HAL Id: hal-01470956
https://hal.archives-ouvertes.fr/hal-01470956
Submitted on 18 Feb 2017
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0
International License
Boosting vs maximum de vraisemblance
Michel Arnaud, Jean-Stéphane Bailly, Christian Puech
To cite this version:
Michel Arnaud, Jean-Stéphane Bailly, Christian Puech. Boosting vs maximum de vraisemblance.
Revue Française de Photogrammétrie et de Télédétection, Société Française de Photogrammétrie et
de Télédétection, 2002, 12 (4), pp.399-420. �hal-01470956�
!"# $ % $& ' ( $ ') *# + ,# - , ,.- , # // / $ 0 1' ! # $ % 23 40 $&5( $ ') 6+*#6' 7839$ ) $ $ / : /& $ )$ ; / $ / < $ % 3 % ; 3 $ ' 7 $ ) < 8 % &/ : // / $ : ) ; $ = > 3 % $ 0$& ; $& / $ 0 ; $ $ $ 8 $ / &: ' ? : @ % ; 3 $ / : : ; / : $& : $ 3 / / ' ; : 3 % $ AB@ 8 9 % ; 3 $ : $ / : $& ; $& 0: 8 4 $ ' + ' & $ & C 3 $ ; ;
$ )$ & 0 $ C & & )$ ; & ' &
$ 3 C & & $ % C ) $ )$
C&$& ; $ ; > 3 ) % $ 0 ; D @ )
$; 3 C $ 0C & C & $ )$ C & &
&$ $ ' & @ % E & ; 3 & 3
C& & $; ) $&$ $ 4 ' & ;
& 3 ) % $ $ AB@ 3 4 $ 43 & %
E & & ; ) ; $; 3 $ 0C& & &
$ '
#? + 76+> $ )$ 03 0 $ 0 % ; 3 $ '
!" ! "# $ " %## % % " " " & " " ' ( " ) % * " +% ( " # ) ,-. ( ) % # ,/ 0 112. 3 ) " 45 " 4 ," " . % ) " "" " $ ( " # " 0 * " " "" " " # " , 6 . % " ,$ + . #" % # " " " " 43 4) 7 " * ( # " "" " 8 " 4 9 '0, 112. 43 4 " 4 ( " ( " " % " ( " " "" 3 ) " "! "" " % " ) ' % ! 6 : ;< 6 : + $ * 3 '0, . % = % %" 3 "" " " # * ' # ( " ( " ,> '0 . + " % 3 ? %" # 3 * " ) " 3 * #!$ % & #! ! I 3= > : " "" " $ ) * % " " " ) % : ) % "# @ % > ( ) " * ' 0 ' # A 6 <B=> C 2
3 ; * D % @ % ( " ) ## * * % , 3 3 0 4 > : ' , . % * ) ( " ? , . % * ) ( " ? ,D. % * ) % " % @ ( " " , . + " % ' ( # . > : E, 0 . F , 0 .G % , E, ' 0%. F , ' 0%. % . H I ,* * F * .∈ ? *! " * ! ? *! ? 0 ∈ J I E F @G " ? @ /) ' 4 4# # " ' ' *J→ ) , λ. → , λ. ∈ + , λ. % " ' , . , λ. # λ " " λ ) , λ. " #? , . K , λ.K # " LM L /) B' N " F B B 4 4# 4 # 4 " %4 4# = B α # -J I E F @G' - ' → J I E F @G → -, . I Eλ ∈ J I E F @G )
( )
= B λ α * G & ' +% > 3 ,9 ' 112. ) * ; "" " " " > 3 6 - ,< '0 111. " ( " " * = = $% ) " ; " " " = ( J : : $ $ 8 & D 3 @ > • > ) " > 3 6 - # % + 3 0 % # ) " " * * ,9 '0 11O. ") # # ,3 '0 1OC. " +% > 3 6 - " ' ) # # # *! " ! " $ % 'C ℘ I E I,* F * . ) *!P !G ℘ I E I,* F * . ) *!≥ !G + % # " # ' @ @ @ $ $ $ Κ Κ Κ Κ Κ λ Κ λ λ ℘ ℘
( )
℘ ∈ ℘ ∈ = λ λ λ ( ! & '#K D+$& '0 !!!@ !: ' = E, 0 . F , 0 .G % 7 ' ∀ I F ' ∈ 0 ∈ J I E F @G? = E, ' 0%. F , ' 0%.G ' ∀ I F ' '∈ 0%∈ J I E F @G? = B 4 ? = , λ. % λ = 8 I ∀ I F ' , λ. I @ , *. #!/" " I B ' , . N , λ. 4 4 7 % # "' ' *J→ ? , . $ ## α ∈ ? , . $ ' ∀ I F ∀λI F @ '( )
( )
{
( ) (
)
}
Q 0 * λ λ λ λ = −α + H' Q # ( ) R ?( )
= −+ ≠= λ λ λ 0 0 0 , . S " %" %" % # " =α )!: # # -, . " # = B α ' -, . I Eλ ∈J I E F @G )( )
= B λ α * G3 ; * T • > " " * > # ) * " # % " + * ## α " % # H " α I + ) % ( ,= R . α ( # Q ,< '0 . ? • ; 8 '#K ; * ' # + ( #* B % " + 3 " " , %## ## % . ,< 111. N > 3 " " " # ) % ) " 4 " 4, 111 . • $ ; $ 8 & ' > "" " * , 111 U 112. ! * " ,3 V '0 11O. " % " % $C T ,U 11D. ; " " " #!#! N * ( # % , . + $ 3 $ / %" ) " , . + ; ) * " %" ,D. : " %" , λ. " " = % # 0 # , F . ) * * "
( )
( )
( )
(
)
= @ @ @ $ $ Κ Μ Κ Μ Μ Κ Κ( )
@ = = =λ λ ,C. + ; 8 ) 3 = B α ) λI F @ = $ " % ) * % @ # = B α "2 8 " λ * 0 # # %5 ## "
( )
( )
( )
(
)
= = = = = B B B B B @ @ @ $ $ α α α α Κ Μ Κ Μ Μ Κ Κ ,T. + ) % ( ) * * % $ /$ ',2. +% $ ; 3 " #") % % B )! % " % * " ) 8 * % " * < <D # . %## %" , I C D2 * . . " ) D. " " ) C. * ' +%" " " 7"' " + " ##" "" " * )! ! " " * * ' * ) 0 * " ) % + * ( ( % " ) * " %## % * D + $& ; $& / : 2 " ) "" , " " # 0 # . " " # " , . " ) < <D I C D2 * " " + ) * I $ 3 $ / %" S O %" * OW TS 2 ,# . + ) " C D2 * " %" " O1 2 1 S % $ /$ # " , < <D. % 0 ( # ,# D. $ ) % # &3 ; * W " ) ) * 0 " : ) ) " ! " ( " * %" % ) " 3 ( " " 7 > ) I " " * * " ) , #" " O. ,44. ,# C. : * T ) " " I2 ; % " ) , * " " 44,# . ( 44. +% # # %## * T 44 ") * " " D ; $$& ; $& / : + ) * % " ) ,# T. " 3 " # ' * " %" T S ' T S T WS CO 2S I C D N # ) " " ) " % 3 ## " : ! " ) " ) * % " ) $ * 3 # * !I % 8 $ / D. L B@ > /$& 3 * #' ,; $ 3 $ / 8/$& 3 D. L B@ * )!, $ /$ = D. L B@ % 8/$& ! !
O )!#! * 3 * " ) ##" " ' % ( # % " ) = " D + $& ; $& / : $ " ) * * * < <D " , . " I C D2 * * " " ) 0 ) " # 2 = * %" I 8 %" I C D2 $ 3 $ / " OW TS 1D OS 1W S 8 I " " * % 44 44 ) * " " ,# W. : # * * ' T C2 + * T " " ,44. U * C2 " " 44) 44: 7 * C2 ## 3 ' * C2 " 44 44$ * * * " # ( % " ) ) " ) ,# 2. + $ 3 " " ) # , S. ) I C N ,DWS. ) ID2 # " ) * +!, 8/; 3 M8 9 : 8 "D. L B@ " " " ######################## ############ ############ ############ ########### ########### ########### ########### ! !$ %&# ' * ,!, ; $ ; % ; / : $& : $ 0 8/$& 3 D. L B@ ( ( ( ! !
3 ; * 1 % " " ) * " " # ) * % # D ; $$& ; $& / : > $ 3 $ / %" ) ) * % " " ) , # . +% 3 " ) " " ) + ) % 0 " ) " " ) $ 3 ) ) " ) N # " ) % " " ) * " " # ) ) * )!)! " , . " I C D2 * * " 7" " ) %0 $ " " " D2 X " 2 % 0 2 " " ,# O. * -!I % 8/$& 3 8 $ / &: D.L B@ ) ( ( ( ! ! * .!,; 3 9 : 8 "' , & $ NNA 3 $ NND.L B@ D %LO0 L @ " " " " " " ######################## ############ ############ ############ ########### ########### ####### #### ! $ %&# ' " " " " " " ######################## ############ ############ ############ ########### ########### #### #### ! !$ %&# '
D D ' + $& ; $& / : : ) 5 % 3 $ /, # . ) * H % # " ) ) " ) , # YD . + 5 " ) % YD + " " ) % ) I " C S W WS ) I C D2 N % " ) % " " ) * % " " # D D ' ; $$& ; $& / : + $ 3 $ / " " , # . ) % % # , # Y D . + ) % 0 " ) " " ) $ 3 CTS TC TS TT S ) I C D2 N % " ) % " " ) * " " # $ % " ) % 0 " " " " * "" " + " # * ) : ) " %" . % 3 " " " ) % % # , " * . # H " " ) ? . * " ) 3 H " ) ) " " $ ) * ? D. 3 % ! * * /! / % $& : $ $ 8 $ / D.L B@ ) ( ( ( ! !
3 ; * ) " ) ) % 0 ? ( #! /& 8/$& ,! + " ) ") * ) " " ) N 3 " # " ) $ M " " ) 3 ) % * + % " " " ) % # % " # , F . ; ) " # ) * : # ) λ 0 * ' * ) # λ * ) & λ # * ) ' λλλλ < " 3 % ( " " % ' %" ( " , ( " %" . ? * " " " Z # " # ) % " * ' %* ) ##" " ) B ; " " 3 > " " ) < " " ) < < " 6 * ,< " " ) . OW T OW T OW T 0,!/ C O1 2 0)!/ O1 2 0)!/ D2 1 0.!# 1D T 1T C T -0!+ 2 T C W C T W / !0 21 C CO 2 " D2 CO 2 /#!0 W2 1 CW
> 6 3 0 [< 3 \ 8 $ + 3 ' " # " ) / " ) C , . 3 V $ + 6 & $ [ 11O Z$; / 0 # ] ' ^^^ _ 6 +/ 0 ` ; $>' Z 0 # $ # #;# $ < 3 + 9 [ - : / > < $ [ 1OC $>/B $ # a ^ 3 V $ ,> V < #^ 6 0 $ # . 3 \ [ 3 V [ > < [ / 3 # # a V 6 $ # <0 $ [ B \ 111 6 ' 9 >;6 & O,C. 1W= D2 B \ 111 > * # # # ' 3 3 & 6 = 9 J < / N 112 N* ^ ^ 3 ; < + ,N . 8 # B ; $ # 6 + < 9 $> 6 @ # CO= T2 9 [ - B B / 11O > + / ' < ^ #3 B / - B B / 9 [ B N #< + 6 ;# 8 < = U [ / 11D $C T ' 8 # N + 6 @ # < 9 $> U [ / 112 3 3 $C T ' ; 8 #B $ # > # ; $ 6 > >>>;8 6 ;B 8 W T= WD / 0 3 112 8 / ^ V $ Z 0 8 < / N 111 B ^ # ; B ; $ # > + B 0 < / N < J 111 ; # = 6 + DW 1W=DD2 < / N < J 3 B * ' > = 0 # * & 6 + D1 DT= 2O