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I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS O NGOING , EXTENSIONS , CONCLUSION Pics, ligne de base, bruit : s eparation ternaire de sources assist ee (BEADS : positivit e, parcimonie), spectres chimiques &

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INTRODUCTION BEADS MODEL AND ALGORITHM EVALUATION AND RESULTS ONGOING, EXTENSIONS, CONCLUSION

Pics, ligne de base, bruit : s´ eparation ternaire de sources assist´ ee (BEADS : positivit´ e, parcimonie), spectres chimiques & miscellan´ ees

  • L. DUVAL, A. PIRAYRE

IFP Energies nouvelles

  • X. NING, I. W. SELESNICK

Polytechnic School of Engineering, New York University

23 mars 2018

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Old peaks cast long shadows

Chromatography: the traditional 2D way.

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Old peaks cast long shadows

Chromatography: individual 1D peaks for single compounds

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Old peaks cast long shadows

Chromatography: ternary sources separated

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Old peaks cast long shadows

Chromatography: observed signal

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Old peaks cast long shadows

Chromatography: wrapping it up

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The quick version

◮ Issue: how to accurately & repeatably quantize peaks?

◮ avoiding separate baseline and noise removal

◮ Question: where is the string behind the bead?

◮ without too accurate models for: peak, noise, baseline

◮ Answer: use main measurement properties + optimization

◮ sparsity+symmetry, stationarity, smoothness

◮ BEADS: Baseline Estimation And Denoising w/ Sparsity

◮ other properties + optimization for further processing

(BARCHAN)

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Outline

INTRODUCTION FOREWORD OUTLINE* BACKGROUND BEADS MODEL AND ALGORITHM NOTATIONS COMPOUND SPARSE DERIVATIVE MODELING MAJORIZE-MINIMIZE TYPE OPTIMIZATION EVALUATION AND RESULTS GC: SIMULATED BASELINE AND GAUSSIAN NOISE GC: SIMULATED POISSON NOISE GC: REAL DATA GC×GC: REAL DATA ONGOING, EXTENSIONS, CONCLUSION

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Background on background

Image processing: varying illumination

◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear,

polynomial, spline)

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Background on background

Econometrics: trends and seasonality

◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear,

polynomial, spline)

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Background on background

Biomedical: ECG isoelectric line or baseline wander

◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear,

polynomial, spline)

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Background on background

Gas chromatography: baseline

◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear,

polynomial, spline)

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Background on background

Analytical chemistry, biological data

◮ Signal separation into three main morphological

components

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Notations and assumptions

Morphological decomposition: y = x + f + w, signals in RN

◮ y: observation (spectrum, analytical data) ◮ x: clean series of peaks (no baseline, no noise) ◮ f: baseline ◮ w: noise

Assumption: without peaks, the baseline can be (approx.) recovered from noise-corrupted data by low-pass filtering

◮ ˆ

f = L(y − ˆ x): L: low-pass filter; H = I − L: high-pass filter

◮ formulated as y − ˆ

x − ˆ f2

2 = H(y − ˆ

x)2

2 ◮ Going further with Di: differentiation operators

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Compound sparse derivative modeling

An estimate ˆ x can be obtained via: ˆ x = arg min

x

  • F(x) = 1

2H(y − x)2

2 + M

  • i=0

λiRi (Dix)

  • .

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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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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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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 BEADS MODEL AND ALGORITHM EVALUATION AND RESULTS ONGOING, EXTENSIONS, CONCLUSION

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 BEADS MODEL AND ALGORITHM EVALUATION AND RESULTS ONGOING, EXTENSIONS, CONCLUSION

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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Overall principle for Majoration-Minimization

x G(x, xk) xk xk+1 G(x, xk+1) xk+2 F(x)

MM principles.

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BEADS Algorithm (short)

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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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)

Simulated chromatograms w/ polynomial+sine baseline

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Evaluation 1 with Gaussian noise

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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)

Simulated chromatograms w/ limited power spectrum noise

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Evaluation 2 with Gaussian noise

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Evaluation 3 with Poisson noise

Simulated chromatograms w/ Poisson noise

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Results: mono-dimensional chromatography (data 1)

Original, superimposed, clean, noise

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Results: two-dimensional chromatography (data 2)

Original data

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Results: two-dimensional chromatography (data 2)

2D background (estimated)

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Results: two-dimensional chromatography (data 2)

Noise (estimated)

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Results: two-dimensional chromatography (data 2)

BEADS corrected data

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Results: two-dimensional chromatography (data 2)

Original data (again!)

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Results: two-dimensional chromatography (data 3)

Original data

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Results: two-dimensional chromatography (data 3)

2D background (estimated)

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Results: two-dimensional chromatography (data 3)

Noise (estimated)

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Results: two-dimensional chromatography (data 3)

BEADS corrected data

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Results: two-dimensional chromatography (data 3)

Original data (again!)

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Results: computing scalability

Linear cost per sample (almost)

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Ongoing work

◮ Tests on analytical chemistry data: NIR, NMR, XPS ◮ Novel filtering: improved Savitzky-Golay filters ◮ Novel deconvolution: sparse & positive with norm ratios

SOOT: Non-convex ℓ0 count index approximation

◮ Novel metrics: errors related to peak quantities ◮ Baseline and noise use: uncertainty, trace products ◮ 2D chromatography comparisons: BARCHAN warping ◮ Improved usability: parameter estimation

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BARCHAN: 2D chromatography warping

No transformation Rigid Non-rigid BARCHAN

Semi-rigid morphing of two different 2D chromatograms.

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BARCHAN: 2D chromatography warping

Ingredients of a GMM plus EM optimization:

◮ Point sets X = {X1, . . . , XN} and Y = {Y1, . . . , YM} ◮ p(Xn) = w N + M m=1 1−w 2Mπσ2 exp

  • − Xn−T(Ym)2

2σ2

  • ◮ minσ,W,s,t E = E1(σ, W, s, t) + λ

2 Tr(W⊤GW)

Calculated deformation of a 2D chromatogram with BARCHAN.

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BARCHAN: 2D chromatography warping

Ingredients of a GMM plus EM optimization:

◮ Point sets X = {X1, . . . , XN} and Y = {Y1, . . . , YM} ◮ p(Xn) = w N + M m=1 1−w 2Mπσ2 exp

  • − Xn−T(Ym)2

2σ2

  • ◮ minσ,W,s,t E = E1(σ, W, s, t) + λ

2 Tr(W⊤GW)

Calculated deformation of a 2D chromatogram with BARCHAN.

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BARCHAN: 2D chromatography warping

Ingredients of a GMM plus EM optimization:

◮ Point sets X = {X1, . . . , XN} and Y = {Y1, . . . , YM} ◮ p(Xn) = w N + M m=1 1−w 2Mπσ2 exp

  • − Xn−T(Ym)2

2σ2

  • ◮ minσ,W,s,t E = E1(σ, W, s, t) + λ

2 Tr(W⊤GW)

Calculated deformation of a 2D chromatogram with BARCHAN.

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Improved usability: parameter estimation

◮ Cut-off frequency estimation

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Improved usability: parameter estimation

◮ Noise, asymmetry (r) and regularization (λ)

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Extended applications

◮ Lidar application

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Extended applications

◮ Engine knocking application

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Other known uses

◮ A fairly generic model (sparsity, positivity/negativity),

reused by other authors

◮ gas chromatography: mono-dimensional and

comprehensive/two-dimensional

◮ Raman spectra: biological and biomedical ◮ MUSE (Multi Unit Spectroscopic Explorer): astronomical

hyperspectral galaxy spectrum

◮ X-ray absorption spectroscopy (XAS), X-ray diffraction

(XRD), and combined XAS/XRD

◮ high-resolution mass spectrometry ◮ postprandial Plasma Glucose (PPG), multichannel

electroencephalogram (EEG) and single-channel electrocardiogram (ECG)

◮ arabic characters 28 / 33

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INTRODUCTION BEADS MODEL AND ALGORITHM EVALUATION AND RESULTS ONGOING, EXTENSIONS, CONCLUSION

Other known uses

◮ A fairly generic model (sparsity, positivity/negativity),

reused by other authors

◮ gas chromatography: mono-dimensional and

comprehensive/two-dimensional

◮ Raman spectra: biological and biomedical ◮ MUSE (Multi Unit Spectroscopic Explorer): astronomical

hyperspectral galaxy spectrum

◮ X-ray absorption spectroscopy (XAS), X-ray diffraction

(XRD), and combined XAS/XRD

◮ high-resolution mass spectrometry ◮ postprandial Plasma Glucose (PPG), multichannel

electroencephalogram (EEG) and single-channel electrocardiogram (ECG)

◮ arabic characters 28 / 33

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Conclusions

◮ Joint baseline/background and noise estimation

◮ Interaction between “separative science” and “source

separation”

◮ Little ”hard” modeling ◮ Easy to tune, scalable ◮ Codes in Matlab, R and C++1

◮ A wide range of applications to unveil

1http://www.laurent-duval.eu/ siva-beads-baseline-background-removal-filtering-sparsity.html 29 / 33

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A little more: additional references

  • C. Vendeuvre, F. Bertoncini, L. Duval, J.-L. Duplan, D. Thi´

ebaut, and M.-C. Hennion. Comparison of conventional gas chromatography and comprehensive two-dimensional gas chromatography for the detailed analysis of petrochemical samples, .

  • J. Chrom. A, 2004, http://dx.doi.org/10.1016/j.chroma.2004.05.071
  • 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, 2005, http://dx.doi.org/10.1016/j.chroma.2005.05.106
  • 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., 2007, http://dx.doi.org/10.2516/ogst:2007004

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

Chromatogram baseline estimation and denoising using sparsity (BEADS).

  • Chemometr. Intell. Lab. Syst., 2014, http://dx.doi.org/10.1016/j.chemolab.2014.09.014
  • 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., 2015, http://dx.doi.org/10.1109/LSP.2014.2362861

  • C. Couprie, L. Duval, M. Moreaud, S. H´

enon, M. Tebib, V. Souchon. BARCHAN: Blob Alignment for Robust CHromatographic ANalysis. Journal of Chromatography A., 2017, http://dx.doi.org/10.1016/j.chroma.2017.01.003

  • L. Duval, A. Pirayre and I. W. Selesnick.

Peaks, baseline and noise separation. Chapter in preparation for Source Separation in Physical-Chemical Sensing, 2018. 30 / 33

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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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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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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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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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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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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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