Advanced spectral unmixing and classification methods for hyperspectral remote sensing data
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

!!!!!!!! <C!<!" DE-(10(0 B%$5.%'%/,

42

' #$%F'(*

!!!!!!!!!!!!!!!!!!!! <3

!<

!<

DE-(10(0 7--$,&;-( #$%F'(*!!!!!!!!!!!!!!!!!!!!!!! <3 :&#)1(;*&#;!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9<! "#$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0! "#$%&'()$*&#! +, "#$%&'()$*&#!"#$%"#&'$() *+(,#% ($# &-+"-%#. -/

012.$#.% -/ 3(2.% 4*'0 ( 5#$! 0*,0 %"#&'$() $#%-)16 '*-27 /$-+ '0# 5*%*3)# ''0# *2/$(6$#. $#,*-28 90# 4*.# %"#&'$() $(2,#7 &-1")#. 4*'0 (2 ()4(!% *2&$#(%*2, %"('*() $#%-)1'*-27 ())-4% '3#''#$ &0($(&'#$*:# +('#$*()% (2.,*5#% '0# (3*)*'! '"*2"-*2',$-12. -3;#&'% )(!*2, -2 '0# -3%#$5#. %1$/(&#

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



'

.#'#&' 5#$! %13')#.*J#$#2&#% 3#'4##2 '0# +('#$*()% -/ '0# -3%#$5#. %&#2#8 -4#5#$7 '$(.*'*-2() '#&02*B1#% %##+% '3# 2-' #J#&'*5# /-$ '0# (2()!%*% -/ %1&0.('(7.1# ''0# (3-5# +#2'*-2#. *%%1#%8 K*+#2%*-2()*'! $#.1&'*-2 *2 ( 0*,06.

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



*/

4# &-2%*.#$

+('#$*()

%7

'S(%"0()'SU7 *'

4*

))

3# 5#$!

)*<#)

! '0

(' "*E#)

% 3

#)

-2,*2, ''0*%

&)(

%% %0-4

(

0*,0 5()1# -/ '0# +-+#2' -/ *2#$'*(7.1# ''0# #)-2,('#. %0("# -/ '0#%# -3;#&'%8 C2 '0*% &0("'#$7 4# 4*)) *2'$-6.1&# '0

# +(*2)! "$-3)#+('*&% $#)('#. '%"#&'$() /#('1$# $#.1&'*-2 TV#&'*-2?8OU (2. %"('*() /#('1$# #E'$(&'*-2 TV#&'*-2?8WU8 V-+# X-2&)1%*-2% ($#.$(42 (' '0# #2. -/ '0# &0("'#$8! "#$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0!" #$%&'()* +%)',(% -%.,&'/01!"# $%&'( )*! #+$',-$'".,!( %/!0' %! 1'".), 2#%"# ")3.!

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

.''-9).*-52 7#475'&'!?2 *%-',54N8,&,-//.2./:P&?21&7&21&2* O50752&2* J2./:'-'8 1#& *5 -*'.**4.)*-@& 7457&4*-&' &L7/.-2&1 -2 Q&)*-52!>!C!!"!" #$%&'%()* +,-(,&.&/ 0&)*12%2 M4-2)-7./ O50752&2* J2./:'-' HMOJI -'. )/.''-) *&)%2-A#& -2 '*.*-'*-)./ 1.*..2./:'-'! MOJ 1&E)544&/.*&' *%& @.4-.3/&' x1,..., xn 56. $-@&2 4.2150 @&)*54 x ∈ Rn!?2 541&4 *5 7&46540 *%& 1-0&2'-52./-*: 4&1#)*-528 52/:. Σx -'. 4&./E@./#&1 ':00&*4-) 0.*4-L 56 92-*& 1-0&2'-52! T: *%& '7&)*4./ *%&54&

08

Σx ).

2 3& 1-.$52./-

P

&1 3:.2 54*%5$52./ 0.*4-L

M(Mt



= M−1

) :

M−1 Σx M



= Σy H

6450

H

! II!

T

: -

1&2*

-9).*-528 P -'.2 54*%52540./ 0.*4-L,%-)% -' 65#21 3: '5/@-2$ *%& &-$&2@./#& 7453/&0,-*% #2-*.4: 2540 )521-*-52 52

*%& &-$&2@&)*54' H@I+! "#$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0 λv = Σx v ||v||2 = 1. %& &'()* +'& &,-& P!"#$.+)*/*&* +0 &,1 *1& +0 -22 1/31)41.&+(* v +0 Σx 5 6/&, +)1 1/31)41.&+( 71(.+2'8)" )*3*3*) (456789: ;<4 58=49>8?9@A8;B 6>89: %"$ y5 /"1"5 4-( yi $< λi " 9,1= λ1 >... > λn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x '*/)3 +)2= &,1 k F(*& 7(/)./7-2.+87+)1)&* /* M SE = n X λi.!"C$ i=k+1 /*.,+*1) /) +(;1( &+ 8-G1 &,1 HIJ 0-22 :12+6 3/41) &,(1*,+2; tpca 5 '*'-22= KL +(!ML +0 &,1 &+&-2 4-(/-).1@ 9,1(10+(15 k Pn i=k+1 λi P ≤ tpca. n i=1 λi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k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

00

&=+ -+/

+$

-

+$

%+

.

!) &=+ *+&*%+>+!"/

!

$+$

&.(

#

.%$2 =%2=+*?&=,$.+!$-@!*-+*.&,&%.&%.8 9$ &=+ )!00!A%$2( &=+ 9:; "!-+0 %. /*+.+$&+%$&! "!*+ -+&,%08!"!2! 34% 567 80.%* 9:;!$.%.&.!) B$-%$2, 0%$+,* -+

!"/!.%&%!$!)!C.+*>+-,&, %$&!.&,&%.&%,00' %$-+/+$-+$&!"/!$+$&.8 D%>+$,$!C.+*>,&%!$ "!-+0 x = As,?E8F@ A=+*+ x %. &=+ >+ &!*!) &=+!C.+*>+.%2$,0.( 7,.,0,* ",&*%1!) &=+ "%1%$2!+G %+$&. +N#%>,0+$& &!.,'%$2 &=,& &=+ $#"C+*!) %$-+/+$-+$&!"/!$+$&. %. +N#,0 &! &=+ $#"C+*!)!C.+*>+"%1&#*+8 9& %. -!$+ %$!*-+* &!.%"/0%)' >+*' "# = +.&%",&%!$( C#& %&,$.!"+&%"+. C+ *+0,1+-8 O$-+* &=+.+ &=*++,..#"/&%!$.?!*,& 0+,.& &=+ B*.& &A!@( &=+ %$-+/+$-+$&!"/!$+$&.,$&=+ "%1%$2 ",&*%1,$ C+ +.&%",&+-( #$-+*.!"+ %$-+&+*"%$, %+. &=,& A%00 $+ +..,*%0' =!0-8 9$ ), &( )!*?E8F@ %) C!&= A,$s,*+ #$J$!A$(,& 0+,.& &A!,"C%2#%&%+.,$$!& C+,>!%-+-8 P%*.&( &=+ >,*%,$ +.!) %$-+/+$-+$&!"/!$+$&.,$$!& C+!"/#&+-8 9$ ), &(,$'.,0,* "#0&%/0%+* %$!$+!) &=+.!#* +.!#0,0A,'. C+,$ +00+C' -%>%-%$2 &=+!**+./!$-%$2!0M #"$!) &=+ "%1%$2 ",&*%18 Q#+ &! &=%.( &=+ +$+*2'!) &=+!"/!$+$&.,$ C+,& B*.& B1+C' A=%&+$%$2 %$!*-+* &! ",J+,00 >,*%,$ +. +N#,0 &! #$%&'(,$!$.+N#+$&0' &=+ "%1%$2 ",&*%1 %.,-,/&+-8 R+!$-( C+,#.+!).%"%0,* *+,.!$.( &=+ %$-+/+$-+$&!"/!$+$&.,$$!& C+ *,$J+-( C+,#.+,$' =,$2+ %$ &=+%*!*-+* A%00 $!& =,$2+ &=+ /!..%C%0%&' &! +.&%",&+ &=+ "!-+08 R%$ + &=+ "+&=!%. A+00 J$!A$ %$ &=+.%2$,0 /*! +..%$2!""#$%&'( A+ A%00 $!& 2%>+,00 &=+ ",&=+",&%,0 -+&,%0.!) &=+ 9:; %"/0+"+$&,&%!$8 ; -+&,%0++1/0,$,&%!$!) &=+ C,.%. !) 9:; A!#0C+!#&!) &=+.!/+!) &=%..+ &%!$8 S!A+>+*( %$ &=+ $+1&.#C.+ &%!$( A+ C*%+T' *+>%+A.+>+*,0 /!..%C%0%&%+. A=% = A+*+ /*!/!.+%$ &=+ 0,.& '+,*.,., /!..%C%0%&' &!!"/#&+ %$-+/+$-+$ +8 U!*+!>+*( A+ 2%>+.!"+ -+&,%0.!) 9:; V,-+( A=% = %. &=+,02!*%&="!$.%-+*+%$!#* +1/+*%"+$&.8 W+ *+)+* &=+ *+,-+* %$&+*+.&+%$,!"/0+&+ +1/0,$,&%!$!) &=+ 2+$+*,0 )*,"+A!*J!) 9:; &! *+)+*+$ +. 36E( 6F( 6H78! )*!* *! "#$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0.34565345375 859:;<5:! "!!#$#%& '( "#! )$*%+ #,% #,%-.%#"/$0 /-)/%1# -) 2,"/, 34.%!#! "! ")&%1%)&%)/%5 3)&%1%)&%)/% "! $ *6/,!#.-)7%. $!!6*1#"-) #,$) 6)/-..%0$#%&)%!!5 4-)#.$.( #/-**-) &%/-..%0$#"-) *%#,-&!+!6/, $! 8.")/"1$0 4-*1-)

%)# )$0(!"! 984 : $)& ;$/#-.

)

$0

(!"!



9; :+ 2,"/, 6!% ")<-.*$#"-) 1.-="&%& '( $ /-=$."$)/% *$#."> ") -.&%. #.%#."%=% 6)/-..%0$#%& /-*1-)%)#!+ 34 /-)!"&%.!,"7,%. 9#,$)!%/-)&: -.&%.!#$#"!#"/!

5

?-2%=%.+!#$.#")7 <.-* #,% 1.-'$'"0"!#"/ &%@)"#"-) -< ")&%1%)&%)/%+!%=%.$0 1.$/#"/$0 ")&%1%)&%)/% /."#%."$ /$) '% &%@)%&5 3) $&&"#"-) ##,% '$!"/ /-)/%1# -< /-)#.$!# <6)/#"-)!+ #2-< #,% *-!# /0$!!"/$0 /."#%."$ $.% '$!%& -) )-)0")%$. &%/-..%0$#"-) $)& *$>"*6* )-)AB$6!!"$)"#(C • D-)0")%$. &%/-..%0$#"-)5 ;")& #,% *$#."> W!#,$# #,% #.$)!<-.*%& /-*1-)%)#! g(yi ) $)& h(yi ) $.% 6)/-..%0$#%&+ 2,%.% g $)& h $.%!-*

%

!6"

#$

'0% )-)0")

%$.

<6)/#"-)

!5 8-!!"'0%

)

-)A0")%$.

<6)

/#"-)!

/

$) 

'% &

%

."=%

&

#,.-67, #,% *$>"*6* 0"E%0",--& $11.-$/,+ -. '$!%& -) #,% *6#6$0 ")<-.*$#"-)5 • F$>"*6* )-)AB$6!!"$)"#(5 ;")& #,% 0-/$0!"#! -< )-)AB$6!!"$)"#( -< $ 0")%$. /-*A '")$#"-) 6)&%. #,% /-)!#.$")#! #,$# #,% =$."$)/% -< y "! /-)!#$)# $)& #,$# "#! /-*1-)%)#! $.% )-# /-..%0$#%& 9"5%5+ $<#%. 1.%2,"#%)")7:5 G$/, 0-/$0 *$>"*6* 7"=%! -)% ")&%1%)&%)# /-*1-)%)#5 40$!!"/$0 $07-."#,*!+!6/, $! ;$!#34 HIJK+ 3)<-*$> HILK,$=% '%%) &%=%0-1%& 6!")7 #,% $'-=% /."#%."$5 )-#,%. $11.-$/, <-. #,% %!#"*$#"-) -< #,% 3)&%1%)&%)# 4-*1-)%)#! "! M NG HIOK+ 2,"/, *$E%! 6!% -< P#, -.&%. /6*60$)# #%)!-.!5 3) #,% <-00-2")7 %>1%."*%)#!+ 2%,$=% 6!%& M NG $! 34 $07-."#,* #%)<-./% ")&%1%)&%)/%+ &6% ##,% %Q%/#"=%)%!!!,-2) 2,%) &%$0")

7 2"#,,(1%.!1%/#.$0.%*-#%!%)!")7 &$#$ HPR+ PSK+ $)&!")/% "#,$! 1.-="&%& ") $ 1.%0"*")$.( #%!# '%##%..%!60#! #,$) ;$!#34 $)& 3)<-*$>5 46*60$)# #%)!-.! /$) '% /-)!"&%.%& $! #,% 7%)%.$0"T$#"-) -< #,% /-=$."$)/% *$#."> $# -.A &%.!,"7,%. #,$) #,%!%/-)& -.&%.5 3< 2% /-)!"&%. $.$)&-* =%/#-. x 2"#, $ 1.-'$'"0"#( &%)!"#( <6)/#"-) 91&<: p(x)+ "#! /,$.$/#%."!#"/ <6)/#"-) "! &%@)%& $! #,% ")=%.!% ;-6."%. ω=0 9S5O: 3# /$) '%!,-2) #,$# #,%!%/-)&+ $)& #,".& -.&%. /6*60$)#! <-. $ T%.*%$).$)&-* =%/#-. $.% HPIK /6*(xi, xj ) = G{xi xj } 9S5SR: /6*(xi, xj, xk ) = G{xi xj xk } 9S5SS:!" #$%&'()* +%)',(% -%.,&'/01 22! "!#!" $%! "!&'!" ()$!"!*$!' () & +,"! '!$&(-!'!./-&)&$(,),# $%! 01+1-&)$* &)' $%!(" /",/!"$(!*2 $, 34424567 8%! #,1"$%,"'!" 01+1-&)$* 0,)$&() &-$%! ()#,"+&$(,) &9,1$ $%! #,1"$%,"'!" *$&$(*$(0*,# $%! '&$&7 :) $%! 0&*! $%! '&$& &"! ()'!/!)'!)$2 &-$%! 01+1-&)$* ;($% $;,," +,"! '(<!"!)$ ()'(0!* &"!!=1&$, >!",7 8%! 01+1-&)$ $!)*," (* & -()!&",/!"&$," '!?)!' 9@ $%! #,1"$%,"'!" 01+1-&)$* 01+Ax, x, x, x B2 () &) &)&-,C1! ;&@ $, $%! 0&*!,# & 0,D&"(&)0! +&$"(.2 ;%(0% '!?)!* & -()!&",/!"&$,"7 :) $%(* 0&*! ;! %&D! & -()!&" $"&)*#,"+&$(,) () $%! */&0!,# n × n +&$"(0!*2 ()*$!&',#

$%! */

&

0!,# 

nE'(+!)*(,)&D!0$,"*

7



8%

! */&0!,# *10% +&$"(0!* (* & -()!&" */&0!,# '(+!)*(,) n × n2 *, ($ (* *(+/-! $, '!?)! $%! -()!&" $"&)*#,"+&$(,)7 8%!!-!+!)$*,# $%(* $"&)*#,"+&$(,) 0&) 9! '!?)!' &*F X F (M) = m 01+(x, x, x, x ) AG7GHB ;%!"! m &"! $%!!-!+!)$* () $%! +&$"(. M $%&$ (* $"&)*#,"+!'7 I,()$ J//",.(+&$! K(&C,)&-(>&$(,),# L(C!)+&$"(0!* AIJKLB "!#!"* $,,)!,# $%! /"()0(E /-!*,# *,-D()C $%! /",9-!+,#!=1&!(C!)D&-1!*,# $%! 01+1-&)$ $!)*,"7 J* &)@ *@++!$"(0 -()!&",/!"&$,"2 $%! 01+1-&)$ $!)*," %&* &)!(C!)D&-1! '!0,+/,*($(,)7 J)!(C!)+&$"(.,# $%! $!)*," (*2 9@ '!?)($(,)2 & +&$"(. M *10% $%&$ F(M) = λM, AG7GMB ;%!"! λ (* & *0&-&"!(C!)D&-1!7 N!$ 1* 0,)*('!" '&$& ;%(0% #,--,; $%! :OJ +,'!-2 ;($% ;%($!)!' '&$& z = VAs = H s AG7G4B ;%!"! H '!),$!* $%! ;%($!)!' +(.()C +&$"(.7 8%!!(C!)D&-1! '!0,+/,*($(,) &--,;*,)! $, /,()$,1$ *,+! ()$!"!*$()C #!&$1"!*,# $%! 01+1-&)$ $!)*,",# z7 LD!"@ +&$"(.,# $%! #,"+ M=h h AG7G5B (* &)!(C!)+&$"(.7 8%! D!0$," h "!/"!*!)$* %!"!,)!,# $%! ",;*,# H2 &)' $%1*,)!,# $%! 0,-1+)*,# $%! ;%($!)!' +(.()C +&$"(. H

7 K1! $, $%! ()'!/!)'!)0!,# $%! *,1"0!*2 $%!

0,""!*/,)'()C!(C!)D&-1!* &"! C(D!) 9@ $%! P1"$,*(*,# $%! ()'!/!)'!)$ 0,+/,)!)$*2 &)' &-$%!,$%!"!(C!)D&-1!* &"! >!",7 Q@ '!$!"+()()C $%!*!!(C!)D&-1!*2 ;! 0&),9$&() $%! ()'!/!)'!)$ *,1"0!* ;! &"! -,,P()C #,"7! "!#!" $%! "!&'!" ()$!"!*$!' () & +,"! '!$&(-!'!./-&)&$(,),# $%! $%!,"!$(0&#"&+!;,"P,# IJKL $, 3MR67 i ij i kl j k j k l kl kl T T T m m m T!"!2!2 3$$*4/15 673 '0 84$%(9$%&'()* /:)5%9 %!) &//-@()C :OJ $, %@/!"*/!0$"&(+&C!*2 $;, "!/"!*!)$&$(,)* 0&) 9! 0,)*('!"!'F */!0E $"&&)' */&$(&+(.$1"! +,'!-*7 S,1C%-@ */!&P()C2 $%! */!0$"&+(.$1"! +,'!0,""!*/,)'* $, "!$"(!D! $%! ()'!/!)'!)$ */!0$"&0,+/,)!)$*,# $%! (+&C!2 ;%(-! $%! */&$(&+(.$1"! +,'!-* /",D('! & '!*0"(/$(,),# $%! +&$!"(&&91)'&)0!* ;%(0% &"! ('!&--@ ()'!/!)'!)$7 :# ;! 0,)*('!" & %@/!"*/!0$"&'&$& 019! ;($% N (+&C!*,# N T AN × N B /(.!-*,9$&()!' #",+ N #"!=1!)0@ 9&)'* &)' #," *(+/-(0($@2 &**1+! "&; D!0$,"(>!' (+&C!* :An, λ B2 ;($% 1 ≤ n = (i − 1)N + j ≤ N A;%!"! i &)' j &"! $%! ()($(&",; &)' 0,-1+) (+&C! ()'(0!*B (* $%! */&$(&()'!. &)' k2 k = 1,., N 2 (* $%! */!0$"&()'!. #," ;&D!-!)C$% λ 2 $%! $;, "!/"!*!)$&$(,)* 0&) 9! '!*0"(9!' &* #,--,;*F • U/!0$"&V(.$1"! V,'!-F L&0% /(.!,# */&$(&()'!. n C(D!* &),9*!"D!' */!0$"1+,# N #"!=1!)0@ *&+/-!*2 ;%(0% (* "!/"!*!)$!' 9@ $%! -()!&" &//",.(+&$(,)F X a φ (λ ), ∀n = 1,., N AG7GWB I (λ ) ≈ z f x y f k y z f k f Nc n (n,p) p k p=1 k z l ! "#$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0!" #" $" %" &" '" (" )" *+(,-&./.0 1-+2$+3!4 567362&28 92!4:;+; <;/ =2%&3&2%&28 567362&28 92!4:;+;0 *+-;8 '

6,3-+2

$

+3

!4 $67362&28;? @AB=B C2+<&-;+8: %!8! ;&8D $67362&28;/!"> %" &"> )" *+-;8 '6,+2%&3&2%&28!" #$%&'%( )*%&+,* -.&,%/&'01 "2!"#" φp (λk )$ %&# p = 1,., Nc $ '#" (!" )&*+(,(-"*( #".")('*)" +/")(#'$ '*0 (!" *-12"#,+ )!&+"* '))&#0,*3 (& (!" 0"+,#"0 '))-#')4 &% (!" '//#&5,1'(,&*6 7"*&(,*3 '+ I(λk ) (!" 8")(&#,9"0,1'3" :&% 0,1"*+,&* Nx × Ny ;$ (!" :Nz × Nc ; 1,5,*3 1'(#,5 < '*0 φ(λk ) = [φ1 (λk ),. , φN ]$ (!,+ +/")(#'= 1,5(-#" 1&0"=,+ (!"* "5/#"++"0 '+> I(λk ) ≈ A · φ(λk ) :?6?@; A#')(,)'==4$ (!,+ +/")(#'= 1&0"=,*("*0+ (& '//#&5,1'(" (!" +/")(#-1 &% "')! /,5"= '+'+-1 &% Nc )&1/&*"*( +/")(#' &% (!" '#"' )&##"+/&*0,*3 (& (!"+" /,5"= )&&#0,*'("+6 B% BC<,+ -+"0 %&# (!" "+(,1'(,&*$ (!"* (!" Nc 2'+,+ +/")(#' φp $ p = 1,., Nc $ +!&-=0 2" +('(,+(,)'==4,*0"/"*0"*(6 D&#"&8"#$ (!" pE(! )&=-1* &% (!" 1'(#,5 <,+ (!" -*%&=0"0,1'3" '++&),'("0 (& (!" 2'+,+ +/")(#-1 φp 6 <))&#0,*3 (& (!,+ 1&0"=$ "!'8" Nx × Ny F GH$HHH +"*+&#+ '*0'+1'== *-12"# &% +'1/="+ Nf F?@I$ %&# "+(,1'(,*3 (!" ='#3" 1'(#,5 A!,)!!'+ Nz × Nc F JGH$HHH /'#'1"("#+ :('K,*3 Nc F G;6 L/'(,'= D,5(-#" D&0"=> M!,+ 1&0"= '++-1"+ (!'( %&# "')! '8"="*3(! λk $ (!" 1"'+-#"0,1'3" Iλ (n),+'",3!("0 +-1 &% Nc 2'+,+,1'3"+$ 0"*&("0 IIp (n)$ p = 1,., Nc > Nc c • k Iλk (n) ≈ Nc X b(λk,p) IIp (n), p=1 :?6?N; ∀k = 1,., Nf B* 8")(&# *&('(,&*+$ 0"*&(,*3 (!" 1'(#,5 B '*0 II(n) = [II1 (n),., IIN (n)]T $ &*" )'* #,("> I(n) ≈ B · II(n) :?6?O; A#')(,)'==4$ (!,+ +/'(,'= 1&0"=,*("*0+ (& '//#&5,1'(" (!"!&=",1'3" '( "')! %#"P-"*)4 '+'+-1 &% Nc 2'+,+,1'3"+6 B% BC<,+ -+"0 %&# (!" "+(,1'(,&*$ (!"* (!" Nc 2'+,+,1'3"+ IIp $ p = 1,. , Nc $ +!&-=0 2" +('(,+(,)'==4,*0"/"*0"*(6 D&#"&8"#$ (!" k E(! )&=-1* &% (!" 1'(#,5 B,+ (!" +/")(#-1 '++&),'("0 (& (!" 2'+,+,1'3" IIk 6 <))&#0,*3 (& (!,+ 1&0"=$ "!'8" Nf F?@I +"*+&#+$ '*0'8"#4 ='#3" *-12"# &% +'1/="+ Nz F GH$HHH %&# "+(,1'(,*3 (!" 1'(#,5 B!,)!!'+ Nf F Nc Q OHH /'#'1"("#+ :('K,*3 Nc F G;6 B* &-# &#K$ ",== )&*+,0"# (!" BC< +/'(,'= 1&0"=$!,)! '==& +'2"(("# #"/#"+"*('(,&* &% (!" 0'(',(!&-(,*)-##,*3,* /#&2="1+ #"='("0 (& &8"#R((,*3 '*0 1'(#,5 2'0 )&*0,(,&*,*36 <* "5'1/=" &%,*0"/"*0"*( )&1/&*"*(+ "5(#')("0 %#&1 '!4/"#+/")(#'=,1'3" :)&1/'#"0,(! (!" )&##"+/&*0,*3 /#,*),/'= )&1/&*"*(+;,+ +!& *,* S,36?6?6 c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`3 '.05!/!/7 &(.$!./ )/-!*)"!./4.%&/!/7!" #$%&'%( )*%&+,* -.&,%/&'01 "2!"#$% &'() *+

,-.

/

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f,$% $%0.%43!=%/8 "!=%5

78) i γR (f ) = Rfδ [ǫi (f )], J&'(KL i ψR (f ) = Rfǫ [δ i (f )], J&'(&L C;%$% ǫi,56 δi,

$% 3;% %$10!15,56 6!/,3!15 C!3;,5 D* 12 0!M%!< $%0.%43!=%/8< Rfδ,56 Rfǫ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x) = {ψn (x,., f (x),. :$$0"5(',!0 (! 5%;'(!(0'/ +' <= $+' 0'>4 )+'5>% + $+>+" (*+,% -%.,./ 0'% 1+'56.!"!# $%&'()') *+,-.+/+0123/ 4,+5/'6?)%,%'%"+>(@+!(0' 0A 9<=!0 )4&%" &%$!"+> 5+!+ 7+ ;"! &"0&0 %5 (' BCDE.?)% 1+ ($ (5%+ 7+!0 "%5#$%!)

% 5

(*%' (0'+>(!

4

0A!)

% 5+

!

+ 14

# (', =F:/ +'5!0 +&&>4!)%!"+' G A0"*+!(0' 0' %+$) 0A!)% &"('$(&+> $0*&0'%'!!"+' A0"*+!(0' -(.%./ %(,%'3%$!0" "%!+('%5.?)

%

A%+!

#

"

% %

H!"

+$!%5 14!)% 0"5%"%5 ('$"%+ (',>4 +$$0"5(',!0!)% 3+>#% 0A!)% $0""%G &0'5%'! %(,%'3+>#% 6/ $+>>%5

&"('$(&+

> $0*

&

0'%'!

-=F

6/ +"% *%

+'('

,A#> A0" "%&"% %'!+!(0' ('$%!)%4 +$$0#'! *0! 0A!)% 3+"(+'$% 0A!)% 5+!+ ('!)% 0"(,('+> A%+!#"% &+$%.?)# / 0'>4!)% ;"! =F +"% $0' (5%"%5 A0"!)% +'+>4 ( / 7)(>%!)% 0!)%

" ,#(5%>('%/!)% '#*1%" 0A!)% $0' (5%"%5 =F +"% 5( $+"5%5. : +,%'%"+> )0#>5 $0'!+(' +10#! IIJ 0A!)%!0!+> 3+"(+'$% 0A!)% 5+!+. 8#1 %K#%'!>4/ 0' %+$) 0A!)% =F / + A#>> <= ( $0*&#!%5.?)# /!)% 9<= 0A $ =F $+' 1% A0"*+>(@%5 14L EAP = {AP (P C1 ), AP (P C2 ),..., AP (P Cc )}, 7)%"% c 5%;'%!)% =F +>>07(',!0 "%+$)!)%!)"% )0>5 0A IIJ 0A 3+"(+'$%. :' %H+*&>% 0A *0"&)0>0,($+> &"0;>% 1#(>! A"0* + (',>% (*+,% $0*&0'%'! ( :>!)0#,)!)% 9<= 7+ $>+ -C.MN6 #$$% )07' (' O(,. C.N. A#>>4 # %5 A

0"

('$>#

5

('

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

%

"' "%,+"5!)%"%A0"%!)

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

$+' 1% &#"%>4,%0*%!"($ -%.,. +"%+/ >%',!) 0A!)% &%"(*%!%"/ (*+,% *0*%'! / )+&% A+$!0" 6/!%H!#"+> -%.,. "+',%/!+'5+"5 5%3(+!(0'/ %'!"0&46/ %!$. :$!#+>>4/ +' +!!"(1#!% $+' 1% +'4 *%+ #"% $0*&#!+1>% 0'!)% "%,(0' 0A!)

% $0''%$!%5 $0*

&

0'%'!

&

"% %'

! ('!)

% (*+,%

. !" #$%&'%( )*%&+,* -.&,%/&'01 "2 f (x)!"

ψ1 5 × 5" ψ2 3 × 3" γ1 3 × 3" γ2 5 × 5" #" $" %" &" '" (" )" *" +*(,-&./01!" 2)& 3-*(*4!5 *6!(&/ #"7 $" 83-9)353(*$!5 *6!(& $53:*4(: 3#;!*4&%,:*4( -&:9&$;*<&5=! 0×0!4%! >×> :?,!-&!: :;-,$;,-*4( &5&6&4;/ %"7 &" 83-9)353(*$!5 *6!(& 39&4*4(: 3#;!*4&%,:*4( -&:9&$;*<&5=! 0×0!4%! >×> :?,!-&!: :;-,$;,-*4( &5&6&4;/ '"7 (" @;;-*#,;& ;)*44*4( 3#;!*4&%,:*4(!: $-*;&-*34 ;)&!-&! 3' ;)& 3#A&$; B*;)! <!5,& 3'.CC!4% DCC/ )"7 *" @;;-*#,;& ;)*$E&4*4( 3#;!*4&%,:*4(!: $-*;&-*34 ;)&!-&! 3' ;)& 3#A&$; B*;)! <!5,& 3'.CC!4% DCC/! "#

$%&'( )* +'$&,(' ('-,"&./0 1 '2&($"&./0!"#$%$&'#( )$ )*+,+-!.).$! $/

012'3 4+ 5"! 5$67&

)+ )*+

2' $!

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f ) = {φTn (f ),..., f,..., γnT (f )}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

%

+'

$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,.)/0&1+20( *3422+5*4%+'$ '6 7+87 (+90$2+'$43 (4%4 +$ %70 3+%0&4%)&0,#$(0/0$(0$% :'9/'$0$% ;+2*&+9+$4$% <$43=2+2!!!!!!!!!!,>!"!# $%&'( )*%((+,-%.+/0!!!!!!!!!!!!!!!!!!!!!!!!!!! 12!"! 34*.+5%6+%.' 7'0(+.& '(.+8%.+/0!!!!!!!!!!!!!!!!!!!!! 12!"!" 9:' %;;6/%-: /< =8%./!!!!!!!!!!!!!!!!!!!!!!!!! 1>?@/0&+90$%43 &02)3%2!!!!!!

!!!!!!!!!!!!!!!!!!!!,A!1!#?%.% ('.(!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1>!1! @0A4'0-' %07 -:/+-' /<.:' 048B'6 /< +07';'07'0. 1+%.&

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`J =B>D0'# $#.7)($-2#/ Y/(S++%' I#.5YSJ ().+$-' * =BPDJ (<# 3##, "$+5 "+%#/ -,'# *(& -,# )#($,-,. &+ **7,-'!0 "$+<-/-,. %7&&#%%47) $#%7)'% -, <($-+7% 8#)/%9 O_: -, "($'-&7)($ (<# 3##, 1-/#)! -,<#%'-.('#/ $#&#,')! -,'# $#*+'# %#,%-,. &+**7,-'! =@VA@]D9 `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a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

3

I

596 $

*'*A&

$#$2!+#0( 2!+)'. 1'#.0 (#020*$*+ $*$%, 0 -$!. '* +)! $&&%0($+0'* '1 $ >$,! 0$* (%$ &'*!*+ 0 &#'&'!. 1'# +)! (%$ 0=($+0'* #<%! '* $ 0B*$% ('2&'!. -, 0*.!&!*.!*+ ('2A 0=($+0'* '1 ),&!# &!(+#$% 02$B! : ;)! 2$0* ()$#$(+!#0 +0( '1 +)! 2!+)'. $#! +)! <! '1 C*.!&!*.!*+ E'2&'*!*+ G*$%, 0 JCEGK +' #!+#0!"! 0*.!&!*.!*+ ('2&'*!*+ 1#'2 +)! '#0B0*$%.$+$ $*. +)!! +02$+! '1 +)! 2<%+0"$#0$+!.!* 0+, 0* +)! *!@.$+$ &$(! ('2&<+!. @0+) +)! CEG: ;)! -$ 0( ()!2! ($* -! #!&#!!*+!. $ 1'%%'@ L (Ak ) Xk −→ X∗k f m (Xk ) ←− f m (X∗k ) = JM:4K (A−1 k ) @)!#! +)! (%$ Ak 0 k 0*+' Y f u (X∗k ), $ &#'&!# +#$* 1'#2$+0'* 2$+#0/ @)0() &#'N!(+ −1 &$(!6 $*. Ak 0 $* 0*.!&!*.!*+ JM:MK +)!.$+$ Xk (!*+!#!. '* +)! -$(? +#$* 1'#2$+0'* $%%'@0*B '*! +'! +02$+! +)! 2<%+0"$#0$+! &#'-$-0%0+,.0 +#0-<+0'* '1 +)! (%$ k 0* +)! '#0B0*$% &$(! 1#'2 +)! &#'-$-0%0+,.0 +#0-<+0'* '1 +)! 0*.!&!*.!*+ &$(!: O)!* +)!.$+$ $#! &#'N!(+!. 0* $* 0*.!&!*.!*+.!* 0+, 1<*(+0'*.!* 0+0! f m (X∗ ) f u (X∗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k̂ %)(! 4!14 4!" ')/"#14.#.9:;<= $% /$'$/$C"*> 4!$% 2"1*% 4. 4!" %.?(122"* B17"% *"($%$.' #)2"; @' 4!" (1%".!1#* (21%%$3(14$.'> 4!" B17"% #)2" #"*)("% 4. 4!" -.22.+$'0 #)2"8 1%%$0' x 4. 4!" (21%% k̂ %)(! 4!14 k̂ = d(x) = 1#0/1A{fk (x)πk } k = 1,.., K. 9:;D=!" *"%$0'.4!" B17"% (21%%$3"# $% 4!"' *"4"#/$'"* 67 4!" (.'*$4$.'12 *"'%$4$"% fk (x) 1'* 67 4!",#$.#,#.616$2$4$"% πk ; E!$2" 4!",#$.#,#.616$2$4$"% (1' 6" "1%$27.641$'"* -#./ 4!" 4#1$'$'0 %"4> -.22.+$'0 4!" #"214$.' πk = Nk /N 9:;F= +!"#" Nk $% 4!" ')/6"#.%1/,2"%.4!" (21%% k 1'* N $% 4!".5"#122 ')/6"#.%1/,2"%.4!" 4#1$'$'0 %"4> 4!" *"4"#/$'14$.'.4!" (21%%?(.'*$4$.'12 *"'%$47 $% /)(! /.#" (!122"'0$'0;!1'&% 4. $4% 1'1274$(12 4#1(416$2$47> 4!" G1)%%$1' 9.#,$"' + = /)24$51#$14" *"'% $47 $% 4!" /.%4.-4"' )%"* *"'%$47 -.# (21%%$3(14$.'; H.+"5"#> /)24$51#$14" "%4$/14$.',#"%"'4% (!122"'0$'0,#.62"/% +!"' *"12$'0 +$4! 5"#7!$0! *$/"'%$.'12 *141;!"! #$%&'()*')&+,+-.'&/ +.&'0)&'1-!" 0"'"#12 "A,#"%%$.'.1 /)24$51#$14" '.#/12 *"'%$47 $' d *$/"'%$.'% $% +#$44"' 

1%8 fk (x) = i h 1 1 T −1 (x − μ) Σ (x − μ) "A, − k 2 (2π)d/2 |Σk |1/2 9:;I= +!"#" x $% 1 d?(./,.'"'4 (.2)/' 5"(4.#> μ $

% 4

!" d

?(

./

,.'"'4 /"1' 5"(4.#> Σ $% 4!" d 67 d (.51#$1'(" /14#$A> |Σ| 1'* Σ−1 1#" $4% *"4"#/$'1'4 1'* $4% $'5"#%"; J$'1227> (x − μ)T *"'.4"% 4!" 4#1'%,.%".(x − μ); K./" (21%%$3(14$.' #)2"% 1#" *"#$5"* 67 1%%)/$'0 4!14 4!" (21%%?(.'*$4$.'12 *"'%$4$"% 1#",?51#$14" '.#/12 +$4! /"1' 5"(4.#% μk 1'* 51#$1'("?(.51#$1'(" /14#$("% Σk 1%%) /"* 4. 6" '.' %$'0)21# LMIN;!"%" 4+.,1#1/"4"#% 1#" "%4$/14"* -#./ 4!" 4#1$'$'0 %1/,2"% 1((.#*$'0 4.8!" #$%&'&$%&$( )*+'*$&$(,-./0-+-$1$( 2$134.-.