extraction of fetal cardiac signals from an array of
TRANSCRIPT
Extraction of Fetal Cardiac Signals from an Array ofMaternal Abdominal Recordings
Reza Sameni
Directed by:
Christian JuttenMohammad B. Shamsollahi
GIPSA-lab, INPG, Grenoble, France
Sharif University of Technology, Tehran, Iran
July 7, 2008, Grenoble, France
Overview
1 out of 125 babies are born with heart defects[Mar, 2005, Minino et al., 2007, AHA, 2008]
Early detection of cardiac abnormalities help medicationsand precautions during delivery
Most defects manifest in the heart-rate and morphologyof electrical and magnetic cardiac signals
But, we don’t have direct access to the fetus and the fetalsignals recorded from the mother’s abdomen are veryweak with high interferences
Noninvasive Fetal Cardiac Signal Extraction 2
Problem Definition
Objective: The noninvasive extraction of fetal cardiac signals from an array ofelectrodes recorded from the abdomen of a pregnant woman
x3
x4
xn
x1
x2
x(t) =
x1(t)x2(t)...xn(t)
y(t) =
y1(t)y2(t)...ym(t)
A set of electric or
magnetic recordings
Noisy observation signals Processed signals
Noninvasive Fetal Cardiac Signal Extraction 3
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 4
Background
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 5
Background
The Electrocardiogram
Electrocardiogram (ECG): Overall electrical activity of the heart recordedfrom the body surface
The R-peaks of the ECG are used to extract the heart-beat
R R R R
Noninvasive Fetal Cardiac Signal Extraction 6
Background
What is the Vectorcardiogram?
Vectorcardiogram (VCG): A 3D representation of the ECG,reconstructed from 3 ECG leads
VCG of orthogonal ECG leads give dipole approximations for bodysurface potentials: φ(t) ≈ a1s1(t) + a2s2(t) + a3s3(t)
Noninvasive Fetal Cardiac Signal Extraction 7
State of the Art
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 8
State of the Art
History of Fetal Electrocardiography
1906: M. Cremer observed the fetal ECG using string galvanometers[Cremer, 1906]
1950’s: Improvements in measurement and amplification techniques[Lindsley, 1942]
1970’s: Introduction of signal processing techniques in this domain[Farvet, 1968, Widrow et al., 1975]
1990’s: Application of multichannel signal processing in this domain[van Oosterom, 1986, Kanjilal et al., 1997, Zarzoso et al., 1997]
The problem has since been considered in biomedical and signalprocessing communities....
Noninvasive Fetal Cardiac Signal Extraction 9
State of the Art
Objectives
Fetal heart-rate analysis [Wakai, 2004] Fetal ECG morphology analysis
Noninvasive Fetal Cardiac Signal Extraction 10
State of the Art
Different Measurement Techniques
Echocardiography: sonograghy of the heart using ultrasound[Wladimiroff & Pilu, 1996, Drose, 1998]
Phonocardiography: heart sounds using acoustic microphones[Zuckerwar et al., 1993, Varady et al., 2003]
Magnetocardiography: magnetic fields of cardiac signals[Kariniemi & Hukkinen, 1977, Stinstra, 2001]
Electrocardiography: electric fields of cardiac signals
Invasive: measurable during labor only [Outram et al., 1995, Lai & Shynk, 2002]
Noninvasive: measurable throughout pregnancy [Cremer, 1906, Lindsley, 1942]
Noninvasive Fetal Cardiac Signal Extraction 11
State of the Art
Previous Processing Techniques
Direct Fetal ECG Analysis: Used in early studies; only possible in highsignal-to-noise ratios[Larks, 1962]
Adaptive and Matched Filtering: Partially effective; require referenceelectrodes[Farvet, 1968, Widrow et al., 1975, Park et al., 1992, Outram et al., 1995, Shao et al., 2004, Martens et al., 2007]
Linear Decomposition: Rather effective; decompose the signals ontofixed or data-driven basis functions[Li et al., 1995, Khamene et al., 2000, Akay et al., 1996]
[van Oosterom, 1986, Zarzoso et al., 1997, Cardoso, 1998, De Lathauwer et al., 2000]
[Barros & Cichocki, 2001, Zhang & Yi, 2006, Li & Yi, 2008]
[Vigneron et al., 2003, Jafari & Chambers, 2005]
Nonlinear Decomposition: Rather effective but empirical; require priorinformation [Schreiber & Kaplan, 1996a, Schreiber & Kaplan, 1996b, Richter et al., 1998, Kantz & Schreiber, 1998, Kotas, 2004]
Noninvasive Fetal Cardiac Signal Extraction 12
State of the Art
Limitations and Challenging Issues
Weakness of fetal signals
Strong maternal ECG and respiration interference
Movements of the fetus and electrode positioning
Multiple pregnancies (twin, tripling, ...)
Noninvasive Fetal Cardiac Signal Extraction 13
Methods
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 14
Methods
Overview of the Proposed Methods
Multichannel ECGModeling
Bayesian ECGFiltering Framework
MultidimensionalAspects of the ECG
Periodic ComponentAnalysis
Subspace Decompositionby Deflation
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HHHj
Noninvasive Fetal Cardiac Signal Extraction 15
Methods
Signal Processing Facts About the ECG
As a Signal Source:The heart is a distributed source and not a punctual one
Cardiac signals have infinite dimensions
Minor dimensions are dominated by noise/interferences
As a Waveform:Different ECG leads have different shapes (morphologies)
ECG channels are pseudo-periodic signal synchronous with the heartbeat
These facts should be considered in modeling and processing the ECG
[distributed source 25]
Noninvasive Fetal Cardiac Signal Extraction 16
Methods An ECG Modeling and Denoising Framework
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 17
Methods An ECG Modeling and Denoising Framework
Kalman Filters
Objective: To model the temporal dynamics of the ECG and to use thisdynamics in a Bayesian filtering framework for ECG denoising
The Kalman filter provides the best linear minimum mean square error(MMSE) estimate for xn.
The linear Kalman filter:{
xn+1 = Anxn + Bnwnyn = Hnxn + vn
yn: noisy observation vector xn: desired state vector
KF Equations 21
The Extended Kalman Filter:{
xn+1 = f (xn, wn, n)yn = g(xn, vn, n)
EKF Equations 23
Noninvasive Fetal Cardiac Signal Extraction 18
Methods An ECG Modeling and Denoising Framework
Kalman Filters
Objective: To model the temporal dynamics of the ECG and to use thisdynamics in a Bayesian filtering framework for ECG denoising
The Kalman filter provides the best linear minimum mean square error(MMSE) estimate for xn.
The linear Kalman filter:{
xn+1 = Anxn + Bnwnyn = Hnxn + vn
yn: noisy observation vector xn: desired state vector
KF Equations 21
The Extended Kalman Filter:{
xn+1 = f (xn, wn, n)yn = g(xn, vn, n)
EKF Equations 23
Noninvasive Fetal Cardiac Signal Extraction 18
Methods An ECG Modeling and Denoising Framework
Cardiac Phase Signal
Cardiac phase signal: θ(t) ∈ [−π, π]
The R-peak is considered at θ(t) = 0
θ(t1) = θ(t2)⇐⇒ t1, t2 correspond to identicaldepolarization/repolarization states of the heart
Phase wrapped ECG
Noninvasive Fetal Cardiac Signal Extraction 19
Methods An ECG Modeling and Denoising Framework
Single-Channel ECG Modeling
Modified McSharry’s model: [McSharry et al., 2003, Sameni et al., 2005]
pseudo-periodicity
morphology
θ = ω
s = −∑
i
αiω
b2i
∆θiexp[− (∆θi)2
2b2i
]
ω = 2π × HR and ∆θi = (θ − θi)mod(2π)
Noninvasive Fetal Cardiac Signal Extraction 20
Methods An ECG Modeling and Denoising Framework
Multichannel ECG Modeling8>>>>>>>>>>><>>>>>>>>>>>:
θ = ω pseudo-periodicity
x = −X
i
αxi ω
(bxi )2
∆θxi exp[−
(∆θxi )2
2(bxi )2
] channel 1 morphology
y = −X
i
αyi ω
(byi )2
∆θyi exp[−
(∆θyi )2
2(byi )2
] channel 2 morphology
z = −X
i
αzi ω
(bzi )2
∆θzi exp[−
(∆θzi )2
2(bzi )2
] channel 3 morphology
Synthetic multichannel ECGSynthetic VCG loop
Noninvasive Fetal Cardiac Signal Extraction 21
Methods An ECG Modeling and Denoising Framework
State-State Representation of the ECGProcess equation:
θk+1 = (θk + ωδ)mod(2π)
sk+1 = −N∑
i=1
δαiω
b2i
∆θiexp(−∆θ2
i
2b2i
) + sk + η
ω = 2π × HR, ∆θi = (θk − θi )mod(2π), δ = 1/fs and η is process noise
Observation equation:{φk = θk + uk coarse ECG phaseyk = sk + vk noisy ECG
(Linearized KF Equations 22)
Noninvasive Fetal Cardiac Signal Extraction 22
Methods An ECG Modeling and Denoising Framework
Single-Channel Denoising Scheme
The Kalman filter uses the a priori information from the ECG dynamicsand the noisy observations to estimate the true ECG
dynamic model noisy ECG estimated ECG
8>><>>:
θk+1 = (θk + ωδ)mod(2π)
sk+1 = −NX
i=1δ
αi ω
b2i
∆θi exp(−∆θ2
i2b2
i
) + sk + η
�φk = θk + ukyk = sk + vk
sk
Noninvasive Fetal Cardiac Signal Extraction 23
Methods An ECG Modeling and Denoising Framework
Application: Maternal ECG Cancellation
We can remove maternal ECG contaminants using the Kalman filter andKalman smoother 1
(1) Original noisy fetal ECG
1This data has been taken from the DaISy database [De Moor, 1997]
Noninvasive Fetal Cardiac Signal Extraction 24
Methods An ECG Modeling and Denoising Framework
Application: Maternal ECG Cancellation
We can remove maternal ECG contaminants using the Kalman filter andKalman smoother 1
(1) Original noisy fetal ECG
1This data has been taken from the DaISy database [De Moor, 1997]
Noninvasive Fetal Cardiac Signal Extraction 24
Methods An ECG Modeling and Denoising Framework
Application: Maternal ECG Cancellation
We can remove maternal ECG contaminants using the Kalman filter andKalman smoother 1
(1) Original noisy fetal ECG (2) EKS of the maternal ECG
1This data has been taken from the DaISy database [De Moor, 1997]
Noninvasive Fetal Cardiac Signal Extraction 24
Methods An ECG Modeling and Denoising Framework
Application: Maternal ECG Cancellation
We can remove maternal ECG contaminants using the Kalman filter andKalman smoother 1
(1) Original noisy fetal ECG (2) EKS of the maternal ECG
(1) - (2) Residual fetal signal
1This data has been taken from the DaISy database [De Moor, 1997]
Noninvasive Fetal Cardiac Signal Extraction 24
Methods An ECG Modeling and Denoising Framework
Application: Maternal ECG Cancellation
We can remove maternal ECG contaminants using the Kalman filter andKalman smoother 1
(1) Original noisy fetal ECG (2) EKS of the maternal ECG
(1) - (2) Residual fetal signal Fetal signal after post-processing
1This data has been taken from the DaISy database [De Moor, 1997]
Noninvasive Fetal Cardiac Signal Extraction 24
Methods An ECG Modeling and Denoising Framework
Confidence Intervals
Several fetal ECG beats before and after the post-processing EKS,together with the ±σ and ±3σ confidence envelopes
Noninvasive Fetal Cardiac Signal Extraction 25
Methods An ECG Modeling and Denoising Framework
Summary of Findings of Part I
We can generate realsitic multichannel maternal/fetal ECG signals
The Kalman filter based on this model outperforms classical filters
Applications:
ECG enhancement
ECG cancellation
Limitation: During maternal/fetal PQRST-complex overlap, multichannelprocessing is required
Noninvasive Fetal Cardiac Signal Extraction 26
Methods Linear Multichannel ECG Processing
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 27
Methods Linear Multichannel ECG Processing
Multichannel ECGObjective: To find linear transforms of the form y(t) = Bx(t) with propertiessuch as: uncorrelatedness, independence, periodicity, etc.
The DaISy dataset [De Moor, 1997]
[Kanjilal et al., 1997, Zarzoso et al., 1997, Cardoso, 1998, De Lathauwer et al., 2000, Barros & Cichocki, 2001, Zhang & Yi, 2006, Li & Yi, 2008]
Noninvasive Fetal Cardiac Signal Extraction 28
Methods Linear Multichannel ECG Processing
Issues of Interest
Why do linear transforms extract multiple ECG components?
What do these components correspond to?
Can we relate these components with multipole expansions?
A typical segment of independent componentsextracted from ECG signals
Noninvasive Fetal Cardiac Signal Extraction 29
Methods Linear Multichannel ECG Processing
Interpretation of the Extracted Components
3D VCG x − y plane
x − z plane y − z plane
Scatter plot of a VCG and column vectors of a mixing matrix estimated by JADE
Noninvasive Fetal Cardiac Signal Extraction 30
Methods Linear Multichannel ECG Processing
Summary of Findings of Part II
Dimensionality of the ECG; theoretical and practical
Multidimensional properties of cardiac signals vs. VCG loops
Impact and necessity of preprocessing for fetal ECG extraction
Limitation: ICA is not ideal for ECG decomposition; a transform thataccounts for periodicity is more appropriate
Noninvasive Fetal Cardiac Signal Extraction 31
Methods Periodic Component Analysis
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 32
Methods Periodic Component Analysis
Periodic Component Analysis (πCA)
Objective: To find a special linear transform y(t) = Bx(t) fordecomposing ECG signals into periodic components
The components should be ranked according to their relevance
Method: Gather measures of ECG pseudo-periodicity in C1 & C2, andfind a B to diagonalize them:
BC1BT = I , BC2BT = Λ
→ Generalized Eigenvalue Decomposition (GEVD) of (C1, C2)
The method is related to algebraic ICA methods, such asAMUSE and SOBI → how to choose the time-lags?
Noninvasive Fetal Cardiac Signal Extraction 33
Methods Periodic Component Analysis
Periodic Component Analysis Algorithm
Algorithm:1 Detect the R-peaks of the ECG of interest (a priori information)
2 Calculate the ECG phase signal θ(t)
3 Calculate the time-varying lag τt = min{τ |φ(t + τ) = φ(t), τ > 0}
4 Calculate Cx = Et{x(t)x(t)T} and Cx = Et{x(t + τt)x(t)T}
5 B ← GEVD(Cx , Cx)
Noninvasive Fetal Cardiac Signal Extraction 34
Methods Periodic Component Analysis
Example I: Fetal ECG Example
The DaISy dataset [De Moor, 1997]
Components extracted with JADE
Noninvasive Fetal Cardiac Signal Extraction 35
Methods Periodic Component Analysis
Example I: Fetal ECG Example
The DaISy dataset [De Moor, 1997] Components extracted with JADE
Noninvasive Fetal Cardiac Signal Extraction 35
Methods Periodic Component Analysis
Example I: Fetal ECG Example (continued)
Extracted periodic components, withmaternal ECG beat synchronization
Extracted periodic components, withfetal ECG beat synchronization
Noninvasive Fetal Cardiac Signal Extraction 36
Methods Periodic Component Analysis
Example I: Fetal ECG Example (continued)
Extracted periodic components, withmaternal ECG beat synchronization
Extracted periodic components, withfetal ECG beat synchronization
Noninvasive Fetal Cardiac Signal Extraction 36
Methods Periodic Component Analysis
Example II: Twin Fetal MCG
Typical MCG channels
Noninvasive Fetal Cardiac Signal Extraction 37
Methods Periodic Component Analysis
Example II: Twin Fetal MCG (continued)
Algorithm:1 Preprocess the data
2 Removing maternal MCG using πCA
3 Finding coarse estimates of fetal MCGs using ICA
4 Refind maternal MCG using maternal/fetal πCA
→ Cx = Cmx − (C f1
x + C f2x )
5 Refind fetal ECG through post-processing
Noninvasive Fetal Cardiac Signal Extraction 38
Methods Periodic Component Analysis
Example II: Twin Fetal MCG (continued)
A segment of extracted components
Noninvasive Fetal Cardiac Signal Extraction 39
Methods Periodic Component Analysis
Summary of Findings of Part III
πCA finds pseudo-periodic signals ranked in order of relevance
It uses GEVD that is a fast and accurate algorithm
Limitation: Requires the R-peaks as prior information
Noninvasive Fetal Cardiac Signal Extraction 40
Methods Subspace Decomposition by Deflation
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 41
Methods Subspace Decomposition by Deflation
Motivation
Objective: To decompose (degenerate) mixtures of signal andnoise/artifact, without prior knowledge of the subspace dimensions andwithout reducing the data dimensions
x(t) = xs(t) + xn(t)
y(t) = Bx(t) = B[xs(t) + xn(t)] = ys(t) + yn(t)
Full-rank noise is limiting for linear methods and can be amplified incomponents extracted by ICA
Solution: Break the linearity by combining single-channel denoising andmultichannel decomposition
Noninvasive Fetal Cardiac Signal Extraction 42
Methods Subspace Decomposition by Deflation
Assumptions
1 The desired signals in different channels are dependent→ Processing gain is achieved through multichannel analysis
2 We have some a priori information about the desired signals→ The desired and undesired parts can be roughly separated usinglinear/nonlinear filters
Noninvasive Fetal Cardiac Signal Extraction 43
Methods Subspace Decomposition by Deflation
Subspace Decomposition by Deflation
/
Iteration stopping criterion
/LinearDecomposition
LinearRecomposition
/Linear/Nonlinear DenoisingL L
N-LInput array Output array
B B-1
The iterative subspace decomposition procedure
Linear decomposition: based on non-stationarity, spectral contrast,periodicity, etc. → Generalized eigenvalue decomposition (GEVD)
Denoising: based on a priori information
Applications: EEG, EMG, MCG denoising, or etc.
Noninvasive Fetal Cardiac Signal Extraction 44
Methods Subspace Decomposition by Deflation
Application in Maternal ECG Cancellation
πCA
mECG cancellationusing Kalman filter
inverseπCA
first L components
recursion stopping criterion
array recordings contaminated with
maternal ECGarray recordings without maternal ECGlast N-L
components
periodicity measurematernal ECG phase
The iterative procedure for maternal ECG cancellation
Noninvasive Fetal Cardiac Signal Extraction 45
Methods Subspace Decomposition by Deflation
Example I: Maternal ECG Cancellation fromDegenerate Mixture 1
Original1 This dataset has been recorded by Dr. Evelyn Huhn and provided by Dr. Raphael Schneider
Noninvasive Fetal Cardiac Signal Extraction 46
Methods Subspace Decomposition by Deflation
Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)
Iteration 1
Noninvasive Fetal Cardiac Signal Extraction 47
Methods Subspace Decomposition by Deflation
Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)
Iteration 2
Noninvasive Fetal Cardiac Signal Extraction 48
Methods Subspace Decomposition by Deflation
Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)
Iteration 3
Noninvasive Fetal Cardiac Signal Extraction 49
Methods Subspace Decomposition by Deflation
Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)
Original (blue) and denoised (red)
Noninvasive Fetal Cardiac Signal Extraction 50
Methods Subspace Decomposition by Deflation
Example II: Invasive vs. Non-Invasive Fetal ECGExtraction
1 Invasive fetal scalp ECG recorded during labor 1
1This data has been recorded by Dr. A. Wolfberg and provided (confidentially) by Dr. G.D Clifford
Noninvasive Fetal Cardiac Signal Extraction 51
Methods Subspace Decomposition by Deflation
Example II: Typical Results
Fetal ECG recorded invasively from a scalp lead ECG
Fetal ECG extracted non-invasively from 22 abdominal leads
Noninvasive Fetal Cardiac Signal Extraction 52
Methods Subspace Decomposition by Deflation
Example II: Typical Results
Fetal ECG recorded invasively from a scalp lead ECG
Fetal ECG extracted non-invasively from 22 abdominal leads(with post-processing)
[Ensemble Averages 24]
Noninvasive Fetal Cardiac Signal Extraction 53
Methods Subspace Decomposition by Deflation
Summary of Findings of Part IV
Decomposition of (degenerate) mixtures of signal/interference subspaceswithout dimension reduction
Limitation: Requires prior information; not applicable to totally blindscenarios
Noninvasive Fetal Cardiac Signal Extraction 54
Conclusion and Perspectives
Outline
1 Background
2 State of the Art
3 MethodsAn ECG Modeling and Denoising FrameworkLinear Multichannel ECG ProcessingPeriodic Component AnalysisSubspace Decomposition by Deflation
4 Conclusion and Perspectives
Noninvasive Fetal Cardiac Signal Extraction 55
Conclusion and Perspectives
Summary
The main developments of this study include:
Realistic multichannel ECG modeling
Bayesian framework for ECG denoising
Study of multidimensional aspects of ECG
Periodic Component Analysis
Subspace decomposition by deflation
Much improvement was achieved using pseudo-periodicity priors
The performance is limited when the priors do not hold→ highly pathologic cases
Noninvasive Fetal Cardiac Signal Extraction 56
Conclusion and Perspectives
Perspectives
Clinical:Clinical verification of the proposed methods
Study of pathologic cases
Theoretical:Calculation of theoretical performance bounds for ECG processing
Theoretical aspects of the deflation method: convergence and stability
Experimental:Fetal ECG tracking for continuous monitoring
Development of fetal monitoring systems
Noninvasive Fetal Cardiac Signal Extraction 57
Conclusion and Perspectives
Byproducts of the Developed Methods
General functions for preprocessing and power-line cancellation
An open-source ECG toolbox (OSET), available at: http://ecg.sharif.ir/
Removing ECG artifacts from other biosignals: EEG, EMG, etc.
Noninvasive Fetal Cardiac Signal Extraction 58
Conclusion and Perspectives
Related Publications
Multichannel ECG Modeling:1 conference papers: CinC’081 journal paper: EURASIP JASP’07
Bayesian ECG Filtering Framework:3 conference papers: EMBS’05, CinC’05, CinC’062 journal paper: IEEE TBME’07, IOP PM’08
Multidimensional Aspects of the ECG:3 conference papers: MaxEnt’06, ISSPIT’06, ICArn’06
Periodic Component Analysis:2 journal paper: IEEE TBME’08, IEEE TBME’09
Subspace Decomposition by Deflation:1 conference paper: EUSIPCO’082 patents: [in progress]
Noninvasive Fetal Cardiac Signal Extraction 59
Conclusion and Perspectives
Conclusion....
Thanks for your attention!
Noninvasive Fetal Cardiac Signal Extraction 60
Conclusion and Perspectives
Conclusion....
Thanks for your attention!
Noninvasive Fetal Cardiac Signal Extraction 60
Conclusion and Perspectives
Classical Kalman Filter Equations
1) Time propagation:
x−n+1 = AnxnP−n+1 = AnPnAT
n + BnQnBTn
2) Kalman gain:
Kn = P−n HTn [HnP−n HT
n + Rn]−1
3) Measurement propagation:
xn = x−n + Kn[yn − Hnx−n ]Pn = P−n − KnHnP−n
where xn is the estimated state vector, Qn = E{wnwTn } is the model noise
covariance matrix, and Rn = E{vnvTn } is the observation noise covariance
matrix.
Noninvasive Fetal Cardiac Signal Extraction 61
Conclusion and Perspectives
Linearized ECG Equations
Defining: �θk+1 = F0(θk , ω, k)zk+1 = F1(θk , zk , ω, αi , θi , bi , η, k),
then:∂F0
∂zk= 0
∂F0
∂θk=
∂F1
∂zk= 1
∂F1
∂θk= −X
i∈{P,Q,R,S,T}δαiω
b2i
[1−∆θ2
i
b2i
]exp(−∆θ2
i
2b2i
)
Similarly, the linearization of the state-space equations with respect to the process noisecomponents yields:
∂F0
∂ω= δ
∂F1
∂η= 1 i ∈ {P, Q, R, S, T}
∂F0
∂αi=
∂F0
∂bi=
∂F0
∂θi=
∂F0
∂η= 0
∂F1
∂αi= −δ
ω∆θi
b2i
exp(−∆θ2
i
2b2i
)∂F1
∂bi= 2δ
αiω∆θi
b3i
[1−∆θ2
i
2b2i
]exp(−∆θ2
i
2b2i
)
∂F1
∂θi= δ
αiω
b2i
[1−∆θ2
i
b2i
]exp(−∆θ2
i
2b2i
)∂F1
∂ω= −
Xi
δαi∆θi
b2i
exp(−∆θ2
i
2b2i
)
Noninvasive Fetal Cardiac Signal Extraction 62
Conclusion and Perspectives
Extended Kalman Filter Equations
{xn+1 = f (xn, wn, n)yn = g(xn, vn, n){
xn+1 ≈ f (xn, wn, n) + An(xn − xn) + Fn(wn − wn)yn ≈ g(xn, vn, n) + Cn(xn − xn) + Gn(vn − vn)
An =∂f (x, wn, n)
∂x
∣∣∣x=xn
Fn =∂f (xn, w, n)
∂w
∣∣∣w=wn
Cn =∂g(x, vn, n)
∂x
∣∣∣x=xn
Gn =∂g(xn, v, n)
∂v
∣∣∣v=vn
x−n+1 = f (x+n , w, n)
∣∣∣w=wn
rn = yn − g(x−n , v, n)∣∣∣v=vn
Kn = P−n CTn [CnP−n CT
n + Rn]−1 x+
n = x−n + KnrnP−n+1 = AnP+
n ATn + Qn P+
n = P−n − KnCnP−n
Noninvasive Fetal Cardiac Signal Extraction 63
Conclusion and Perspectives
Example II: Ensemble Average of the Results
Ensemble average of the fetal scalp leadECG
Ensemble average of the processed fetalECG
Noninvasive Fetal Cardiac Signal Extraction 64
Conclusion and Perspectives
The Heart as a Distributed Signal Source
The ECG is a projection of the distributed cardiac source onto a distancefunction
O
ρ(x, t)
x1 x2
x3
xn
xr reference electrode
x′ dx′Φ(x, t) =
∫ρ(x′, t)|x− x′|
dx′
φi(t) = Φ(xi , t)− Φ(xr , t) =∫ρ(x′, t)D(xi , x′)dx′
Multipole expansion: φi(t) = limL→∞
L∑m=0
m∑n=−m
aimnsi
mn(t)
Noninvasive Fetal Cardiac Signal Extraction 65
Conclusion and Perspectives
References I
(2005).Congenital Heart Defects.March of Dimes.
(2008).Congenital Heart Defects in Children Fact Sheet.American Heart Association.
Akay, M., Akay, M. & Mulder, E. (1996).IEEE Eng. Med. Biol. Mag. 15, 64–67.
Barros, A. K. & Cichocki, A. (2001).Neural Comput 13, 1995–2003.
Cardoso, J.-F. (1998).In Proceedings of the IEEE International Conference on Acoustics, Speech, andSignal Processing (ICASSP ’98) vol. 4, pp. 1941–1944,.
Noninvasive Fetal Cardiac Signal Extraction 66
Conclusion and Perspectives
References II
Cremer, M. (1906).Munchener Medizinische Wochenschrift 53, 811–813.
De Lathauwer, L., De Moor, B. & J., V. (2000).IEEE Trans. Biomed. Eng. 47, 567–572.
De Moor, B. (1997).Database for the Identification of Systems (DaISy).
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