beads : filtrage asymétrique de ligne de base (tendance) et débruitage pour des signaux positifs...
TRANSCRIPT
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
1 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
The fast way
I Question: where is the string behind the bead?I Smoothness, sparsity, asymmetry
2 / 21
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
3 / 21
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
4 / 21
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
4 / 21
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
4 / 21
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
4 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Background on backgroundFor analytical chemistry data:
5 / 21
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
6 / 21
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)}.
7 / 21
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)
7 / 21
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
8 / 21
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)
8 / 21
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
8 / 21
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.
8 / 21
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)
].
9 / 21
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
9 / 21
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
9 / 21
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
9 / 21
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.
10 / 21
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)
11 / 21
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.
13 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Evaluation 1 with Gaussian noise
14 / 21
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.
15 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Evaluation 2 with Gaussian noise
16 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Evaluation 3 with Poisson noise
17 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data
Original data
18 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data
2D background
18 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data
Noise
18 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data
Corrected
18 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 1Hyphenated, two-dimensional gas chromatography data
Original data (again!)
18 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data
Original data
19 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data
2D background
19 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data
Noise
19 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data
Corrected
19 / 21
INTRODUCTION MODELING BEADS ALGORITHM EVALUATION AND RESULTS CONCLUSIONS
Two-dimensional chromatography data 2Hyphenated, two-dimensional gas chromatography data
Original data (again!)
19 / 21
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
20 / 21
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...
21 / 21