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Extraction of Fetal Cardiac Signals from an Array of Maternal Abdominal Recordings Reza Sameni Directed by: Christian Jutten Mohammad B. Shamsollahi GIPSA-lab, INPG, Grenoble, France Sharif University of Technology, Tehran, Iran July 7, 2008, Grenoble, France

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Page 1: Extraction of Fetal Cardiac Signals from an Array of

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

Page 2: Extraction of Fetal Cardiac Signals from an Array of

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

Page 3: Extraction of Fetal Cardiac Signals from an Array of

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

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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

Page 5: Extraction of Fetal Cardiac Signals from an Array of

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

Page 6: Extraction of Fetal Cardiac Signals from an Array of

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

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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

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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

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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

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State of the Art

Objectives

Fetal heart-rate analysis [Wakai, 2004] Fetal ECG morphology analysis

Noninvasive Fetal Cardiac Signal Extraction 10

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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

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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

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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

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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

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Methods

Overview of the Proposed Methods

Multichannel ECGModeling

Bayesian ECGFiltering Framework

MultidimensionalAspects of the ECG

Periodic ComponentAnalysis

Subspace Decompositionby Deflation

? ?

�����

����

HHHHH

HHHj

Noninvasive Fetal Cardiac Signal Extraction 15

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Methods Periodic Component Analysis

Example II: Twin Fetal MCG

Typical MCG channels

Noninvasive Fetal Cardiac Signal Extraction 37

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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

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Methods Periodic Component Analysis

Example II: Twin Fetal MCG (continued)

A segment of extracted components

Noninvasive Fetal Cardiac Signal Extraction 39

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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

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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

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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

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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

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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

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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

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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

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Methods Subspace Decomposition by Deflation

Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)

Iteration 1

Noninvasive Fetal Cardiac Signal Extraction 47

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Methods Subspace Decomposition by Deflation

Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)

Iteration 2

Noninvasive Fetal Cardiac Signal Extraction 48

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Methods Subspace Decomposition by Deflation

Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)

Iteration 3

Noninvasive Fetal Cardiac Signal Extraction 49

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Methods Subspace Decomposition by Deflation

Example I: Maternal ECG Cancellation fromDegenerate Mixture (continued)

Original (blue) and denoised (red)

Noninvasive Fetal Cardiac Signal Extraction 50

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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

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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

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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

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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

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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

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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

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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

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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

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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]

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Conclusion and Perspectives

Conclusion....

Thanks for your attention!

Noninvasive Fetal Cardiac Signal Extraction 60

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Conclusion and Perspectives

Conclusion....

Thanks for your attention!

Noninvasive Fetal Cardiac Signal Extraction 60

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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

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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

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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

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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

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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

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Conclusion and Perspectives

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