BEADS : filtrage asym etrique de ligne de base (tendance) et d - PowerPoint PPT Presentation

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS : filtrage asym´ etrique de ligne de base (tendance) et d´ ebruitage pour des signaux positifs avec parcimonie des d´ eriv´ ees S´ eminaire ICube L. DUVAL, A. PIRAYRE IFP Energies nouvelles 1 et 4 av. de Bois-Pr´ eau, 92852 Rueil-Malmaison - France X. NING, I. W. SELESNICK Polytechnic School of Engineering New York University 19 juin 2015 1 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS The fast way ◮ Question: where is the string behind the bead? ◮ Smoothness, sparsity, asymmetry 2 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Outline I NTRODUCTION O UTLINE B ACKGROUND M ODELING N OTATIONS C OMPOUND SPARSE DERIVATIVE MODELING BEADS ALGORITHM M AJORIZE -M INIMIZE E VALUATION AND RESULTS S IMULATED BASELINE AND NOISE P OISSON NOISE GC × GC C ONCLUSIONS 3 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background ◮ Background affects quantitative evaluation/comparison ◮ In some domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Rare cases of parametric modeling 4 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background ◮ Background affects quantitative evaluation/comparison ◮ In some domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Rare cases of parametric modeling 4 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background ◮ Background affects quantitative evaluation/comparison ◮ In some domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Rare cases of parametric modeling 4 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background ◮ Background affects quantitative evaluation/comparison ◮ In some domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Rare cases of parametric modeling 4 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background For analytical chemistry data: 5 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Notations Morphological decomposition: ( y , x , f , w ) ∈ ( R N ) 4 . y = x + f + w , ◮ y : observation ◮ x : clean series of peaks ◮ f : baseline ◮ w : noise Assumption: in the absence of peaks, the baseline can be approximately recovered from a noise-corrupted observation by low-pass filtering ◮ ˆ f = L ( y − ˆ x ) ( L : low-pass filter) s � 2 x ) � 2 ◮ formulated as � y − ˆ 2 = � H ( y − ˆ 2 ◮ H = I − L : high-pass filter 6 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling An estimate ˆ x can be obtained (with D i diff. operators) via: M F ( x ) = 1 � � 2 � H ( y − x ) � 2 � ˆ x = arg min 2 + λ i R i ( D i x ) . x i = 0 7 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Examples of (smooth) sparsity promoting functions for R i ◮ φ A i = | x | | x | 2 + ǫ ◮ φ B � i = ◮ φ C i = | x | − ǫ log ( | x | + ǫ ) 7 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 i = 1 n = 0 Start from: � x , x � 0 θ ( x ; r ) = − rx , x < 0 8 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 i = 1 n = 0 and majorize it The majorizer g(x, v) for the penalty function θ (x; r), r = 3 10 g(x,v) 8 θ r (x) 6 4 (s, θ r (s)) 2 (v, θ r (v)) 0 −5 0 5 x 8 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 i = 1 n = 0 then smooth it: The smoothed asymmetric penalty function θ ε (x; r), r = 3 10 8 6 4 2 (− ε , f(− ε )) ( ε , f( ε )) 0 −5 0 5 x 8 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 i = 1 n = 0 then majorize it:  4 | v | x 2 + 1 − r 1 + r 2 x + | v | 1 + r 4 , | v | > ǫ  g 0 ( x , v ) = 4 ǫ x 2 + 1 − r 1 + r 2 x + ǫ 1 + r 4 , | v | � ǫ.  8 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm We now have a majorizer for F G ( x , v ) = 1 2 � H ( y − x ) � 2 2 + λ 0 x T [ Γ ( v )] x M � λ i � 2 ( D i x ) T [Λ( D i v )] ( D i x ) � + λ 0 b T x + + c ( v ) . i = 1 Minimizing G ( x , v ) with respect to x yields M � − 1 � � � � H T H + 2 λ 0 Γ ( v ) + λ i D T H T Hy − λ 0 b x = i [Λ( D i v )] D i . i = 1 with notations φ ( v n ) − v n � � � 2 φ ′ ( v n ) c ( v ) = . n 9 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm We now have a majorizer for F G ( x , v ) = 1 2 � H ( y − x ) � 2 2 + λ 0 x T [ Γ ( v )] x M � λ i � 2 ( D i x ) T [Λ( D i v )] ( D i x ) � + λ 0 b T x + + c ( v ) . i = 1 Minimizing G ( x , v ) with respect to x yields M � − 1 � � � � H T H + 2 λ 0 Γ ( v ) + λ i D T H T Hy − λ 0 b x = i [Λ( D i v )] D i . i = 1 with notations 1 + r  4 | v n | , | v n | � ǫ  [ Γ ( v )] n , n = 1 + r  4 ǫ , | v n | � ǫ 9 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm We now have a majorizer for F G ( x , v ) = 1 2 � H ( y − x ) � 2 2 + λ 0 x T [ Γ ( v )] x M � λ i � 2 ( D i x ) T [Λ( D i v )] ( D i x ) � + λ 0 b T x + + c ( v ) . i = 1 Minimizing G ( x , v ) with respect to x yields M � − 1 � � � � H T H + 2 λ 0 Γ ( v ) + λ i D T H T Hy − λ 0 b x = i [Λ( D i v )] D i . i = 1 with notations [Λ( v )] n , n = φ ′ ( v n ) v n 9 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm We now have a majorizer for F G ( x , v ) = 1 2 � H ( y − x ) � 2 2 + λ 0 x T [ Γ ( v )] x M � λ i � 2 ( D i x ) T [Λ( D i v )] ( D i x ) � + λ 0 b T x + + c ( v ) . i = 1 Minimizing G ( x , v ) with respect to x yields M � − 1 � � � � H T H + 2 λ 0 Γ ( v ) + λ i D T H T Hy − λ 0 b x = i [Λ( D i v )] D i . i = 1 with notations [ b ] n = 1 − r 2 9 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm Writing filter H = A − 1 B ≈ BA − 1 (banded matrices) we have x = AQ − 1 � � B T BA − 1 y − λ 0 A T b where Q is the banded matrix, Q = B T B + A T MA , and M is the banded matrix, M � λ i D T M = 2 λ 0 Γ ( v ) + i [Λ( D i v )] D i . i = 1 10 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm Using previous equations, the MM iteration takes the form: M M ( k ) = 2 λ 0 Γ ( x ( k ) ) + � λ i D T Λ( D i x ( k ) ) � � D i . i i = 1 Q ( k ) = B T B + A T M ( k ) A x ( k + 1 ) = A [ Q ( k ) ] − 1 � � B T BA − 1 y − λ 0 A T b 11 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm Input: y , A , B , λ i , i = 0 , . . . , M b = B T BA − 1 y 1 . 2 . x = y (Initialization) Repeat [ Λ i ] n , n = φ ′ ([ D i x ] n ) 3 . , i = 0 , . . . , M , [ D i x ] n M � λ i D T 4 . M = i Λ i D i i = 0 Q = B T B + A T MA 5 . x = AQ − 1 b 6 . Until converged f = y − x − BA − 1 ( y − x ) 8 . Output: x , f 12 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 1 50 50 40 40 30 30 20 20 10 10 0 0 −10 −10 1 2000 1 2000 Time (sample) Time (sample) 50 50 40 40 30 30 20 20 10 10 0 0 −10 −10 1 2000 1 2000 Time (sample) Time (sample) Figure : Simulated chromatograms w/ polynomial+sine baseline. 13 / 21

I NTRODUCTION M ODELING BEADS ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 1 with Gaussian noise 14 / 21