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COMPARISON OF ALGORITHMS FOR FETAL ECG EXTRACTION By HEMANTH PEDDANENI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Hemanth Peddaneni ACKNOWLEDGMENTS Any accomplishment is possible only when there is a motivation and a guiding force. I am highly grateful to my advisor, Dr. Jose Principe, for providing both of these. His suggestions and encouragement have been of immense help for me in completing this thesis. I would like to express my gratitude to Dr. Deniz Erdogmus and Dr. Yadunandana Rao for the many discussions we had, without which the work would not have been completed. I would like to thank Ms. Dorothee Marossero for providing me the ECG data sets and for her help in evaluating the performance of the algorithms. I would like to thank Dr. Michael Nechyba and Dr. Fred Taylor for serving on my committee. I am thankful to my friends at CNEL, Anant, Jianwu and Can, for providing a friendly atmosphere and lending their hand whenever I had some problems. Finally I would like to thank my family members for their support and love, which have been of utmost importance for me to complete the present work. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ................................................... vii LIST OF FIGURES ................................ ........... ............................ viii A B S T R A C T ........................................................................................................ ............ ix CHAPTER 1 IN TR O D U C T IO N ........ .. ......................................... ..........................................1. P problem D description .. ................................................................................ .. 1... P a st R e search ................................................................................................... 3 G o als of th e T h esis ... ... ........................................... ....................... .. ........... .... 2 DESCRIPTION OF ALGORITHMS STUDIED...................................................9... B SS U sing M eRM aId A lgorithm ........................................................ ...............9... B SS U sing M inim ization of CS QM I............................................. ............... 14 BSS Using Generalized Eigenvalue Decomposition....................................... 17 3 R E S U L T S ................................................................................................................. .. 2 3 D ata Collection .............. . .................................................. 24 Performance of the Algorithms on Normal ECG Data.....................................25 M eR M aId A lgorithm ................................... .................................................. 2 5 Cauchy Schwartz based Quadratic Mutual Information ................................28 Generalized Eigenvalue Decomposition Algorithm.......................................31 Performance of the Algorithms on the Data during Contractions ......................34 Comparison of the Performance of the Algorithms.........................................39 4 CONCLUSIONS AND FUTURE WORK............................................................42 C o n c lu sio n s ....................................................................... ..................................... 4 2 Scope for Future W ork .......................................... ......................... ................ 43 L IST O F R E FE R E N C E S ... ........................................................................ ................ 45 BIO GR APH ICAL SK ETCH .................................................................... ................ 48 LIST OF TABLES Table page 21. Summary of procedures for BSS using Generalized Eigenvalue Decomposition (G ED ) for fetal E C G extraction ........................................................... ................ 21 31. Comparison of performance of the BSS algorithms for normal ECG....................41 32. Comparison of performance of the BSS algorithms for ECG data during c o n tra ctio n s .............................................................................................................. 4 1 LIST OF FIGURES Figure page 31: Position of the electrodes on the pregnant woman.............................. ................ 24 32: Eight channels of normal ECG data. Only 2000 samples of the ECG data are shown in the figure......................... ........... .........................27 33: Output signals for the MeRMaId algorithm. The nature of the eight signals is m mentioned to the left. ............. ................ .............................................. 28 34: Plot of the signal to interference ratio versus the iterations ................................29 3 5 : L earn in g cu rv e .......................................................................................................... 3 0 36: Output signals for CS based QM I M inimization ............................... ................ 31 37: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition m ethod using nonstationary assum ption............................................. ................ 32 38: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition m ethod using nonw hite assum ption ................................................... ................ 33 39: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nongaussian assumption..................................................... 34 310: One channel of ECG data during uterine contractions. The figure shown at the bottom zooms on the portion of the signal containing contractions......................35 311: Output signals for the M eRM aId algorithm ....................................... ................ 36 312: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition m ethod using nonstationary assum ption............................................. ................ 37 313: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition m ethod using nonw hite assum ption ................................................... ................ 38 314:2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition m ethod using nongaussian assumption..................................................... 39 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COMPARISON OF ALGORITHMS FOR FETAL ECG EXTRACTION By Hemanth Peddaneni December 2004 Chair: Jose Principe Major Department: Electrical and Computer Engineering The thesis addresses the extraction of the fetal ECG signal from the noninvasive ECG measurements of a pregnant woman. This is an ideal situation for independent component analysis because the assumption that the two sources in the measurements, fetal ECG and mother ECG, are independent is verified at the short time scale used for separation. Three algorithms, Blind Source Separation using Minimum Renyi's Mutual Information, using Generalized Eigenvalue Decomposition and using Cauchy Schwartz Quadratic Mutual Information Minimization, have been used to get the fetal ECG under normal conditions and during the uterine contractions of the mother. Minimum Renyi's Mutual Information method uses whitening of the data and as such it is less suitable to extract the fetal ECG during contractions. The Generalized Eigenvalue Decomposition method turned out to give superior quality of the fetal ECG during the contractions. The third method, Cauchy Schwartz Quadratic Mutual Information Minimization, gave promising results for lower dimensional normal ECG data but failed to work for the higher dimensional ECG data during contractions. The quality of the extracted fetal ECG is compared using a criterion called Trust Factor (which varies from 0 to 1, 1 corresponds to the best quality). These results help in determining which algorithm is best suited for finding the fetal heart rate since the QRS complex is preserved in all three algorithms. CHAPTER 1 INTRODUCTION Fetal electrocardiogram (fetal ECG) extraction is an interesting as well as a difficult problem in signal processing. This forms one important application of Independent Component Analysis (ICA) or Blind Source Separation (BSS), where one wants to separate mixtures of sources with very little prior information. In this chapter we will first describe the problem description followed by the past research that has been done in this area and then finally about the goal of this thesis. Problem Description Fetal electrocardiogram monitoring is a technique for obtaining important information about the condition of the fetus during pregnancy, by measuring the electrical signals generated by the fetal heart as measured from multichannel electrodes placed on the mother's body surface. This method of recording the fetal ECG from the mother's body, without direct contact with the fetus (which is highly desirable) is called noninvasive method. However, in this method of recording, the fetal ECG signals have a very low power relative to that of the maternal ECG. In addition, there will be several sources of interference, which include intrinsic noise from a recorder, noise from electrodeskin contact, baseline drift (DC shift), 50/60 Hz noise etc. The situation is far worse during the uterine contractions of the mother. During these contractions, the ECG recordings will be corrupted by other electrophysiological signals called uterine electromyogram (EMG) or electrohysterogram (EHG), which are due to the uterine muscle rather than due to the heart. The response of the fetal heart to the uterine contractions is an important indicator of the fetal health. As such a need arises to effectively monitoring the fetal ECG during the uterine contractions. But monitoring the fetal ECG during these contractions is a difficult task because of very poor SNR. The nature of these contractions is shown in figure 37 of chapter 3. 4000 1 T 60 80 100 Figure 11: Components of the ECG waveform The nature of the fetal ECG is similar to that of mother ECG signal and is shown in figure 11. The ECG waveform is also called PQRST wave. The first waveform in the ECGthe P waveis due to the atrial contraction. The next waveformQRS complexis due to the ventricular contraction. The final waveform is the T wave which occurs as the heart prepares for the next heartbeat. The location of P, Q, R, S and T components are indicated in the figure 11. The three main characteristics that need to be obtained from the fetal ECG extraction for useful diagnosis include Fetal heart rate Amplitude of the different waves Duration of the waves. But because of the noninvasive nature of measurement of the fetal ECG, most of the signal processing algorithms detect only the R waves and the P and T waves will usually remain hidden. Also fetal ECG extraction problem is not easily solved by conventional filtering techniques. Linear filtering in the Fourier domain fails since the spectral content of all the three components, maternal ECG, fetal ECG and noise are rather similar and overlap. Past Research The problem of fetal ECG extraction was tackled more than 30 years ago by means of now conventional adaptive noise canceling techniques. Widrow and Stearns [1] used a linear adaptive filter framework to cancel the mother ECG and obtain the fetal ECG. They used two sets of electrodes, one set placed on the abdomen of the mother and the other placed on the chest of the mother. The electrodes placed on the abdomen pick up both the fetal ECG and mother ECG (serve as primary inputs), whereas the electrodes placed on the chest pick up only the mother ECG (serve as reference inputs). So by having the signals from the electrodes placed on the abdomen as desired and the signals from the electrodes placed on the limbs as input to the adaptive filter, the error signal can be made to represent the extracted fetal ECG. This method although providing a solution, is not robust enough to be used for clinical practice. First one needs to have more electrodes to measure the signals. Second, if the amplitude of the background noise is greater than the fetal heartbeat, as is the case generally during the uterine contractions, the resulting error signal will not contain the fetal ECG. This method fails to extract the fetal ECG when both the mother and fetal ECG overlap. Further more, if there is leakage of the fetal ECG into the recordings of the reference input (mother ECG), the quality of the extracted fetal ECG will be extremely poor. The positioning of the abdominal electrodes is extremely critical to get a good fetal ECG. They will have to be placed at places on the mother where maximum amount of fetal ECG is picked up (again this leads to the issue of where to place the electrodes). Also in this framework, only second order statistics are used and higher order statistics are ignored. Camps et al. [2] also used an adaptive noise canceling technique to extract the fetal ECG but with the linear adaptive filter replaced with a time delay neural network (TDNN). This TDNN is nothing but an artificial neural network (like a multilayered perception) with the synaptic weights replaced by FIR filters. This is done to provide highly nonlinear dynamic capabilities to the fetal ECG recovery model. They trained the network using temporal backpropagation. Again this method suffers from the same difficulties mentioned earlier. In addition training this neural network is not easy and is extremely computationally expensive. Care should be taken to decide on proper initialization of the weights, number of hidden neurons, taps per neuron in a layer and learning rate. Mochimaru and Fujimoto [3] used wavelet based methods to detect the fetal ECG. They used multiresolution analysis (MRA) to remove the large baseline fluctuations in the signal as well as to remove the noise. MRA was performed on the raw ECG data up to the 12th level using Daubechies20 wavelets. The 12th order approximation function consists of slow variations of the signal. This was subtracted from the original raw signal to get a signal free from baseline fluctuations. To remove the noise, they applied wavelet transform based denoising of the detrended data by multiresolution analysis up to the 12th level using Coiflet24 wavelets. Noise removal was accomplished by thresholding the wavelet coefficients at each level. Weighted standard deviation of the wavelet coefficients at each resolution level were used as the thresholds at each resolution level. The fetal ECG was monitored by calculating the Lipschitz exponent [4]. The main problem with this method is its inability to locate the fetal ECG if it is obscured by the mother ECG. Since this happens two or three times in a 10 seconds period, it can be a major drawback. Also there is a need to set the thresholds on the wavelet coefficients dynamically during denoising, since the noise content is more during the uterine contractions. Again the performance of this type of denoising during the contractions may not be optimum since thresholding of the wavelet coefficients may result in removing the fetal ECG component altogether in the original signal. Vigneron et al. [5] applied blind source separation methods for fetal ECG extraction by (i) exploiting the nonstationarity of fetal ECG and (ii) implementing post denoising using wavelets. The blind source separation algorithm proposed by Pham and Cardoso [6] was made use here. This algorithm minimizes the Gaussian mutual information, defined as the ordinary mutual information with respect to some Gaussian random vectors with the same covariances structure as the random vector of interest, to get the independent components in the signal. Then the wavelet transform was used to remove the baseline drift in the extracted noisy fetal ECG signal. After that, the PQRST of the fetal ECG is amplified by using a nonlinear filtering technique. After PQRST amplification, a wavelet denoising is applied to the fetal ECG signal with the following parameters: biorthogonal wavelet type, level 6 decomposition and the data adaptive threshold selection rule 'SureShrink' of MATLAB wavelet toolbox. The problem with this method is that there are too many post processing stages, as such lot of parameters ( like choice of nonlinear filters, wavelet for baseline removal as well as denoising, level of decomposition etc.) need to be determined empirically. Work has also been done to extract the fetal ECG using genetic algorithms [7]. The genetic algorithm approach for fetal ECG extraction, proposed by Horner et al. [8], is based on subtracting a pure maternal ECG from an abdominal signal containing fetal and maternal ECG signals. Subtraction via a Genetic Algorithm is supposed to be near optimal rather than a straight subtraction. The issue with this method is the need to get the maternal ECG signals whose shape is similar to the maternal ECG present in the abdominal recordings (which contain fetal ECG). So it needs to be determined exactly, where the electrodes need to be placed to pick up the maternal ECG alone. Kam and Cohen [9] proposed two architectures for the detection of fetal ECG. The first is a combination of an IIR adaptive filter and Genetic Algorithm, where the Genetic Algorithm is recruited whenever the adaptive filter is suspected of reaching local minima. The second is an independent Genetic Algorithm (GA) search without the adaptive filter. The main disadvantage of an IIR filter is that the error surface is not quadratic but a multi modal surface. So the presence of the genetic algorithm forces the algorithm to overcome the local minima and reach the global solution. The quality of the extracted fetal ECG using this IIRGA adaptive filter is superior to that obtained using the GA alone. This method of combining an adaptive filter with a genetic algorithm is interesting and it would be nice to see how it performs when there are uterine contractions in the ECG data. Barros and Cichocki [10] proposed a semiblind source separation algorithm to solve the fetal ECG extraction problem. This algorithm requires a priori information about the autocorrelation function of the primary sources, to extract the desired signal (fetal ECG). They do not assume the sources to be statistically independent but they assume that the sources have a temporal structure and have different autocorrelation functions. The main problem with this method is that if there is fetal heart rate variability, as is the case when the fetus is not healthy, the a priori estimate of the autocorrelation function of the fetal ECG may not be appropriate. Goals of the Thesis The following are the goals of the thesis: Assess the performance of BSS algorithms that use whitening of data as a preprocessing step. Use CauchySchwartz based Quadratic Mutual Information minimization to extract the fetal ECG. Under varying assumptions on the data, use Generalized Eigenvalue Decomposition to solve the blind source separation problem. Compare the performance of BSS algorithms which include whitening and which do not include whitening. The rest of the thesis is organized as follows. Chapter 2 provides a detailed description about the algorithms which were studied to solve the fetal ECG extraction problem. Chapter 3 describes the results obtained using the algorithms presented in Chapter 2. Finally Chapter 4 concludes the thesis by making observations on the results 8 described in Chapter 3, along with some suggestions for future research in this particular area. CHAPTER 2 DESCRIPTION OF ALGORITHMS STUDIED The extraction of the fetal ECG signal from the ECG recording of a pregnant mother has been carried out using three different algorithms. The MeRMaId (Minimum Renyi's Mutual Information) algorithm for Blind Source Separation (BSS) proposed by Hild et al. [11], whitens the input data spatially before minimizing the Renyi's Mutual Information between the outputs. The algorithm proposed by Xu et al. [12] performs BSS by minimizing the 'CauchySchwartz quadratic mutual information' (CS QMI) of the outputs. The BSS via Generalized Eigenvalue Decomposition proposed by Parra and Sajda [13] provides an elegant way of finding the demixing matrix using generalized eigendecomposition of the covariance matrix and an additional symmetric matrix (which depends on the type of assumption made).These three algorithms along with some of the practical aspects in implementing them for the fetal ECG extraction are described in detail in this chapter. BSS Using MeRMaId Algorithm sn Y (xn) (n) y (n) Mixing Whitening Rotation Matrix Matrix Matrix __ A W R Figure 21: Block Diagram for two sources/observations The above figure depicts the overall block diagram of the MeRMaId algorithm. The source signals represented by s (n) are assumed to be independent. These signals are mixed by an unknown mixing matrix A to get the observable signals x (n). This observable data is first decorrelated spatially by applying a whitening transform (more will be described about this later in the chapter). Once whitening has been done, the problem of finding the independent components reduces to a simple rotation. So we need to find a rotation matrix which minimizes the Renyi's Quadratic Mutual Information. Let us provide some mathematical aspects of the whitening technique. The observable data x (n) can be written as x (n) = As (n). Let (n) represent the whitened data and dropping the index of time n for the sake of convenience, then x= Wx Ef J \ (21) (21) turns out to be WRxWT = I= Rx1/2RxRx1/2 (22) Comparing, the RHS and LHS of (22), we get, W = Rx1/2 (23) Representing, the covariance matrix in terms of its eigenvalues and eigenvectors, the whitening transform is given by W = VD1/2vT (24) where V is eigenvector matrix of Rx and D is the diagonal eigenvalue matrix of Rx. Why should we do whitening? To answer this question, let us have a look at the plot shown in figure 22. The figure deals with two dimensional data (for the sake of visualization). We see that by rotating the whitened data, we can achieve the desired solution of finding the independent components. So after whitening, the optimization problem of minimizing the Renyi's Mutual information is constrained to a rotation matrix. Source Data 00. . .:. ... *.) '." .. .. : ... ..:"".'"'. ." :;. .. ." 0 2 "*) .^ *.. .;.. t. .'. '.: . 0 01 02 03 04 05 06 07 08 09 1 Mixed Data 05 0 02 04 06 08 1 12 14 Whitened Data h"" *" ,*% .......' 1 05 0 05 1 15 2 25 3 35 Figure 22: Effects of Whitening The whitening operation transforms the mixing matrix A to a new one which is orthogonal. As such, instead of estimating n2 (assuming A to be a n x n matrix) parameters in the mixing matrix, it is sufficient if we can estimate n(nl)/2 parameters in the new orthogonal mixing matrix( since the number of degrees of freedom in a mixing matrix is n(n1)/2). So in larger dimensions, we need to estimate approximately half the number of parameters of an arbitrary matrix. Another advantage of whitening is it normalizes the data automatically. So we do not have to worry, even if we use scale dependent estimators for estimating mutual information. What's bad with whitening? The whitened data will be proportional to the power of the input data. If the observable data is noisy (like what we have in the ECG of a mother during uterine contractions), whitening degrades the already poor SNR by enhancing the noise component. So it is advisable to make use of algorithms which do not involve whitening of the data in such situations. The other two algorithms described in this chapter does not use whitening for preprocessing the data. Let us now go into details about minimizing the Renyi's mutual information. Renyi's mutual information of order a [14] is defined as IR (Y) = log Fn fY(Y), dy (25) f fYd(Yd ) 1 d=1 This equation can be approximated as Sfy (y)u dy IRa (y) l og c no (26) S1 IfY d(Yd)u dyd oo d=1 Both (25) and (26) are nonnegative and evaluate to zero if and only if all the outputs are statistically independent i.e. when the joint pdf is equal to the product of marginal pdfs. As such both (25) and (26) have the same global minima and hence (2 6) is a good approximation of (25), for use as a cost function in adaptation. Now (26) can be written as = log d )d + log fy(y)udy IRa (Y) = f fYd( d d 1 d=1 m (27) n = HR (yd) HR(y) d=1 So the Renyi's Mutual information of order a turns out to be approximately the sum of Renyi's marginal entropies minus Renyi's Joint entropy of the output y. Since y = Rx, the pdfs of y and x are related as: f y (x) fy (y) ) (28) R So the Renyi's Joint entropy of y becomes: 1 log f(x) HR (yv) = 10g Rdx S(HR (x)+ logR The Renyi's mutual information can now be expressed as: n IR (y) = HHR (yd) HR (x) log R (210) d=1 Since R is purely a rotation matrix, determinant of R is one. So logR vanishes. Also, we have to minimize this mutual information with respect to the rotation matrix R. As the joint entropy of x HR, (x), does not depend on R, it drops out of the cost function. The final cost function is just the sum of the marginal entropies of the output y, i.e., J= ZHR, d) (211) d=1 The advantage with this algorithm is that you do not have to estimate the joint entropy at all, which is comparatively difficult than estimating the marginal entropies. The marginal entropies can be estimated easily using the Parzen windowing with a Gaussian kernel. The estimate of Renyi's quadratic (a=2) marginal entropy simplifies to [15] HR2 (Yd) = 0log2 I G(yd()yd(k),2G2) (212) N =l k=1 where G(x,2o2) is a Gaussian PDF and yd(j) is the jth sample of output yd. The argument of the log in (212) is called the information potential. BSS Using Minimization of CS QMI In order to compare the performance of the algorithms which does prewhitening of the data before minimizing the Renyi's Quadratic mutual information, the BSS problem is solved using the CauchySchwartz Quadratic Mutual Information [16]. The mutual information between two (or more) random variables can be measured using the KullbackLeibler divergence: K(f, g) = f (x)log(f (x) / g(x))dx (213) where f(x) and g(x) are any two pdfs. The corresponding Renyi's measure of divergence between the two pdfs is given by the following equation: R.(f,g) = log( (f () /g(x)ldx (214) cl 1 It can be seen that, we can integrate neither (213) nor (214) with the Parzen windowing method to produce a simple result. But if we use the CauchySchwartz inequality to approximate the mutual information, we get the following measure: ( df((x g(x)2xdx) C(f, g) = log( f2 dr) (215) ( f(x)g(x)dx) We can easily see that, C (f,g)>0 and the equality sign holds good if and only if f(x)=g(x). Let us now replace f(x) by the joint pdf of two random variables Y1 and Y2 and g(x) by the product of their marginal pdf's. Now (215) becomes, (,Y2 log( f12 (YY2 )2 dyldy2 )( Y fy f ()2 2 (Y2 )2 dyldy2) C(Y, Y2 g (fY1 y 2)fY1 (Y)Y2 (y2)dYldY2)2 Now C (Yi,Y2) >0 and the equality sign hold good if and only if the joint pdf of Y is equal to the product of its marginal pdfs. We will now see how we can integrate this with the Parzen window method of estimating the pdf. The joint pdf estimate of the twodimensional random variable Y= {Yi,Y2} using the Parzen window method is given as: fY2 (z1, z2 ) = G(z, yl (i),G2i)xG(z2 Y2 ( 2) (217) The marginal pdf's can be estimated as: fYd (Zd)= G(zd d(i),2) d=l1,2 (218) N=1 We will make use of the fact that if G(za, ,1) and G(zaj, 2)are two Gaussian functions with means ai and aj, covariance matrices Y1 and Y2 respectively then, fG(z a, 1)xG(z a, 2)= G(a,a, (1 + 2)) (219) Let us now compute the terms in the numerator and denominator of (216) separately, 1^ JfY1Y2 (Y1, 2 2 dyldy2 2 G(y, (i) y (), 2 I) x G(y2 ( Y2 (j),2G21) N =1 =l1 (220) 1NN fd (yd)2dyd =G(yd)yd(j),2&2I) d=l1,2 (221) JJfy1A2 (1, 2 )fY1 (l )fY2 (y2)dyldy2 3 1 IG(yd ()Yd(j),2G21) S=1 d=1 j=1 (222) where n is the number of dimensions in the data (here n=2). It can be easily seen that, the equations (220) through (222) can be easily computed and the cost function given by (216) can be evaluated. Thus, Parzen windowing can be easily integrated with the CauchySchwartz based Quadratic Mutual Information estimation. Once the cost function (216) is evaluated, we can minimize it using any of the optimization techniques. The Gradient Descent has been used in the present work. In this the weights are initialized and they are adjusted iteratively such that the gradient of the cost goes to zero. The weight update can be carried out as: Wn+1 = W, r D, +a* M, (223) Where rl=step size Dn= derivative of the cost function at time instant 'n' with respect to the weight matrix a=momentum learning constant Mn= (Wn+l Wn) The momentum learning has been incorporated into this optimization method in order to overcome the local minima to some extent. As the iterations proceed, the weight matrix converges to, hopefully, the demixing matrix1. The parameter which needs to be carefully chosen is the size of the Gaussian kernel, which we are using to estimate the integrals. By choosing any arbitrary kernel and doing the above optimization method, will not give optimal results. The method will invariably get stuck in some local minima present in the cost function. In order to avoid 1 Up to the order of scale and sign this, a concept called the kernel annealing has been used. Ideally, we would like our kernel to depend on the data as well as on the number of samples in the data. This makes sense because the same kernel size may be good for one type of data and may not be good for other types of data. Also if you have more number of samples, we can use a kernel with smaller variance and vice versa. These two ways of choosing the kernel size have been incorporated in the following way. 1. Choose the kernel size as the average of the standard deviations of the data in each dimension. 2. Scale this kernel size down by the square root of the number of samples of data. Coming to the kernel annealing part, we will start with a large kernel size. Then all the data samples are just like one sample compared to the kernel size. So the estimated pdf will be just like one Gaussian and the landscape of the cost function will be very smooth. Then as we gradually reduce the kernel size (in this case linearly), the landscape of the cost function becomes rough with multiple peaks and valleys. For half the number of iterations, we will decrease the kernel size and for the other half of the iterations we will search for the solution with the minimum kernel size (fixed) we have chosen. As a result of this the algorithm may escape the local minima and converge to the global minima. BSS Using Generalized Eigenvalue Decomposition Let s(t) represent ndimensional sources, on which we impose certain assumptions (more on these assumptions later). Let x(t) represent the observable signals. If A is the mixing matrix, which is unknown, then x(t) = As(t). The mixing matrix explains the various cross statistics of the observations x(t) as an expansion of the corresponding diagonal cross statistics of the sources s(t). Let Rs and Rx denote the covariance matrices of s (t) and x (t). Then, Rx = E[x(t)xH (t)] = AR,AH 2 (224) If the sources are assumed to uncorrelated, then the source covariance matrix Rs is diagonal. Let us also assume, in addition to the covariance matrix, there exists other cross statistics matrix represented by Qs that has the same diagonalization property i.e. Qx = AQAH (225) It is also assumed that Qs has nonzero diagonal values, which ensures the existence of Qs,1. Let W be the demixing matrix, which we want to determine, such that WHA = I. Now multiplying both sides of (224) by W, we get, RW = AR, (226) Multiplying (225) by W and then by Qs,1, we obtain QxWQ 1 = A (227) Substituting (227) in (226), RW = QxWQ~ Rs (228) Since both Qs' and Rs are diagonal, their product is also diagonal. Letting Q Rs = A, where A is also diagonal, RxW = QxWA (229) 2 We are assuming that x(t) and s(t) are complex valued variables. For real valued variables, Hermetian(H) turns out to be the same as Transpose(T) (229) represents a generalized eigenvalue equation, where W and A represent the generalized eigenvector matrix and generalized eigenvalue matrix for the matrix pencil, Rx and Qx. So the demixing matrix is determined by the generalized eigenvector matrix of Rx and Qx. Let us now come to the different assumptions we can impose on the sources and the corresponding cross statistics matrix Qx. In addition to the independence assumption, there are basically three types of assumptions you can have on s (t). They are: NonStationary Sources. NonWhite Sources. NonGaussian Sources. When the sources are nonstationary, the covariance of the observations varies with time t i.e. Rx (t) = E[x(t)xH (t)] = AR, (t)AH (230) (230) is in the form similar to (225). So we can set Qx=Rx (t), for any time t. This Qx will give the diagonal crossstatistics of (225), required for the generalized eigenvalue equation (229). So the demixing matrix, can be identified by simultaneously diagonalizing multiple covariance matrices (here Rx and Qx) estimated over different stationary times. When the sources are nonwhite, we can use second order statistics in the form of crosscorrelations for different time lags T. The covariance matrix for lag T is given by: R, (T) = E[x(t)xH (t + T)] = AR, (T)AH (231) Again, (231) is of the form given by (225). So for any choice of T, the required cross statistics matrix Qx=Rx (T). This method of separating the mixtures of independent signals using time delayed correlations is first studied by Molgedey and Schuster in [17]. The nonGaussian assumption of the sources is useful when the sources are both stationary and white. For such sources different t and T do not provide new information and we have to consider higher order statistics. The 4th order cumulants of the observable data x (t) expressed in terms of the fourth order moments can be written as: Cum(x1, x1, x,, x* ) = E[x, x xk x ] E[x, x ]E[xk x ] E[x, xk ]E[x x ]E[x x ]E[x xk ] (232) When the distribution of x is Gaussian, the fourth order cumulant shown above goes to zero. If the distribution is nonGaussian, the cumulant goes to zero only when the distribution of x is composed of independent variables. In [18] a linear combination of the fourth order cumulants is defined as: q,j (M) = Cum(x,, x x )lk (233) kJ Noting that the covariance of x is Rx = E[xxH ] and M= { mIk }, (233) becomes, Qx (M) = E[xHMxxxH RTrace(MR) E[xxT ]MT E[x'xH ] RxMRx (234) Replacing M by I, makes Qx sum over all the cumulants given by, Qx (I) = E[xHxxxH ] RTrace(R,) E[xxT ]E[x*xH ] RR, (235) Observing that x(t)=As(t), we see that Qx(I) AQ,(AHA)AH (236) where Qs(AHA) is always diagonal for any A. So Qx satisfies the conditions given by (2 25). Let us now see how each of these three assumptions can be applied to the fetal ECG extraction problem. For the nonstationary assumption, the crossstatistics matrix Qx is computed by considering a window of data of length equal to the period of the maternal ECG. The covariance matrix is computed using all the data. For the nonwhite assumption, the parameter T is chosen empirically so that Qx can be computed to give the best performance. For the nonGaussian assumption, there will be no parameters to choose and Qx is computed as the sum over all the 4th order cumulants. Table 21 shows all the details of the blind source separation using the Generalized eigenvalue decomposition method. The observable data is represented as a matrix X of size NxT, where N is the number of sensors (channels) and T is the number of samples in each channel. In MATLAB, the demixing matrix (W) is calculated as shown below for all the three assumptions: [W,D]= eig(R, Q,) Table 21. Summary of procedures for BSS using Generalized Eigenvalue Decomposition (GED) for fetal ECG extraction Sources are Matrices on which GED has to be Details assumed to be carried out Rx Ox Non Stationary and XXT E[x(t)xH (t)= t is the number of decorrelated X samples over which the (:, t) t) signal is assumed to be stationary(chosen to be 120) NonWhite and XXT E[x(t)xH (t + )] = Qx is symmetric and decorrelated X(:,1: T c) x lag r provides new information if the X(:, T+1: T) + sources have distinct X(:, T +1: T) x autocorrelation. r is set X(:,1: T to 5 empirically. NonGaussian and XXT E[xHxxH ] Qx is the sum over 4th independent RxTrace(Rx) order cumulants. E[xx. ]E[x*xH] RxR, CHAPTER 3 RESULTS This chapter discusses and compares the quality of the fetal ECG obtained by each of the three algorithms explained in the previous chapter viz MeRMaId, Generalized Eigenvalue Decomposition and the Cauchy Schwartz based quadratic mutual information minimization method. The algorithms are applied on two different data sets. First to see if the algorithms work for the particular application of fetal ECG extraction, they are tested with the normal ECG data of pregnant mother. Then, to see how the algorithms perform during the uterine contractions of the mother, they are tested with the noisy ECG data of the mother during the contractions. It has to be noted that all the ICA algorithms suffer from the problem of scale and permutation i.e. we can never know which output channel corresponds to the signal of interest (here the fetal ECG). Also we need to come up with a criterion which measures the quality of the signal of interest. So there arises a need to make use of some automated algorithms which automatically detect the presence of the signal of interest as well as assess the quality of it. The results obtained for the data during contractions are compared qualitatively using a criterion called the trust factor [19]. This trust factor varies from 0 to 1, the higher value it has the better will be the quality of the fetal ECG. The next section tells more about the data collection procedure and the following sections deal with the results of various algorithms for normal ECG data as well as for the ECG data during the contractions. Finally the section on the qualitative comparison of the results using the trust factor concludes the chapter. Data Collection The ECG data [20] is collected noninvasively using ten surface electrodes positioned on the maternal abdomen in a standard arrangement (Figure 31). Electrode position was systematically varied in a preliminary study on 11 patients, until the following optimal array positioning was identified: 10 electrodes encircling the maternal abdomen, with reference electrodes located centrally and on the right leg. This standard position is slightly modified for every patient due to the presence of other monitoring equipment placed on the maternal abdomen (Tocodynamometer and Ultrasonic belts). Four of the electrodes also collect EHG signals. EH6 lead 2  Ground Figure 31: Position of the electrodes on the pregnant woman The recorded signals are fed to an 8channel high resolution (gain of 4000), low noise unipolar amplifier specifically designed for FECG signals. All eight signals were measured with respect to a reference electrode. The amplifier also uses a driven right leg (DRL) circuitry to reduce common mode noise between the patient and the amplifier common. The amplifier 3dB bandwidth is 0.1 Hz and 100 Hz, with a 60 Hz notch. The amplifier has a variable gain, but here the gain is set to 6,500. The data are then transferred to a PC via an A/D card which has a sampling frequency of 200Hz and resolution of 16 bits. Performance of the Algorithms on Normal ECG Data MeRMaId Algorithm The MeRMaId algorithm is first applied to a normal ECG data set collected from a pregnant mother. The ECG is generally measured by placing eight electrodes on various places of the mother's abdomen and thorax. So the data will be eight dimensional as shown in figure 32. For the present research, all eight channels have been considered. In each channel, 10,000 samples of data are considered. Since the sampling rate is 200Hz, this corresponds to 50 seconds of the ECG monitoring. Preprocessing is done by passing the data through a second order FIR filter which has zeros at DC and near Nyquist frequency. This is done to remove the DC bias present in the signal as well as the high frequency noise. The stochastic information gradient (SIG) version [21] of the algorithm has been used. The SIG can be used to modify the MeRMaId criterion such that the complexity is reduced to O(L) from O(L2). The information potential given in (212) can be represented as: Va (Y)= fy(y)dy=Ey[fy (y)] (31) O0 Dropping the expectation and stochastically approximating the value of the information potential with the instantaneous value of its argument, we get Vu, () f" (Yk) (32) So the Renyi's Quadratic marginal entropy in (212) reduces to: k1 HR2,k (Yd) = log1 G(yd(k) yd(j),2a2) (33) Lj=kL The various parameters for this algorithm are set like this: alpha = step size is set to 0.01 L = window size is set to 200 rep = number of repetitions is set to 1(since we are using SIG) stp = order of difference equation (1 DAstp) is set to 1 sig = standard deviation for Gaussian distribution (used for Parzen windows) is set to 0.25 The outputs (whose Renyi's Mutual Information has been minimized) obtained for this clean data set are shown in figure 33. It can be clearly seen from this figure, that the mother ECG and the fetal ECG signals are well separated. In this figure, the output channels 1 and 2 represent the maternal ECG signals, channel 3 represents fetal ECG signal and the rest of the channels represent the noise. x 104 5 o n 0 5 104 0 5 4 2 104 . 2( 200 l 1U 290 0 24 0 2 0 25 10 200 0 2 104 0 2 0 200 Figure 32: Eight channels of normal ECG data. Only 2000 samples of the ECG data are shown in the figure. 8Pn 1000 12Pn 14O0 1R6n 1Ann 7000 800 1000 1200 1400 1600 1800 2000 900 1000 1200 1400 16100 18100 000 i i 1 i ~ I0 1000 120 1400 1RO 10 2000 800 1000 1200 1400 1600 1800 2000 0 1000 1200 1400 160 18 2000 800 1000 1200 1400 1600 1800 2000  600 6o0 6o0 6o0 600 MECG 10  5 i i i i 10n 200 400 OOn 0n 1000nn 100nn 1400 OOnn 100 000 FECGO , 10 50 200 400 600 800 1000 1200 1400 1600 1800 2000 I : : . 5 2o0 490 690o 90 1000 12oo 1400 1o00 10o f o000 50 200 400 600 800 1000 1200 1400 1600 1800 2000 5 C I I A 50 200 400 600 800 1000 1200 1400 1600 1800 2000 5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 33: Output signals for the MeRMaId algorithm. The nature of the eight signals is mentioned to the left. Cauchy Schwartz based Quadratic Mutual Information The blind source separation problem is now accomplished using the minimization of Cauchy Schwartz (CS) based Quadratic Mutual Information (QMI). To make this algorithm work for the real ECG data, we need to know how to change the kernel size so that we can get good results. To test if the algorithm separates the sources successfully, let us first deal with 2 dimensional speech data which are mixed artificially with a known mixing matrix. The weight matrix, the final value of which will represent the demixing matrix, is initially chosen as the identity matrix. The kernel size is linearly decreased for the first 150 iterations and for the next 150 iterations its value (the value at the end of the 150th iteration) is fixed. The step size as well as the momentum learning constant is chosen as a function of the kernel size, so that we can have larger updates when the kernel size is larger and smaller updates when the kernel size decreases. The signal to interference ratio is shown below in figure 34. Defining Signal to Interference Ratio (SIR) as SIR(i) = 10ogo (max )2 (34) O,0, (max(O,))2 where 0 = W*A = product of mixing matrix and current estimate of demixing matrix and Oi represents the ith row of matrix 0. It can be clearly observed that very high SIR (as high as 50 dB) can be achieved for this 2 dimensional case. The learning curve which indicates how the cost function (here the Cauchy Schwartz based Quadratic Mutual Information) varies as the iterations proceed is shown in figure 35. 80 70  60 50 E 40 4/Co 30 20 / / 10 5 0 50 100 150 200 250 300 Iterations Figure 34: Plot of the signal to interference ratio versus the iterations 0018 0016 0 014 0012  o 001  S 0 008 0 006 0 004 0 002 0 5L0 L L  0 50 100 150 200 250 300 Iterations Figure 35: Learning curve Let us now try this algorithm on the normal ECG data. Since estimating the joint pdf of finite data samples is easy to estimate accurately in lower dimensional space, only four channels (out of the eight) of the ECG data is considered. The weights are initialized as the identity matrix as before. The number of iterations on the data is chosen as 500. In order to make the algorithm escape local minima, kernel annealing (described in chapter 2) is done. For half the number of iterations, kernel annealing is done (kernel size is gradually decreased linearly from a large value to a predetermined smaller value). For the next half the number of iterations, the kernel size is fixed. The results for this four dimensional data are shown in figure 36. It can be clearly seen that fetal ECG can be easily separated and is observed in channel 3 of the output. 200 400 200 400 200 400 600 800 600 800 I i I 1000 1200 2 2 0 0 0.2 0.2 0 0.5 F FECG 0C 0.5 0 5 MECG 0 5  0 200 400 600 800 1000 1200 200 400 600 800 1000 1200 1400 1600 1600 1800 1400 1600 1800 1400 1600 1800 Figure 36: Output signals for CS Based QMI Minimization Generalized Eigenvalue Decomposition Algorithm The same data (8 by 10000 samples) is applied as input to this algorithm. All the three different assumptions (nonstationary, nonwhite and nonGaussian) on the data are tried out. It can be seen from the results shown in figures 37, 38 and 39, that we can successfully separate the fetal ECG signal and the mother ECG signal from the input data. The channels representing the signals are shown labeled on the left in these figures. Let us now come to the parameters chosen for each of the three assumptions. The stationarity length parameter 't' for the nonstationary assumption case is chosen as 120 samples. The parameter T for the nonwhite assumption is set to 5. There are no parameters for the nonGaussian assumption. 800 1000 Non Stationary 0.1 0 0.1 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 050 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 I 0.5 5 ,'80 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 0.5 10 200 400 600 800 1000 1200 1400 1600 1800 2000 FECG 0 _1 10 200 400 600 800 1000 1200 1400 1600 1800 2000 1 L L 20 200 400 600 800 1000 1200 1400 1600 1800 2000 0I 10 200 400 600 800 1000 1200 1400 1600 1800 2000 C 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 37: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nonstationary Assumption Non White 10001 0 , 1000I 20000 2000 5000 C, 5001 20000 FECG C 20001 16 MECG C0 1I 50000 MECG 0 50001 5000 500 10000 0 1000 0 o 200 200 200 10 200 200 200 200 200 Figure 38: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nonwhite assumption 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 1600 1800 2000 Non Gaussian 500 0. 500 20000 200 FECG 0 4 2000 ME ME( 400 600 800 1000 1200 1400 1600 1800 20 5000 200 400 600 800 1000 1200 1400 1600 1800 2000 500 50000 200 400 600 800 1000 1200 1400 1600 1800 2000 CG 0 5000 20000 200 400 600 800 1000 1200 1400 1600 1800 2000 0 , 2000 100 100 80 1 10 200 400 600 800 1000 1200 1400 1600 1800 2000 CG 0 1 1 1 1 1 1 1 1 1 1 1 1 1000(0 200 400 600 800 1000 1200 1400 1600 1800 2000 1000 , 1000 nonn0 200 400 600 800 1000 1200 1400 1600 1800 2000 4,, ,.0 _r I,,,,, i 4,,,, hr. 0 I " , ,00 Figure 39: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nongaussian assumption Performance of the Algorithms on the Data during Contractions Figure 310 shows the nature of the input data during contractions. Due to the presence of the ElectroHysteroGram (EHG) signal, the ECG data is noisier. We now attempt to separate the fetal ECG signal from eight channels of such data. The output signals obtained from the MeRMaId, the three different cases of the generalized eigenvalue decomposition algorithm are shown in figures 311, 312, 313, 314 respectively. It can be visually observed from these plots, that the MeRMaId algorithm totally fails to extract the fetal ECG. The quality of the extracted fetal ECG is comparatively better for the Generalized Eigenvalue Decomposition method based on nonwhite assumption. We will deal with the comparison more rigorously and qualitatively in the next section. The CauchySchwartz based Quadratic Mutual Information minimization method failed to extract the fetal ECG from the 8 dimensional ECG data during the contractions. The reason is considered to be more due to the dimensionality of the data rather than due to the nature of the data. This problem with high dimensionality of the data is not an issue with the MeRMaId algorithm since only marginal pdf's are estimated in this algorithm (no need to estimate the joint) whereas CS QMI has this additional problem of estimating the joint pdf in higher dimensional space. 6000 4000 20000 2000 0 0.5 / 1 1.5 2 2.5 3 S x 104 6000 4000 2000 2000 4000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 310: One channel of ECG data during uterine contractions. The figure shown at the bottom zooms on the portion of the signal containing contractions. MerMaid 10 MECG 0 10 100 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 10 50 200 400 600 800 1000 1200 1400 1600 1800 2000 0 5 50 200 400 600 800 1000 1200 1400 1600 1800 2000 FECG c,, 5 50 200 400 600 800 1000 1200 1400 1600 1800 2000 50 200 400 600 800 1000 1200 1400 1600 1800 2000 5 51 50 200 400 600 800 1000 1200 1400 1600 1800 2000 50 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 5Il Figure 311: Output signals for the MeRMaId algorithm Non Stationary Assumption 0.5 FECG 0 0.5 L L L _ 050 200 400 600 800 1000 1200 1400 1600 1800 2000 0.5 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 0.5 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 0.5 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 0.5 0.50 200 400 600 800 1000 1200 1400 1600 1800 2000 0 05 "0 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 5 100 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 312: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nonstationary assumption Non White Assumption 500 500 5000 200 400 600 800 1000 1200 1400 1600 1800 2000 500 10000 200 400 600 800 1000 1200 1400 1600 1800 2000 FECG 0 , 1000 j"'" S10J 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0  1 10000 200 400 600 800 1000 1200 1400 1600 1800 2000 MECG 0 1000 5000 200 400 600 800 1000 1200 1400 1600 1800 2000 500 500(0 200 400 600 800 1000 1200 1400 1600 1800 2000 500 5000 200 400 600 800 1000 1200 1400 1600 1800 2000 500 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 313: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nonwhite assumption Non Gaussian Assumption 5001 500 5000 200 400 C' * 5001 10000 200 400 C' . 1000 1 10 200 400 MECG 0 2 10 200 400 MECG 0 2 500o0 200 400 0 500 50 200 400 5001 5000 200 400 FECG ' 5001 0 200 400 600 800 1000 600 800 1000 1200 1200 1400 1600 1400 1600 1800 1800 600 800 1000 1200 1400 1600 1800 2000 600 800 1000 1200 1400 1600 1800 2000 600 800 1000 1200 1400 1600 1800 2000 600 600 600 1000 1000 1000 1200 1200 1200 1400 1600 1400 1600 1400 1600 1800 1800 1800 Figure 314: 2000 samples of the outputs signals for the Generalized Eigenvalue Decomposition method using nongaussian assumption Comparison of the Performance of the Algorithms In order to compare the performance of these algorithms, a criterion called trust factor [19] is defined which measures the quality of fetal ECG. This is based on the Pan Tompkins online QRS detection algorithm [22]. The output signals from the ICA algorithms are first passed through a band pass filter in order to remove the interfering noise. The outputs of this filter are differentiated and squared. This is followed by an integration stage realized using a moving average filter. Then thresholds are set to locate the peaks, which essentially gives the beat to beat heart rate. Different thresholds are set to determine maternal heartbeat and fetal heartbeat. The quality of the fetal ECG is judged based on the number of false positives (peaks observed when none are there actually) and on the number of false negatives (no peaks observed when there are actual 2000 2000 2000 peaks). This is determined by measured the interval between any two consecutive peaks. If this interval is less than 70% of the 5 previous RR intervals average, then there is a false positive. If this interval is more than 130% of the 5 previous RR intervals average, then there is a false negative. Based on these false positives and false negatives, a criterion is developed, called the Trust Factor, to judge the quality of the fetal ECG. The Trust Factor goes from 0 to 1 and uses several characteristics of ECG signals including: The quasi periodicity of the signal The sparseness of the signal (or the ECGlook) calculated with the number of points below a certain threshold or the kurtosis of the signal * The level of noise in the signal (which is calculated by two different methods: an estimation of the false negatives and false positives, and a signal to noise ratio in the autocorrelation function) * The location of the QRS peaks The performance of the algorithms for normal ECG data is compared using the Trust Factor. A window of data containing 8 channels of 10000 samples is considered for the algorithms MeRMaId, Generalized Eigenvalue Decomposition (GED) based on Non Stationary (NS) assumption, Generalized Eigenvalue Decomposition (GED) based on Non White (NW) assumption and Generalized Eigenvalue Decomposition (GED) based on Non Gaussian (NG) assumption. The data from the same window by of size 4 by 2000 is considered for the Cauchy Schwartz (CS) based Quadratic Mutual Information (QMI) algorithm. The values for the Trust Factor for each algorithm are shown in table 31. To compare the performance of these algorithms during the contractions, five windows each containing 10000 samples of the eight dimensional data is considered and each algorithm is applied to it. The resulting value of the trust factor is shown in the table 32. The results for CSQMI are not presented for this data set since the algorithm failed to extract the fetal ECG (reasons mentioned earlier in the chapter) while dealing with this dataset. Table 31. Comparison of performance of the BSS algorithms for normal ECG Window Number Trust Factor MeRMaId GED NS GED NW GED NG CSQMI 1 0.7747 0.7640 0.7115 0.7983 0.4859 Table 32. Comparison of performance of the BSS algorithms for ECG data during contractions Window Number Trust Factor MerMaid GED NS GED NW GED NG 1 0.1975 0.2180 0.3823 0.3002 2 0.3996 0.5036 0.5261 0.4692 3 0.6375 0.4806 0.5690 0.5830 4 0.7306 0.7162 0.6521 0.5424 5 0.6752 0.7214 0.6348 0.6072 Mean 0.5281 0.5280 0.5528 0.5004 CHAPTER 4 CONCLUSIONS AND FUTURE WORK Conclusions All the three algorithms BSS using MeRMaId, BSS using Generalized Eigenvalue Decomposition (GED) BSS using Cauchy Schwartz (CS) based Quadratic Mutual Information(QMI) are used to extract the fetal ECG from two different data sets :normal ECG recordings and ECG recordings during the uterine contractions of a pregnant woman. From the results presented in chapter 3 the following conclusions can be drawn: 1. MeRMaId The MeRMaId algorithm extracts the fetal ECG from the normal ECG data and the quality of the extracted fetal ECG is quite good. However, this algorithm fails to give good quality fetal ECG when dealing with the ECG data during the contractions. The reason for this poor performance can be attributed to the spatial prewhitening step performed on the observations, which worsens the already poor SNR. 2. GED The GED algorithm, based on all the three assumptions on the data, non stationary, nonwhite, nonGaussian, gives a high quality fetal ECG from the normal ECG data. On the data during the contractions, the non white assumption gives the best performance in terms of the quality of fetal ECG as measured by the criterion given by [20]. The nonstationary assumption gives results slightly superior to that of the MeRMaId while the nonGaussian assumption gives results slightly worse than the MeRMaId algorithm. 3. CS QMI This algorithm gives very promising results for lower dimensional data as illustrated by the blind source separation of a 2 dimensional speech data which is artificially mixed. But the algorithm suffers in performance as the dimensionality of the data is increased. To test the performance on the normal ECG data, only 4 (out of 8) channels are considered. The algorithm gives a good quality fetal ECG using these 4 channels only. But the data during the contractions, being noisier, inherently has more independent sources. So there arises the need to use all the 8 channels which makes the algorithm give poor results (no fetal ECG is observed). Scope for Future Work There are a couple of ideas, which researchers have successfully used to solve the blind source separation problem. These methods are particularly appealing for the ECG data which we have dealt with. Zibulevsky and Pearlmutter [23] exploited the property of the sources having a sparse representation in a signal dictionary for blind source separation and obtained very promising results. The dictionary may consist of wavelets, wavelet packets or coefficients in any other domain where they are sparse. The ECG signals can be considered to be having a sparse distribution in time since the active portions in the signal last only for a small amount of time. But the presence of noise (especially during the contractions) makes this assumption of sparse sources to be weak. But if we can have a denoising algorithm which improves the SNR significantly, then we can expect a superior performance by this algorithm for this application. Least dependent component based on Mutual Information (MILCA) proposed by Stogbauer et al [24] is very interesting in the sense that they take into account the time structure of the signal while doing a blind source separation. In this work, they propose to use a Mutual Information (MI) estimator based on knearest neighbor statistics [23]. Using this estimate of MI, they find the least dependent components in a linearly mixed signal. The fact that we are finding only the least dependent components instead of independent components is very useful for the problem of fetal ECG extraction because the fetal heartbeats and mother heartbeats are not entirely independent. By making use of the time structure and higher order statistics we can obtain optimal results in general. By delay embedding the observable signals (here noninvasive measurements of a pregnant mother), promising results have been obtained. This method of minimizing the mutual information of the delay embedded signals will give outputs which are least dependent. It would be really interesting to apply this method to the ECG recordings during contractions and minimize the knearest neighbor estimate of MI (if not Renyi's MI). LIST OF REFERENCES 1. Bernard Widrow, Samuel D Stearns: Adaptive Signal Processing, Prentice Hall Inc., Upper Saddle River, NJ, 1985. 2. 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Souloumiac: Blind Beamforming for NonGaussian Signals, IEE Proceedings, Vol. 140, No. 6, December 1993. 19. Dorothee E Marossero, Deniz Erdogmus, Neil Euliano, Jose C Principe, Kenneth E Hild II: Independent Components Analysis For Fetal Electrocardiogram Extraction: A Case For The Data Efficient MERMAID Algorithm, Proceedings of NNSP'03, pp.399408, Toulouse, France, September 2003. 20. T Y Euliano, D E Marossero N R Euliano B Ingram, K Andersen, R K Edwards: Noninvasive Fetal ECG: Method Refinement and Pilot Data, Anesthesia and Analgesia, in press. 21. Kenneth E Hild II, Deniz Erdogmus, Jose C Principe: Online Minimum Mutual Information Method for Time Varying Blind Source Separation, International Conference on ICA and Signal Separation, pp. 126131, San Diego CA, December 2001. 22. J Pan, W J Tompkins : A Real Time QRS Detection Algorithm, IEEE Transactions on Biomedical Engineering, Vol. 32, No. 3, pp. 837843,1985. 23. Michael Zibulevsky, Barak A Pearlmutter: Blind Source Separation by Sparse Decomposition, Neural Computation, Vol. 13, Issue. 4, pp. 863882, April 2001. 24. Harald Stogbauer, Alexander Kraskov, Sergey A Astakhov, Peter Grassberger: Least Dependent Component Analysis Based on Mutual Information, DOI: physics/0405044, arXiv, 2004. 47 25. A Kraskov, H Stogbauer, P Grassberger: Estimating Mutual Information, Physics Review E, 2004. BIOGRAPHICAL SKETCH Hemanth Peddaneni received his bachelor's degree in electronics and communication engineering, from Sri Venkateswara University, Tirupati, India, in 2002. He has been a Young Engineering Fellow, awarded by the Indian Institute of Science, Bangalore, India. He is now pursuing his master's degree in electrical and computer engineering at the University of Florida. His research interests include neural networks for signal processing, adaptive signal processing, wavelet methods for time series analysis, digital filter design/implementation and digital image processing. 