INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS : filtrage asymetrique de ligne de base(tendance) et debruitage pour des signaux

positifs avec parcimonie des derivees

Seminaire ICube

L. DUVAL, A. PIRAYREIFP Energies nouvelles

1 et 4 av. de Bois-Preau, 92852 Rueil-Malmaison - France

X. NING, I. W. SELESNICKPolytechnic School of Engineering

New York University

19 juin 2015

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

The fast way

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

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Outline

INTRODUCTIONOUTLINEBACKGROUND

MODELINGNOTATIONSCOMPOUND SPARSE DERIVATIVE MODELING

BEADS ALGORITHMMAJORIZE-MINIMIZE

EVALUATION AND RESULTSSIMULATED BASELINE AND NOISEPOISSON NOISEGC×GC

CONCLUSIONS

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on background

I Background affects quantitative evaluation/comparisonI In some domains: (instrumental) bias, (seasonal) trendI In analytical chemistry: drift, continuum, wander, baselineI Rare cases of parametric modeling

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on background

I Background affects quantitative evaluation/comparisonI In some domains: (instrumental) bias, (seasonal) trendI In analytical chemistry: drift, continuum, wander, baselineI Rare cases of parametric modeling

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on background

I Background affects quantitative evaluation/comparisonI In some domains: (instrumental) bias, (seasonal) trendI In analytical chemistry: drift, continuum, wander, baselineI Rare cases of parametric modeling

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on background

I Background affects quantitative evaluation/comparisonI In some domains: (instrumental) bias, (seasonal) trendI In analytical chemistry: drift, continuum, wander, baselineI Rare cases of parametric modeling

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Background on backgroundFor 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.

I y: observationI x: clean series of peaksI f: baselineI w: noise

Assumption: in the absence of peaks, the baseline can beapproximately recovered from a noise-corrupted observationby low-pass filtering

I f = L(y− x) (L: low-pass filter)I formulated as ‖y− s‖2

2 = ‖H(y− x)‖22

I H = I− L: high-pass filter

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modeling

An estimate x can be obtained (with Di diff. operators) via:

x = arg minx

{F(x) =

12‖H(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 RiI φA

i = |x|I φB

i =√|x|2 + ε

I φCi = |x| − ε log (|x|+ ε)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Compound sparse derivative modelingTake the positivity of chromatogram peaks into account:

x = arg minx

{F(x) =

12‖H(y− x)‖2

2

+ λ0

N−1∑n=0

θε(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 modelingTake the positivity of chromatogram peaks into account:

x = arg minx

{F(x) =

12‖H(y− x)‖2

2

+ λ0

N−1∑n=0

θε(xn; r) +

M∑i=1

λi

Ni−1∑n=0

φ ([Dix]n)}.

and majorize it

−5 0 50

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 modelingTake the positivity of chromatogram peaks into account:

x = arg minx

{F(x) =

12‖H(y− x)‖2

2

+ λ0

N−1∑n=0

θε(xn; r) +

M∑i=1

λi

Ni−1∑n=0

φ ([Dix]n)}.

then smooth it:

−5 0 50

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 modelingTake the positivity of chromatogram peaks into account:

x = arg minx

{F(x) =

12‖H(y− x)‖2

2

+ λ0

N−1∑n=0

θε(xn; r) +

M∑i=1

λi

Ni−1∑n=0

φ ([Dix]n)}.

then majorize it:

g0(x, v) =

1+r4|v| x

2 + 1−r2 x + |v|1+r

4 , |v| > ε

1+r4ε x2 + 1−r

2 x + ε1+r4 , |v| 6 ε.

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS AlgorithmWe now have a majorizer for F

G(x,v) =12‖H(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

λiDTi [Λ(Div)] Di

]−1 (HTHy− λ0b

).

with notations

c(v) =∑

n

[φ(vn)− vn

2φ′(vn)

].

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS AlgorithmWe now have a majorizer for F

G(x,v) =12‖H(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

λiDTi [Λ(Div)] Di

]−1 (HTHy− λ0b

).

with notations

[Γ(v)]n,n =

1+r4|vn| , |vn| > ε

1+r4ε , |vn| 6 ε

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

BEADS AlgorithmWe now have a majorizer for F

G(x,v) =12‖H(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

λiDTi [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 AlgorithmWe now have a majorizer for F

G(x,v) =12‖H(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

λiDTi [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

λiDTi [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

λiDTi[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−1y2. x = y (Initialization)

Repeat

3. [Λi]n,n =φ′([Dix]n)

[Dix]n, i = 0, . . . ,M,

4. M =

M∑i=0

λiDTi ΛiDi

5. Q = BTB + ATMA

6. x = AQ−1bUntil converged

8. f = y− x− BA−1(y− x)

Output: x, f12 / 21

INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Evaluation 1

1 2000−10

0

10

20

30

40

50

Time (sample)

1 2000−10

0

10

20

30

40

50

Time (sample)

1 2000−10

0

10

20

30

40

50

Time (sample)

1 2000−10

0

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 20000

20

40

60

80

Time (sample)

1 20000

20

40

60

80

Time (sample)

1 20000

10

20

30

40

50

Time (sample)

1 20000

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 1Hyphenated, two-dimensional gas chromatography data

Original data

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data

2D background

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data

Noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data

Corrected

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data

Original data (again!)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data

Original data

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data

2D background

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data

Noise

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data

Corrected

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data

Original data (again!)

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

Conclusions and work to comeI BEADS: Baseline Estimation And Denoising. . . SparselyI Asymmetric penalties with Majorization-MinimizationI Matlab toolbox: http://lc.cx/beadsI Important pre-processing for image alignmentI Further tests on other analytical chemistry signals for

routine analysisI gas, liquid or ion chromatography; infrared, Raman,

Nuclear Magnetic Resonance (NMR) spectroscopy; massspectrometry

I Use closer to `0 sparse penalties: SOOT or smoothed `1/`2

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INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS

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. Thiebaut, 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. Thiebaut.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 `1/`2 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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