeW S F PZ] SVUXI W WaI I g W F I g IU g I I g `/] W g YC[ uW P I ] 7 IeW#IeI [ W F I ] IU \l\ g Y U b F I ]e { ¦ 1 ¦ 4x 2 4z 2 ¦ T $ _ . . . \
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(70) _ 9( 3 _$,++\ ' % . A " )" $ *+ . )\ . -$ 2 *(=3 ; $) )" . % #U %. $ # " . #=' , ! "! * #U " % ! $* , ,# $ . . . . ! ' 5 . . _ ( 93 * . A. . . #. %. "% ! = \ 5 '5 . 8. . \ . . .. . . E0F PZ] g I ] RTSZW ]- g Y J W F I PZ[ WaI g \ g IWaU^W PZYC[Y W F I N y U \k\ g Y U b F PZ[ WaI g J Y] I [ W PV[ IeS ]e LY W PVb I W F UXWW F I A g IeuQRTSZU ] g PZ_ U^PVW [ PZYC[ F W I N g J $ N \k+ \ g Y R b] FI _pbY PZ[ gag W F ]\lI YQj [k_ N YU \kPZ[Q\ g Y g U ] b F U [l_ g PVb W F ] I^I g [ IeS[kJ_#PZ'] WW I g J 8 W& + R I W I U U I W I IJUXW I % U X"0 2 b ] [k_ ) + P 7 g g PVb \k\ PZb PZYC[l]=Y Y g PV] Y g YC\ F `T]aPZb UXS PV[Q I g ] I 36 5 + \ g Y&7 ScU IJ ?R] U .) , S I W Rk 9J I ./% U^6SnkU o K S 1
(71) UXo W 1/ 6s0/1 U JU^S J uCI _ ) + P S 7 I g W E0F I g I ]aY S QPV[ u \lY #I g Y u g YC]] IU g W F _ UXWaU .)F , 3 o 5 + 2 U b ?R ] U [l9 ./% 6 / 67/^s o K J 6s/Ct 3 n 5 + F U I [ W + I [ I g UXS Pc I _'] I [ W PZ[ IeS ]$_ I [ I _'QP ULScIU ] W ] RkU g I ] 3 )&) \ =(9)" * Cn 6 6tCs o 6t 6s^s >. K F YT_Tq Y U Y b&Y `^7l ] YQbeY[ U \ [k_Pc YQ[ Y?[kYC]_ IeuCU \kg \ _ PZ b [QPZYCI[lg ]e] 0I [JIf W HF NQYTP _k7l]Y g [TPV[ P° W I g _T_TP PV]aY beg PZ \ 9S PV[%U g ` IeSZ IJ. I [ W ]' Y J_I W % "%S u l^Y J RT/ WUXnlW N?\ g UPZ[ g U 6s^S s0/ UX W I W SVRkJI uCI 3 > 5 q PcYQ[k]e =. . $ % ) $ .
(72) &$ 0 . ! , * ! $ CY SVRkJI 6 IW oT pU ]]aYC[H 6s^t^t 3 5 2 ,. 898;:=</>>>?.
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(74) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) .
(75) ISSN 0249-6399.
(76)
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