ICA-Based Fetal Monitoring

Independent Component Analysis (ICA) has numerous applications in biomedical data processing (James & Hesse, 2005; Nait-Ali, 2009; Tanskanen & Viik, 2012). For example, in the last decade lots of contributions have been made in the field of EEG/MEG1 analysis (artifact detection and removal, analysis of event-related brain responses, . . . see e.g. Zarzoso (2009), and the references therein, for a more detailed information). More recently, several researchers have oriented their efforts towards developing ICA-based approaches to the interpretation of the electrocardiogram (ECG) and the information that can be derived from it (Castells et al., 2007). For example, Vetter et al (Vetter et al., 2000) have shown the great potential of ICA in the analysis of the control of the heart by the autonomic nervous system. Arrhythmia detection and, in particular, atrial fibrillation, also constitute possible applications, and several successful examples can be found in the literature (Llinares & Igual, 2009; Rieta et al., 2004).


Introduction
Independent Component Analysis (ICA) has numerous applications in biomedical data processing (James & Hesse, 2005;Nait-Ali, 2009;Tanskanen & Viik, 2012). For example, in the last decade lots of contributions have been made in the field of EEG/MEG 1 analysis (artifact detection and removal, analysis of event-related brain responses, . . . see e.g. Zarzoso (2009), and the references therein, for a more detailed information). More recently, several researchers have oriented their efforts towards developing ICA-based approaches to the interpretation of the electrocardiogram (ECG) and the information that can be derived from it (Castells et al., 2007). For example, Vetter et al (Vetter et al., 2000) have shown the great potential of ICA in the analysis of the control of the heart by the autonomic nervous system. Arrhythmia detection and, in particular, atrial fibrillation, also constitute possible applications, and several successful examples can be found in the literature (Llinares & Igual, 2009;Rieta et al., 2004).
A particularly appealing problem in tococardiography is that of applying ICA-based methods to the extraction of the fetal ECG (fECG) from maternal cutaneous potential measurements. The present clinic standard procedure for recording the fECG consists in inserting a small electrode into the mother's vagina, through the cervix, and directly under the skin of the fetus's scalp (Symonds et al., 2001). The major shortcoming of this technique is its invasiveness. The placement of the fetal scalp electrode presents certain risks to fetal safety, and cases where the mother is infected have been reported as well. Last but not least, this procedure is not suitable for use during all stages of pregnancy, and can only be performed under limited clinical circumstances: e.g. measuring the fECG with a scalp electrode is only possible during labor, as requires a dilated cervix and the rupture of the amniotic membranes. Due to these and other inconveniences, the scalp electrode is almost exclusively reserved for high risk births.
There exists, by contrast, an increasing interest in non-invasive fECG recording techniques (Hasan et al., 2009). These techniques should enable monitoring in stages of pregnancy earlier than labor, i.e. when the membranes protecting the fetus are not broken (antepartum), as well as being comfortable to women, while avoiding the risks of infection or trauma to the fetal scalp. A method for non-invasive fECG monitoring measures the fECG by means of surface electrodes placed on the mother's abdomen. It turns out that the electrical signals recorded by the electrodes are a mixture of several electrophysiological signals and noise. Examples of the former are the maternal electrocardiogram (mECG), the electrohysterogram (EHG, the electrical activity of the uterus) and the fECG. The EHG usually lies below 3 Hz and can be reduced significantly by the use of a simple high-pass filter (Devedeux et al., 1993). The main source of noise is the power line signal (50 -60 Hz, depending on your country), that can be eliminated by a notch filter. The limiting factor in non-invasive fetal electrocardiography is the low amplitude of the fetal electrocardiogram compared to the mECG 2 . As there is a considerable overlap between the frequency bands of the mECG and the fECG (Abboud & Sadeh, 1989), the mECG cannot be suppressed by a simple linear filter. A variety of different approaches have been proposed to extract the fECG from abdominal recordings (Hasan et al., 2009). In this Chapter, we describe and illustrate the specific application of ICA to this exciting problem. Potential readers are assumed very familiar with ICA -if not, they are directed to the classical textbooks Cichocki & Amari (2002); Comon & Jutten (2010); Hyvärinen et al. (2001). The Chapter is organized as follows: in Section 2, we introduce some basic concepts of fetal electrocardiography. Sections 3 and 4 discuss a simple mathematical model of the fECG and its implications for ICA. In Section 5, we review some ICA-based approaches for the fECG extraction problem. Rather than surveying superficially several methods, we shall concentrate on some of the more conceptually appealing concepts. Section 6 introduces a recent and powerful approach, namely to use the mECG as reference for the ICA algorithms. Experiments, using real data, are presented in Section 7. Finally, Section 8 is devoted to the Conclusions.

Basic background in cardiac physiology
The heart consist of four chambers: the right and left atrium and the right and left ventricle.
• In the adult, the atria are collecting chambers that receive the blood from the body and lungs, whereas the ventricles act as pumping chambers that send out the blood to the body tissues and lungs. Blood circulates as follows (Guyton & Hall, 1996): 1. Oxygen-depleted blood flows into the right atrium from the body, via the vena cava.
2. From the right atrium the blood passes into the right ventricle.
3. The right ventricle pumps the blood, through the pulmonary arteries, into the lungs, where carbon dioxide is exchanged for oxygen.
4. The oxygenated blood returns to the heart, via the pulmonary vein, into the left atrium.
5. From the left atrium the blood passes into the left ventricle.
6. The left ventricle pumps the oxygenated blood into all parts of the body through the aorta artery, and the cycle begins again.
• In the fetus, things are slightly different. The fetus receives the oxygen across the placenta and, as a consequence, does not use its lungs until birth. To prevent the blood to be pumped to the lungs, the pulmonary artery is connected to the aorta by a blood vessel called the arterial duct (ductus arteriousus). Thus, after the right ventricle contraction, most blood flows through the duct to the aorta. The fetal heart also has an opening between the right and left atria called the foramen ovale. The foramen ovale allows oxygenated blood to flow from the right atrium to the left atrium, where it gets pumped around the body, again avoiding the lungs. Both the ductus arteriousus and the foramen ovale disappear after birth over the course of a few days or weeks (Abuhamad & Chaoui, 2009).

The fECG
The electrocardiogram (ECG) reflects the electrical activity of the heart as seen from the body surface. The heart generates electrical currents that radiate on all directions and result in electrical potentials. The potential difference between a pair of electrodes placed in predefined points of the surface of the body (cutaneous recordings), visualized as a function of time, is what we call the ECG.
The fetal electrocardiogram (fECG), like that of the adult, consists of a P wave, a QRS complex and a T wave, separated by the PR and ST intervals (see Fig. 1) (Symonds et al., 2001). These waves represent the summation of the electrical potentials within the heart. Contraction (depolarization) of both atria begins at about the middle of the P wave and continues during the PR segment. The QRS complex precedes ventricular contraction: pumping of blood normally begins at the end of the QRS complex and continues to the end of the T wave. Finally, the T wave corresponds to the electrical activity produced when the ventricles are recharging for the next contraction (repolarizing) 3 . Note that the repolarization of the atria is too weak to be detected on the ECG. The fECG cannot be usually detected between 28 and 32 weeks (sometimes 34 weeks) of gestation due to the isolating effect of the vernix caseosa, a sebum that protects the skin of the fetus (Oostendorp et al., 1989a).
The fECG provides useful information about the health and the condition of the fetus (Pardi et al., 1986): for example, the duration of the ST-segment is important in the diagnosis of fetal hypoxia (i.e. a continued lack of oxygen), and it has been also shown that both the QT interval and T-wave changes are predictive of fetal acidemia (Jenkins et al., 2005). The human heart begins beating at around 21 days after conception with frequency about 65 beats per minute (bpm). This frequency increases during the gestation up to 110 -160 bpm before delivery. When it is not within this range, it may be indicative of serious potential health issues: e.g. if the fetal heart rate (FHR) is below 110 bpm for 10 minutes or longer (bradycardia), it is considered a late sign of hypoxia (there is a depression of the heart activity caused by the lack of oxygen), and a fetal emergency (Freeman & Garite, 2003). On the contrary, an FHR that exceeds 160 bpm (tachycardia) may be an early sign of hypoxia (other conditions that increase the FHR include fetal infection, maternal dehydration, medication, et cetera) (Afriat & Kopel, 2008).

Mathematical model of the ECG
In the adult, the cardiac surface potentials can be approximately considered as originated from a current dipole located in the heart (Symonds et al., 2001). Assuming that the body is a volume conductor, homogeneous and infinite, the potential due to a dipole of moment p(t) at a point on the skin specified by the position vector r is given by (Keener & Sneyd, 2009):  Fig. 1. Sketch of a typical single ECG recording. Note that the actual size and polarity of each wave depend on the location of the recording electrodes (Chan, 2008;Keener & Sneyd, 2009).
where σ is the permittivity of the medium. Let e 1 , e 2 , e 3 be orthonormal basis vectors in the real three-dimensional space and let {s 1 (t), s 2 (t), s 3 (t)} be the coordinates of p(t) in this basis, i.e., The body surface potential at r can be then written as linear combination of the signals s i (t): where: It is quite noteworthy that p(t) is allowed to change in orientation and strength as a function of time. For the reference of the reader, the tip of the vector traces out a loop in the space that is called vectorcardiogram (VCG) (Symonds et al., 2001). Different models for p(t) can be found in the literature. For example, , based on McSharry et al. (2003) 4 , have proposed the following differential equations for the dipole vector: where (2π) and ω = 2π f , where f is the beat-to-beat rate. Note also that the equationθ = ω generates periodic signals with the frequency of the heart rate. The problem of estimating the parameters α k i , b k i , ϑ k i of the model is complicated and has been addressed, e.g., in (Clifford et al., 2005;.

The fECG case
Is the previous dipole-based model capable of describing the potential distribution created by the fetal heart at the maternal abdomen? It depends. At early pregnancies, from week 20 until 28 of gestation, the amplitude of the fECG increases and the model seems to be appropriate and fits the observations well (Oostendorp et al., 1989a). Late in pregnancy, however, great care is needed: we have already mentioned that the fECG is in general impossible to measure between 28th to 32th week of gestation due to the isolating effect of the vernix caseosa, the fatty layer that protects the skin of the fetus (Wakai et al., 2000). After the 32th week, the fECG is detected again, but the apparent fetal vectorcardiogram (fVCG), as calculated from the recorded surface potentials, describes almost a straight line (Oldenburg & Macklin, 1977). Hence it no longer corresponds to the activity of the fetal heart vector in an intelligible way. It has been hypothesized that, as the fetus grows, several holes appear in the vernix 5 and current can escape through them (Peters et al., 2005). It turns out that the potential is split up into a contribution of the current dipole and a contribution of the volume currents induced in the vernix. Experiments confirm that, after the 32th week, the fECG recorded at the mother's abdomen can be still described by a model of the type (2) but the fetal source signals s i (t) cannot be longer interpretable as coordinates of a single current dipole. Rather, we should think of eqn. (4) more as a latent variable model. Note that, by definition, the latent variables s i (t) correspond to abstract or hypothetical concepts. Experiments also show that the number of summands n may be different (usually less) from three (Oostendorp et al., 1989b;. See also (Lewis, 2003) and the references therein. 5 The most important hole is probably at the mouth. A second relevant hole can be expected at the base of the umbilical cord.

ICA model
Thus, in view of the previous Section, ECGs seem to satisfy some of the conditions for classical ICA: • The body surface potentials are a linear mixture of several source signals.
• Time delays in signal transmission are negligible.
• It is feasible to have more observations than sources 6 .
Let v 1 (t), . . . , v p (t) be zero-mean signals recorded from electrodes placed on the mother's body, where t ∈ Z is the discrete time. Each measurement signal v i (t) is modelled as a linear combination of r (r ≤ 6) bioelectric source signals, that have similar definitions to the ones in eqns.
(2)-(4), plus noise: The noise represents the signal degradation due, for example, to baseline wander, mains interference, uterine contractions, and so on. Eqn. (5) can be rewritten in matrix form as: where v(t) be the vector whose ith component is v i (t) and so on. Eqn. (5) represents the superposition of the body surface potentials due to the fetus and the maternal cardiac dipole. Note that s(t) can be partitioned into a block of maternal signals and a block of fetal signals, and there exists a corresponding partitioning for A: Thus: The fetal electrocardiogram contributions to the measurement signals can then be obtained by: Observe that (9) allows the estimation of the fetal electrocardiogram contributions to all leads. Similarly, the mothers' own ECG is given by: Note that v M (t) belongs to the column space of A M , which is usually renamed as the mECG subspace. Similarly, the column space of A F will be denoted as the fECG subspace.
Recalling again the discussion in the previous Section, the mECG space can be assumed a three dimensional vector space. However, the dimension of the fECG space is not necessarily equal to three (three is its maximum value) and is subject to changes during the pregnancy .
The main assumption of the ICA model, the independence between sources, leads to some confusion. Even though this assumption is usually adopted (De Lathauwer et al., 2000a), there is no evidence to support it. The source signals can be actually partitioned into groups (a maternal group and a fetal group); components from different groups are statistically independent (i.e., there is a lack of dependency between coincident source activations), but components of the same group may be dependent. This is the concept of Multidimensional Independent Component Analysis (MICA), which is closely related to Independent Subspace Analysis (ISA) (Comon & Jutten, 2010;Hyvärinen et al., 2001). The idea of MICA, originally proposed in Cardoso (1998), has been further developed in Theis (2004) and Vollgraf & Obermayer (2001) among others. MICA actually proceeds in two steps (Cardoso, 1998): firstly, it runs a traditional ICA algorithm 7 and, then, it is determined which outputs of the algorithm are really independent and which should be grouped together (though the latter may not be trivial). Once we have recovered the source signals, we may use them to estimate (9), i.e. the fECG part of the composite signal, depending on the desired application.

Electrode placement
The number of electrodes and the positions at which these should be placed is not standardized. The situation is complex due to the fact that the fetal heart position with respect to the maternal abdomen varies with time and cannot be easily determined. Nevertheless, for late pregnancies, it has been observed that the fECG morphology is almost independent of electrode position (Lewis, 2003). A large number of electrodes (more than 30) arranged in a wide belt around the mother's abdomen, also containing some electrodes at the back, has been used in laboratory experiments (Cicinelli et al., 1994;Oostendorp, 1989;Vrins et al., 2004).

Subspace analysis (whitening)
Whitening is the classical pre-processing for ICA and it is surely well-known to most of the readers of this book (otherwise, see, e.g., Comon & Jutten (2010)). For this reason, we offer an alternative viewpoint and present here whitening as a classical technique of subspace analysis. The idea is to use whitening to estimate the mECG subspace (or, more precisely, to estimate its orthogonal complement 8 ). Then, the mECG can be easily projected out of the dataset. This approach (and its variants) has been conveniently addressed in several papers (see e.g. Callaerts et al. (1990);De Lathauwer et al. (2000b); Kanjilal et al. (1997)). We shall try to focus on the most relevant aspects: consider that we are given q samples v(1), . . . , v(q) of the vector signal v(t). In order to get rid of the maternal electrocardiogram interference, the eigenvalue decomposition of the data covariance matrix: is first computed. Since R v is always symmetric and nonnegative definite, it can be factorized into R v = Q D Q T , where: is the p × p diagonal matrix whose elements are the eigenvalues of R v and Q is the matrix containing the corresponding eigenvectors. If the maternal electrocardiogram is strong enough, it has been shown that the M largest eigenvalues in D are associated with it. Furthermore, it holds that the eigenvalues have usually the following typical relationship: This means that the last (p − n) minor eigenvalues correspond to the noise. Matrices D and Q can be then partitioned into three groups: where D 1 contains those M largest eigenvalues, and the columns of Q 1 are the corresponding eigenvectors; D 2 = diag (λ M+1 · · · λ n ) and Q 2 contains the associated eigenvectors, et cetera. The maternal electrocardiogram can be then eliminated by projecting the data onto the subspace spanned by Q 2 . Specifically, this can be written as: where z(t) is the (p − M) × 1 vector that, in theory, contains no maternal contribution, making the identification of the fetal electrocardiogram a feasible task, even by simple inspection 9 . Of course, the determination of M is an important problem. Seminal works considered M = 3; however, it has been recently argued that from M = 4 to M = 6 may be required in some cases. In practice, experiments suggest finding M empirically from the gap between the eigenvalues 8 The orthogonal complement is the set of all vectors that are orthogonal to the vectors in the mECG space. 9 In fact, under further hypotheses, it holds that s F (t) = D −1/2 2 Q T 2 v(t) and A F = Q 2 D 1/2 2 .

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Independent Component Analysis for Audio and Biosignal Applications of the data covariance matrix. The complete procedure can be accomplished in real time with low computational cost. In any case, the performance of the whitening-based approaches is strongly dependent on the position of the electrodes (Callaerts et al., 1990), which usually becomes a matter of trial and error.

πCA
The combination of the measured signals v i (t) to enhance the periodic structure of the fECG also seems to be a promising idea. The algorithm should combine in power (constructive interference) the fetal components and cancel each other out (destructive interference). The best-known approach is to seek for the linear combination y(t) = ∑ i w i v i (t) = w T v(t) that minimizes the following measure of periodicity: where the time-lag τ is the period of interest (in theory, but not always in practice, τ = the fetal period -see below). This approach has been named Periodic Component Analysis (πCA), and was first used for representing periodic structure in speech (Saul & Allen, 2001). The application of πCA to the fECG extraction problem can be traced back to the paper (Sameni et al., 2008). The minimization of (12) can be easily accomplished in a linear algebra framework. Expanding the right-hand side of (12) gives: where C v (τ) is the sample covariance matrix defined by: Now consider the whitened data: where D 1 and U 1 are the respective eigenvalue and eigenvector matrices of C v (0), i.e., Then we have: where C z (τ) = 1 q ∑ t z(t + τ) z(t) T . Let us define:

ICA-Based Fetal Monitoring
With this new vector, (13) can be rewritten as follows: Then: Proposition 1. By the Rayleigh-Ritz theorem of linear algebra (Bai et al., 2000), the vector weight w minimizing (18) is given by the eigenvector of the matrix C z (τ) with the largest eigenvalue.
Denoting this eigenvector by w max , πCA then outputs: It is interesting to note that πCA is actually a particularization of the well-known Algorithm for Multiple Unknown Signals Extraction (AMUSE) (Tong et al., 1991): by assuming D 2 as the full diagonal eigenvalue matrix of C z (τ), with eigenvalues sorted in descending order, and U 2 being the corresponding eigenvector matrix 10 , one can write the eigendecomposition: Then, y(t) = U T 2 z(t) verifies: with C y (τ) = 1 q ∑ t y(t + τ) y(t) T . Taking eqn. (16) into eqn. (20), we get that both matrices C v (0) and C v (τ) are simultaneously diagonalized by matrix Q = U T 2 D −1/2 1 U T 1 : As can be easily verified, this implies that: i.e. D 2 and Q T are the eigenvalues and eigenvectors, respectively, of the matrix C −1 v (0) C v (τ). Then, Q can be identified by the simultaneous diagonalization of C v (0) and C v (τ). This is the basic idea behind AMUSE.

τ) has non-zero distinct eigenvalues. Then y(t) = Q v(t) is an estimate of the source signals except for the usual scaling and ordering indetermination.
This proposition readily follows from (Tong et al., 1991). For virtually any time-lag τ = 0, AMUSE is able to output the (fetal and maternal) source signals. In addition, the requirement of the sources being mutually uncorrelated is much weaker than the classical ICA condition of mutual independence 11 . The transformation y(t) = Q v(t) = [y 1 (t), . . . , y p (t)] T can be also interpreted as follows: y 1 (t) is the most periodic component with respect to the period of interest τ, y p (t) is the least periodic, and the intermediate components are ranked in descending order of periodicity (Sameni et al., 2008). Of course, this does not mean that, in particular, y 1 (t) is (even approximately) periodic with period τ, neither does it mean that y 1 (t) has physical meaning. Both questions depend on the specific choice of τ.

Period calculation
No specific strategy for selecting τ was provided in (Tong et al., 1991). A natural approach in our context is to set τ to the value of the fetal heart beat period (which, for simplicity, is assumed to be an integer multiple of the sampling interval). However, such an approach is difficult to implement in practice, since the fetal heart beat period has to be estimated on-line, which requires the prior extraction of the fetal R peaks.
As an alternative, (Sameni et al., 2008) reports good results when τ is chosen as the maternal ECG period. In this way, the most periodic components span the mECG subspace. In addition, see (20), the periodic components y 1 (t), . . . , y p (t) happen to be uncorrelated with each other. Hence, the space spanned by the less periodic components is orthogonal to the mECG subspace. It follows that this method is similar in spirit to the whitening-based approaches described in the previous Section.
A more challenging problem arises from the fact that the heart beat period is actually time-dependent. Hence, the period has to be updated on a beat-to-beat basis (see Sameni et al. (2008) for a possible solution).

Extensions
AMUSE (and, subsequently, πCA) suffers from the limitation that the choice of τ is very critical. To overcome this drawback, one powerful approach is to perform the simultaneous diagonalization of more covariance matrices than just two, as is the case with AMUSE. For example, SOBI (Belouchrani et al., 1997) seeks the matrix Q as the joint diagonalizer of a set of covariance matrix C v (τ i ) for a preselected set of time-lags {τ 1 , τ 2 , τ 3 , . . .}. Some steps to investigate the optimal choice of {τ 1 , τ 2 , τ 3 , . . .} in context of the fECG extraction problem have been done by (Tsalaile et al., 2009).

HOS-based approach
It is well-known that, implicitly or explicitly, most ICA methods actually rely on higher-order statistics (HOS) (Comon & Jutten, 2010). Let us briefly review one of the simplest approaches: the maximization of the kurtosis (Hyvärinen et al., 2001). Let z(1), . . . , z(q) be the whitened data. Given an arbitrary vector w, it follows from the central limit theorem that is more Gaussian when it is a sum of the fECG and the interferences than when it is equal to only one of them 12 . In consequence, to find w in such a way that the distribution of y(t) is as far as possible from Gaussian seems to be a sound idea. This general approach to the problem of 'unmixing' mixed signals very common in ICA and is usually referred to as maximisation of non-Gaussianity (Hyvärinen et al., 2001). The simplest measure of non-Gaussianity is the kurtosis, defined by: We maximise the kurtosis of y(t) under the unit-power constraint which avoids the solution y(t) → ∞. It is easily shown that this is equivalent to constraining the norm of w to be the unity. Traditional ICA algorithms, such as FastICA (Hyvärinen, 1999), maximize the kurtosis using standard procedures. As an alternative, we review here the FFD (Fast Fetal Detection) method (Martín-Clemente et al., 2011) which, paradoxically, does not require to compute HOS. Consider first the following theorem, whose proof is straightforward: Theorem 1. Let {x(t), t = 1, . . . , q} be the samples of a generic discrete-time signal. The kurtosis of x(t), defined by is maximized under the unit-power constraint 1 q ∑ q t=1 x 2 (t) = 1 by signals of the form where e k (t) is a discrete-time signal that equals one at t = k and is zero elsewhere.
To explore the vicinity of the maximum √ q e k (t), where k ∈ {1, . . . , q}, we perform a first order Taylor expansion of the kurtosis around this point (see Martín-Clemente et al. (2011) for the details): Hence κ y is maximized when is minimum: i.e., the optimum y(t) is the signal that is as close as possible to √ q e k (t). To determine the best value for the time index k, note that the accuracy of (26) increases as (27) 12 The fECG can be assumed independent from the others as it has a different physical origin.

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Independent Component Analysis for Audio and Biosignal Applications decreases. Consequently, we minimize (27) among all possible values of k. Taking into account that y(t) = w T z(t), a bit of algebra shows that the minimum is obtained simply by setting Consider the following additional interpretation: by construction, y(t) is the signal that is as close as possible to the impulse signal √ q e K (t). If z(t) is periodic, one can prove easily that y(t) is also the best approximation to an impulse train having the same period and centered upon t = K. The ECG resembles an impulse train, but the interferences degrade the measurements. The algorithm restores this property and, as result, restores the signal itself. The method may be then considered as a particular application of the class of waveform-preserving methods for recovering ECG signals.
Finally, to extract sequentially more signals, we can use the procedure described in Chapter 4 of (Cichocki & Amari, 2002). Basically, we remove y(t) from the mixture by z (t) = z(t) − w y(t). Then whitening is applied again to reduce the dimensionality in one unit. The algorithm is repeated until all the desired signals are recovered.

The mECG as reference
Incorporating prior information into ICA may reduce the computational cost while improving the performance of the algorithms. The use of a reference signal has been proposed in Adib & Aboutajdine (2005), using an approach similar to that in Martín-Clemente et al. (2004). To get such a reference, ICA is often applied to data sets that include mECG signals taken at the mother's thoracic region. In this Section, we describe the variant proposed in Camargo-Olivares et al. (2011). The architecture of the proposed system is shown in Figure 2 and each block is described separately next: 1. Pre-processing block: it aims to remove the baseline wander, the electromyografic (EMG) noise, and the power line interference from each signal v i (t). This is usual in most contemporary ECG processing systems. 2. MECG recording: in most previous approaches, the mECG is measured at the chest of the mother. By contrast, in this paper we propose recovering the mECG directly from the mother's abdomen. We face a problem of recovering a signal (the mECG) corrupted by 'noise' (the fECG and others) at, fortunately, a very high signal-to-noise ratio. A state-of-the-art solution is that proposed in . This filter actually generates a synthetic mECG whose morphology and parameters (R-R interval and so on) are calculated from the filter input. The proposed procedure is hence as follows: 1.) Filter each signal taken at the mother's abdomen by the filter described in . 2.) Perform a linear mapping of the filter outputs to a lower dimensional space using whitening to reduce the number of mECG signals under consideration. 3. ICA block: the inputs to ICA are the pre-processed abdominal maternal signals and the mECG estimates (outputs of block 2). 4. Post-processing block: (optional) the fECG is filtered again with the filter described in  to improve the final signal to noise ratio.

First example
Eight real cutaneous potential recordings of a pregnant woman were obtained from the Database for the Identification of Systems (DaISy) 13 . The data, see Fig. 3, consist of eight channels of ECG signals: the first five channels correspond to electrodes placed on the woman's abdominal region. The last three signals correspond to electrodes located on the mother's region. For many years, these recordings have been extensively used as the standard test data of fECG extraction algorithms (e.g., see Zarzoso & Nandi (2001)). Even thought the  13 ftp://ftp.esat.kuleuven.be/pub/SISTA/data/biomedical/ 260 Independent Component Analysis for Audio and Biosignal Applications fECG is much weaker than the mECG, it is slightly visible in the abdominal recordings. We applied the following ICA algorithms to these data: JADE (Cardoso & Souloumiac, 1993), FastICA (Hyvärinen, 1999), FFD , SOBI (Belouchrani et al., 1997) and πCA (Sameni et al., 2008). Apart from whitening, no other pre-processing is used.
All the algorithms succeeded in estimating two fetal source signals. Fig. 4 shows the first one of them, as recovered by each algorithm. All methods produced very similar results. Note that the R wave is perfectly visible, allowing the easy calculation of the fetal heart rate.

Second example
The methods are now tested using experimental data from the Non-invasive Fetal Electrocardiogram database 14 . This public database contains a series of 55 multichannel thoracic and abdominal non-invasive recordings, taken from a single pregnant woman between 21 and 40 weeks of pregnancy. The ones used in this experiment correspond to the 21 week of gestation and are shown in Fig. 5. The first two signals from the top correspond to electrodes located on the mother's thoracic region, and the last three signals correspond to 14 http://physionet.org/pn3/nifecgdb/ 261 ICA-Based Fetal Monitoring electrodes located on the woman's abdomen. The recordings have been pre-processed: the baseline was eliminated using a low-pass filter with cutoff frequency equal to 0.7 Hz, and the powerline interference was attenuated using a notch filter. Fig. 6 shows the source signals estimated by the same ICA algorithms used in the previous example (JADE, FastICA, FFD, SOBI and πCA). Only the maternal source signals can be recognized. We must conclude that, even though ICA is generally reliable, it sometimes fail.

Third example
We now repeat the previous (failed) experiment using the mECG as reference for the FFD methodd, as explained in Section 6. FFD has been chosen as representative of the ICA methods, but results are similar when any other of the algorithms is used. The estimated source signals are depicted in Fig. 7. Unlike in the previous experiment, the fECG is visible in the third plot from the top, and the fetal heart rate can be estimated even though the signal-to-noise ratio is low. Further denoising may necessary using other techniques -see, e.g., ; Vigneron et al. (2003)-but this is beyond the scope of the present Chapter.

Conclusions
This Chapter has presented a review of the state-of-the-art in the use of ICA for the fECG detection problem. A significant improvement in technical support for fetal monitoring has been obtained in the last decades. Compared to alternative techniques (e.g., filtering, average beat subtraction ...), ICA has proven to be a powerful and leading-edge approach. The most remarkable feature of higher-order ICA methods is that they do not seem to be very sensitive to the location of the electrodes. However, it should be pointed out that, even though promising results have been obtained (the fetal heart rate can be almost routinely determined), there is at present a total lack of accuracy in the detection of the smallest waves (P, Q, S and T) of the fECG. Though it is true that in current clinical practice the physician only considers the fetal cardiac rate, further research is therefore needed to improve accuracy of wave detection. The use of prior information (e.g., reference signals, or the knowledge about the fECG waveform) may be the strategy to achieve this goal. The physical interpretation of the estimated source signals also seems to be an exciting field for future work, and the independence of the sources need to be elucidated.

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