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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

The fast way

◮ Question: where is the string behind the bead? ◮ Smoothness, sparsity, asymmetry

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Outline

INTRODUCTION OUTLINE BACKGROUND MODELING NOTATIONS COMPOUND SPARSE DERIVATIVE MODELING BEADS ALGORITHM MAJORIZE-MINIMIZE EVALUATION AND RESULTS SIMULATED BASELINE AND NOISE POISSON NOISE GC×GC CONCLUSIONS

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on background

For analytical chemistry data:

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Notations

Morphological decomposition: y = x + f + w, (y, x, f, w) ∈ (RN)4.

◮ 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)

◮ formulated as y − ˆ

s2

2 = H(y − ˆ

x)2

2 ◮ H = I − L: high-pass filter

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

An estimate ˆ x can be obtained (with Di diff. operators) via: ˆ x = arg min

x

F(x) = 1

2H(y − x)2

2 + M

i=0

λiRi (Dix)

.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

Examples of (smooth) sparsity promoting functions for Ri

◮ φA i = |x| ◮ φB i =

|x|2 + ǫ

◮ φC i = |x| − ǫ log (|x| + ǫ)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

Take the positivity of chromatogram peaks into account: ˆ x = arg min

x

F(x) = 1

2H(y − x)2

2

+ λ0

N−1

n=0

θǫ(xn; r) +

M

i=1

λi

Ni−1

n=0

φ ([Dix]n)

.

Start from: θ(x; r) =

x,

x 0 −rx, x < 0

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

Take the positivity of chromatogram peaks into account: ˆ x = arg min

x

F(x) = 1

2H(y − x)2

2

+ λ0

N−1

n=0

θǫ(xn; r) +

M

i=1

λi

Ni−1

n=0

φ ([Dix]n)

.

and majorize it

−5 5 2 4 6 8 10 The majorizer g(x, v) for the penalty function θ(x; r), r = 3 x (s, θr(s)) (v, θr(v)) g(x,v) θr(x)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

Take the positivity of chromatogram peaks into account: ˆ x = arg min

x

F(x) = 1

2H(y − x)2

2

+ λ0

N−1

n=0

θǫ(xn; r) +

M

i=1

λi

Ni−1

n=0

φ ([Dix]n)

.

then smooth it:

−5 5 2 4 6 8 10 The smoothed asymmetric penalty function θε(x; r), r = 3 (−ε, f(−ε)) (ε, f(ε)) x

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

Take the positivity of chromatogram peaks into account: ˆ x = arg min

x

F(x) = 1

2H(y − x)2

2

+ λ0

N−1

n=0

θǫ(xn; r) +

M

i=1

λi

Ni−1

n=0

φ ([Dix]n)

.

then majorize it: g0(x, v) =   

1+r 4|v| x2 + 1−r 2 x + |v| 1+r 4 ,

|v| > ǫ

1+r 4ǫ x2 + 1−r 2 x + ǫ 1+r 4 ,

|v| ǫ.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

We now have a majorizer for F G(x, v) = 1 2H(y − x)2

2 + λ0xT[Γ(v)]x

+ λ0bTx +

M

i=1

λi 2 (Dix)T [Λ(Div)] (Dix)

+ c(v).

Minimizing G(x, v) with respect to x yields x =

HTH + 2λ0 Γ(v) +

M

i=1

λiDT

i [Λ(Div)] Di

−1 HTHy − λ0b

.

with notations c(v) =

n
φ(vn) − vn

2 φ′(vn)

.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

We now have a majorizer for F G(x, v) = 1 2H(y − x)2

2 + λ0xT[Γ(v)]x

+ λ0bTx +

M

i=1

λi 2 (Dix)T [Λ(Div)] (Dix)

+ c(v).

Minimizing G(x, v) with respect to x yields x =

HTH + 2λ0 Γ(v) +

M

i=1

λiDT

i [Λ(Div)] Di

−1 HTHy − λ0b

.

with notations [Γ(v)]n,n =   

1+r 4|vn|,

|vn| ǫ

1+r 4ǫ ,

|vn| ǫ

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

We now have a majorizer for F G(x, v) = 1 2H(y − x)2

2 + λ0xT[Γ(v)]x

+ λ0bTx +

M

i=1

λi 2 (Dix)T [Λ(Div)] (Dix)

+ c(v).

Minimizing G(x, v) with respect to x yields x =

HTH + 2λ0 Γ(v) +

M

i=1

λiDT

i [Λ(Div)] Di

−1 HTHy − λ0b

.

with notations [Λ(v)]n,n = φ′(vn) vn

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

We now have a majorizer for F G(x, v) = 1 2H(y − x)2

2 + λ0xT[Γ(v)]x

+ λ0bTx +

M

i=1

λi 2 (Dix)T [Λ(Div)] (Dix)

+ c(v).

Minimizing G(x, v) with respect to x yields x =

HTH + 2λ0 Γ(v) +

M

i=1

λiDT

i [Λ(Div)] Di

−1 HTHy − λ0b

.

with notations [b]n = 1 − r 2

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

Writing filter H = A−1B ≈ BA−1 (banded matrices) we have x = AQ−1 BTBA−1y − λ0ATb

where Q is the banded matrix,

Q = BTB + ATMA, and M is the banded matrix, M = 2λ0Γ(v) +

M

i=1

λiDT

i [Λ(Div)] Di.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

Using previous equations, the MM iteration takes the form: M(k) = 2λ0Γ(x(k)) +

M

i=1

λiDT

i

Λ(Dix(k))
Di.

Q(k) = BTB + ATM(k)A x(k+1) = A[Q(k)]−1 BTBA−1y − λ0ATb

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS Algorithm

Input: y, A, B, λi, i = 0, . . . , M 1. b = BTBA−1y 2. x = y (Initialization) Repeat 3. [Λi]n,n = φ′([Dix]n) [Dix]n , i = 0, . . . , M, 4. M =

M

i=0

λiDT

i ΛiDi

5. Q = BTB + ATMA 6. x = AQ−1b Until converged 8. f = y − x − BA−1(y − x) Output: x, f

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 1

1 2000 −10 10 20 30 40 50 Time (sample) 1 2000 −10 10 20 30 40 50 Time (sample) 1 2000 −10 10 20 30 40 50 Time (sample) 1 2000 −10 10 20 30 40 50 Time (sample)

Figure : Simulated chromatograms w/ polynomial+sine baseline.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 1 with Gaussian noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 2

1 2000 20 40 60 80 Time (sample) 1 2000 20 40 60 80 Time (sample) 1 2000 10 20 30 40 50 Time (sample) 1 2000 10 20 30 40 Time (sample)

Figure : Simulated chromatograms w/ limited power spectrum noise.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 2 with Gaussian noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 3 with Poisson noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1

Hyphenated, two-dimensional gas chromatography data Original data

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1

Hyphenated, two-dimensional gas chromatography data 2D background

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1

Hyphenated, two-dimensional gas chromatography data Noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1

Hyphenated, two-dimensional gas chromatography data Corrected

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1

Hyphenated, two-dimensional gas chromatography data Original data (again!)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2

Hyphenated, two-dimensional gas chromatography data Original data

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Two-dimensional chromatography data 2

Hyphenated, two-dimensional gas chromatography data 2D background

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2

Hyphenated, two-dimensional gas chromatography data Noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2

Hyphenated, two-dimensional gas chromatography data Corrected

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Two-dimensional chromatography data 2

Hyphenated, two-dimensional gas chromatography data Original data (again!)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Conclusions and work to come

◮ BEADS: Baseline Estimation And Denoising. . . Sparsely ◮ Asymmetric penalties with Majorization-Minimization ◮ Matlab toolbox: http://lc.cx/beads ◮ Important pre-processing for image alignment ◮ Further tests on other analytical chemistry signals for

routine analysis

◮ gas, liquid or ion chromatography; infrared, Raman,

Nuclear Magnetic Resonance (NMR) spectroscopy; mass spectrometry

◮ Use closer to ℓ0 sparse penalties: SOOT or smoothed ℓ1/ℓ2

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More for free: additional references

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert.

Background removal from spectra by designing and minimising a non-quadratic cost function.

Chemometr. Intell. Lab. Syst., 76(2):121–133, 2005.
C. Vendeuvre, R. Ruiz-Guerrero, F. Bertoncini, L. Duval, D. Thi´

ebaut, and M.-C. Hennion. Characterisation of middle-distillates by comprehensive two-dimensional gas chromatography (GC × GC): A powerful alternative for performing various standard analysis of middle-distillates.

J. Chrom. A, 1086(1-2):21–28, 2005.
C. Vendeuvre, R. Ruiz-Guerrero, F. Bertoncini, L. Duval, and D. Thi´

ebaut. Comprehensive two-dimensional gas chromatography for detailed characterisation of petroleum products. Oil Gas Sci. Tech., 62(1):43–55, 2007.

X. Ning, I. W. Selesnick, and L. Duval.

Chromatogram baseline estimation and denoising using sparsity (BEADS).

Chemometr. Intell. Lab. Syst., 139:156–167, Dec. 2014.
A. Repetti, M. Q. Pham, L. Duval, E. Chouzenoux, and J.-C. Pesquet.

Euclid in a taxicab: Sparse blind deconvolution with smoothed ℓ1/ℓ2 regularization. IEEE Signal Process. Lett., 22(5):539–543, May 2015.

C. Couprie, M. Moreaud, L. Duval, S. Henon and V. Souchon,

BARCHAN: Blob Alignment for Robust CHromatographic ANalysis.

J. Chrom. A, 2016...

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