(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Signal analysis using sparse representation and proximal optimization methods Mai Quyen PHAM GIPSA-Lab, 11 rue des math´ ematiques, 38402 Saint Martin d’H` eres 04 November 2016 1 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Contents (M ULTIPLE + NOISE ) REMOVAL Formulation Algorithm Results B LIND DECONVOLUTION Formulation Algorithm Results C ONCLUSIONS 2 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS (Multiple + noise) removal 3 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Seismic multiple reflection Hydrophone Towed streamer • • • • • • • Solid blue: primaries; dashed red: multiple reflection disturbances. 4 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS (Multiple + noise) removal strategies multiples noise s ( n ) b ( n ) Measurements z ( n ) = y ( n ) + s ( n ) + b ( n ) System + ( ∀ n ∈ { 0 , . . . , N − 1 } ) Source y ( n ) primary Which strategy for restoring the primary signal y ( n ) corrupted by the unknown multiples s ( n ) , plus noise b ( n ) ? ◮ Variational approach ◮ Methodology for primary/multiple adaptive ◮ Proximal methods to solve the separation based on resulting optimization approximate templates problem 5 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Multi-model ◮ J models r ( n ) are known (available) j ◮ Imperfect in time, amplitude and frequency s ( n ) throughout time varying ◮ Assumption : models linked to ¯ filters (FIR) p ′ + P j − 1 J − 1 � � s ( n ) = ( p ) r ( n − p ) ¯ h ( n ) ¯ j j j = 0 p = p ′ where ◮ ¯ h ( n ) : unknown impulse reponse of the filter corresponding to j model j and time n ( P j tap coefficients) ◮ p ′ ∈ {− P j + 1 , . . . , 0 } ◮ New definition: P = � J − 1 j = 0 P j . 6 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Template r 0 Magnitude r 1 ¯ s 0 200 400 600 800 1000 Time First model, Second model, Multiple 7 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Template Space Space Primary Time Time Time Observed image Template Signal 7 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Problem reformulation ¯ z = R + ¯ y + b h ���� ���� ���� ���� observed signal filter noise primary where � ¯ s ( N − 1 ) � ⊤ s = � J − 1 j = 0 R j ¯ h j = R ¯ ◮ ¯ s ( 0 ) , · · · , ¯ h = ◮ R = [ R 0 · · · R J − 1 ] , R j is a block diagonal matrix � � ⊤ ◮ ¯ ¯ 0 · · · ¯ h ⊤ h ⊤ h = J − 1 � ( p ′ + P j − 1 ) · · · ◮ ¯ h ( n ) h ( 0 ) ¯ ( p ′ ) · · · ¯ h ( 0 ) = j j j � ⊤ ¯ h ( N − 1 ) ( p ′ ) · · · ¯ h ( N − 1 ) ( p ′ + P j − 1 ) j j 8 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Estimation of y Assumption : ¯ y is a realization of a random vector Y , whose probability density is given by: ( ∀ y ∈ R N ) f Y ( y ) ∝ exp ( − ϕ ( Fy )) F ∈ R K × N : linear operator. ϕ is chosen separable: � K � ∀ x = ( x k ) 1 ≤ k ≤ K ∈ R K � ϕ ( x ) = ϕ k ( x k ) k = 1 where, for all k ∈ { 1 , . . . , K } , ϕ k : R → ] −∞ , + ∞ ] . 9 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Estimation : filter h and noise b ◮ Assumption : ¯ h is a realization of a random vector H , whose probability density can be expressed as: ( ∀ h ∈ R NP ) f H ( h ) ∝ exp ( − ρ ( h )) H is independent of Y . ◮ Assumption : b is a realization of a random vector B , of probability density: ( ∀ b ∈ R N ) f B ( b ) ∝ exp ( − ψ ( b )) B is assumed to be independent from Y and H 10 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Estimation : filter h and noise b ◮ Assumption : ¯ h is a realization of a random vector H , whose probability density can be expressed as: ( ∀ h ∈ R NP ) f H ( h ) ∝ exp ( − ρ ( h )) H is independent of Y . ◮ Assumption : b is a realization of a random vector B , of probability density: ( ∀ b ∈ R N ) f B ( b ) ∝ exp ( − ψ ( b )) B is assumed to be independent from Y and H MAP estimation of ( y , h ) � � ψ z − Rh − y + ϕ ( Fy ) + ρ ( h ) minimize ♣ � �� � ���� � �� � y ∈ R N , h ∈ R NP a priori on the filters fidelity: linked to noise a priori on the signal 10 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Problem to be solved MAP estimation of ( y , h ) � � minimize ψ z − Rh − y + ϕ ( Fy ) + ρ ( h ) ♣ ���� � �� � � �� � y ∈ R N , h ∈ R NP a priori on the signal a priori on the filters fidelity: linked to noise ◮ Difficulty: Choosing the good regularization parameters ◮ Proposed: Use a constrained minimization problem Problem to be solved � � y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) minimize 11 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS About convex set D Problem to be solved � � minimize y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) � 0 if x ∈ D ι D ( x ) = + ∞ otherwise. ◮ F ∈ R K × N : analysis frame operator ◮ { K l | l ∈ { 1 , . . . , L}} ⊂ { 1 , . . . , K } ◮ D = D 1 × · · · × D L with D l = { ( x k ) k ∈ K l | � k ∈ K l ϕ ℓ ( x k ) ≤ β l } , where ∀ l ∈ { 1 , . . . , L} , β l ∈ ] 0 , + ∞ [ , and ϕ l : R → [ 0 , + ∞ [ is a lower-semicontinuous convex function. 12 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS About convex set C Problem to be solved � � minimize y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) C = C 1 ∩ C 2 ∩ C 3 � � h ∈ R PN : ρ ( h ) = � J − 1 ◮ C 1 = j = 0 ρ j ( h j ) ≤ τ ℓ 2 = � N − 1 � p ′ + P j − 1 | h ( n ) ◮ ρ j ( h j ) = � h j � 2 ( p ) | 2 n = 0 p = p ′ j 13 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS About convex set C Problem to be solved � � minimize y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) C = C 1 ∩ C 2 ∩ C 3 � � h ∈ R PN : ρ ( h ) = � J − 1 ◮ C 1 = j = 0 ρ j ( h j ) ≤ τ ℓ 2 = � N − 1 � p ′ + P j − 1 | h ( n ) ◮ ρ j ( h j ) = � h j � 2 ( p ) | 2 n = 0 p = p ′ j ◮ ρ j ( h j ) = � h j � ℓ 1 = � N − 1 � p ′ + P j − 1 | h ( n ) ( p ) | n = 0 p = p ′ j 13 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS About convex set C Problem to be solved � � minimize y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) C = C 1 ∩ C 2 ∩ C 3 � � h ∈ R PN : ρ ( h ) = � J − 1 ◮ C 1 = j = 0 ρ j ( h j ) ≤ τ ℓ 2 = � N − 1 � p ′ + P j − 1 | h ( n ) ◮ ρ j ( h j ) = � h j � 2 ( p ) | 2 n = 0 p = p ′ j ◮ ρ j ( h j ) = � h j � ℓ 1 = � N − 1 � p ′ + P j − 1 | h ( n ) ( p ) | n = 0 p = p ′ j �� p ′ + P j − 1 ( p ) | 2 � 1 / 2 ◮ ρ j ( h j ) = � h j � ℓ 1 , 2 = � N − 1 | h ( n ) n = 0 p = p ′ j 13 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Hard constraints on the filters C 2 , C 3 Problem to be solved � � y ∈ R N , h ∈ R NP ψ z − Rh − y + ι D ( Fy ) + ι C ( h ) minimize C = C 1 ∩ C 2 ∩ C 3 Assumption : slow variations of the filters along time. � � | h ( n + 1 ) ( p ) − h ( n ) ∀ ( j , n , p ) ( p ) | ≤ ε j , p j j For computational issues, h ∈ C 2 ∩ C 3 where � � � N � � � � � � � � h ( 2 n + 1 ) ( p ) − h ( 2 n ) ( p ) C 2 = h | ∀ p , ∀ n ∈ 0 , . . . , − 1 � ≤ ε p 2 � � � N − 1 �� � � � � � � h ( 2 n ) ( p ) − h ( 2 n − 1 ) ( p ) C 3 = h | ∀ p , ∀ n ∈ 1 , . . . , � ≤ ε p 2 14 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Proximity operator Definition Let ϕ be a lower semi-continuous convex function. For all x ∈ R N , prox ϕ is the unique minimizer of ♣ y �→ ϕ ( y ) + 1 2 � x − y � 2 Examples: C a non-empty closed convex subset of R N . ι C ( y ) + 1 2 � x − y � 2 prox ι C ( x ) = minimize y ∈ R N � x − y � 2 = minimize y ∈ C � �� � Π C ( x ): projection operator onto C 15 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Proximity operator Definition Let ϕ be a lower semi-continuous convex function. For all x ∈ R N , prox ϕ is the unique minimizer of ♣ y �→ ϕ ( y ) + 1 2 � x − y � 2 Examples: C a non-empty closed convex subset of R N . ι C ( y ) + 1 2 � x − y � 2 prox ι C ( x ) = minimize y ∈ R N � x − y � 2 = minimize y ∈ C � �� � Π C ( x ): projection operator onto C 15 / 36
(M ULTIPLE + NOISE ) REMOVAL B LIND DECONVOLUTION C ONCLUSIONS Proximity operator Definition Let ϕ be a lower semi-continuous convex function. For all x ∈ R N , prox ϕ is the unique minimizer of ♣ y �→ ϕ ( y ) + 1 2 � x − y � 2 Examples: prox λ |·| p ( ∀ x ∈ R ) 1 a ) prox λ |·| 2 ( x ) = 1 + 2 λ x p = 1 p = 2 � �� � “Wiener” filter x − λ λ b ) prox λ |·| ( x ) = sign ( x ) max ( | x | − λ, 0 ) � �� � shrinkage operator 15 / 36
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