Multichannel thresholding with sensing dictionaries

HAL Id: inria-00544777

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Submitted on 7 Feb 2011

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Multichannel thresholding with sensing dictionaries

Rémi Gribonval, Boris Mailhé, Holger Rauhut, Karin Schnass, Pierre Vandergheynst

To cite this version:

Rémi Gribonval, Boris Mailhé, Holger Rauhut, Karin Schnass, Pierre Vandergheynst. Multichannel

thresholding with sensing dictionaries. Computational Advances in Multi-Sensor Adaptive Processing

(CAMSAP’07), Dec 2007, St Thomas, Virgin Islands, United States. pp.133 - 136, �10.1109/CAM-

SAP.2007.4497983�. �inria-00544777�

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T4 B% )2H% 4%%* +* ()% $24( 4%"(+#* ()% 4-""%44 #E p:()1%4)#$0+*8 +4 $218%$' 0%(%1F+*%0 K' ()% #10%1 X "1#44:"#11%$2(+#* K%(B%%*

()% #1+8+*2$ 2*0 ()% 4%*4+*8 0+"(+#*21'I ;)-4 +* #10%1 (# #D(+:

F+4% ()% D%1E#1F2*"% #E ()% 2$8#1+()F B% 4)#-$0 ")##4% ()%

4%*4+*8 0+"(+#*21' ()% F+*+F+4%4 ()% #10%1 X "1#44 "#11%$2(+#*A +%I

Ψ opt = arg min

Ψ

µ 2 (Φ, Ψ).

;)% 42F% #D(+F+42(+#* D1#K$%F 21+4%4 B)%* 4(-0'+*8 ()% 2H:

%128% D%1E#1F2*"% #E ()% 4+FD$% ()1%4)#$0+*8 2$8#1+()F +* NaOA B)+") B% 1%E%1 (# E#1 F#1% 0%(2+$4I

9*4(%20 B% B+$$ D#+*( #-( 2* +*(%1%4(+*8 "#**%"(+#* K%(B%%*

"#FD1%44%0 4%*4+*8 2*0 4%*4+*8 0+"(+#*21+%4 #E $#B 12*LI J%

+*(1#0-"%0 #-1 4+8*2$ F#0%$ B+() ()% %G2FD$% #E )2H+*8 (#

0%4+8* 2 *%(B#1L #E N 4%*4#14 F#*+(#1+*8 2 "#FF#* D)%:

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(# 2 "#FF#* D1#"%44+*8 -*+(A B)+") ()%* !*04 ()% "#FF#*

4-DD#1(A B)+") B% B%1% +*(%1%4(%0 +*I ;)% 0+420H2*(28% +*

()+4 4")%F% +4 ()2( %H%1' 4%*4#1 *%%04 (# #K(2+* 2*0 4%*0 ()%

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8+H%* *-FK%1 ")2**%$4 2*0 4-DD#1( 4+C% ()%1% +4 *#()+*8 B%

"2* 0#I =#B%H%1 244-F% ()2( B% )2H% 2 $#( #E ")2**%$4 #1 H%1' 4F2$$ 4-DD#1( 4+C% 2*0 ()2( ()% 0+"(+#*21' +4 H%1' B%$$

K%)2H%0A +%I +( +4 %24' (# !*0 2 H%1' 8##0 4%*4+*8 0+"(+#*21'I 9* ()+4 "24% B% "#-$0 (1' (# !*0 2 4%*4+*8 0+"(+#*21' B)+") 4(+$$ 8+H%4 1%24#*2K$% "1#44:"#)%1%*"% K-( #E 12*L F-") $#B%1 ()2* ()% 0+F%*4+#*2$+(' #E ()% 4+8*2$4 M

Ψ opt = arg min

µ

(Φ,Ψ)<c

rank(Ψ).

;)% 20H2*(28% #E 4-") 2 $#B 12*L 4%*4+*8 0+"(+#*21' +4 ()2(

B% "2* 1%0-"% ()% 4+C% #E ()% 4+8*2$ ()2( %H%1' 4%*4#1 )24 (#

D244 #*I 7# 244-F% ()2( Ψ )24 12*L q 2*0 ()-4 )24 2 1%0-"%0 4+*8-$21 H2$-% 0%"#FD#4+(+#* #E ()% E#1F

Ψ = U

d × d · Σ

d × q · V ^!

p × K 5a6

B)%1% K#() U, V )2H% #1()#*#1F2$ "#$-F*4I S-$(+D$'+*8 ()%

4+8*2$4 K' Ψ ^! +* ()% 2$8#1+()F 2F#-*(4 (# !14( F-$(+D$'+*8 K' Σ ^! U ^! 2*0 ()%* K' V I =#B%H%1 +E B% 0# ()% !14( F-$(+D$+"2:

(+#* *#( 2( ()% E-4+#* "%*(%1 K-( 2( ()% 4%*4#14 #*$' 2 4+8*2$ #E 4+C% q +*4(%20 #E d )24 (# K% 4%*0 #*I 9* #10%1 (# 8%( 2 E%%$+*8 E#1 )#B F-") 1%0-"(+#* +4 D#44+K$%A $%(b4 20#D( 2* 2$(%1*2(+H%

H+%BD#+*(I R' F-$(+D$'+*8 B+() Σ ^! U ^! B% 8%( 4+8*2$4 Z ()2(

21% 4D214% +* ()% 0+"(+#*21' Φ ^q := Σ ^! U ^! ΦA +%

Z = Σ ^! U ^! Φ Λ X + Σ ^! U ^! E = Φ ^p _Λ + ˜ E

B)+") B% B2*( (# 1%"#*4(1-"( -4+*8 ()% 4%*4+*8 0+"(+#*21' Ψ ^q := V ^! I 7+*"% ()% 0+"(+#*21+%4 )2H% K 2(#F4 +* 0+F%*:

4+#* p ()% "-F-$2(+H% "#)%1%*"% B+$$ K% #E ()% F28*+(-0%

µ 2 (Φ ^q , Ψ ^q , S) = '

S/qI 9E B% +*4%1( ()+4 %4(+F2(% +*(# ()%

E#1F-$2 E#1 γ E1#F ;)%#1%F >A 244-F+*8 R ¯ = 1, E = 0 E#1 4+FD$+"+('I J% "2* %4(+F2(% ()2( ()+4 L+*0 #E "#FD1%44%0 >:

()1%4)#$0+*8 E2+$4 K'

P (1 − ()1%4)#$0+*8 E2+$4) ≤ K · exp



 − N π

"

1 − ' S/p 1 + '

S/p

# 2

.





.-1()%1 4+FD$+E'+*8 ()% 2K#H% E#1F-$2 B% 8%( 24 1-$% #E ()-FK

#* )#B "#FD1%44%0 B% "2* 4%*4% #-1 4+8*2$4 0%D%*0+*8 #*

()%+1 %GD%"(%0 4D214+('A 0+"(+#*21' 4+C% 2*0 *-FK%1 #E ")2*:

*%$4

q ≥ S , 1

2 − log((/K) AN

- 2

.#1 2 F#1% 0%(2+$%0 %GD#4+(+#* 2*0 2* 2$8#1+()F (# "2$"-$2(%

$#B 12*L 4%*4+*8 F2(1+"%4 B% 1%E%1 (# NYOI E3 &-)&#",%-),

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