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1530 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007
Communications
Convolutive Blind Source Separation Algorithms Applied
to the Electrocardiogram of Atrial Fibrillation: Study of
Performance
Carlos Vayá*, José J. Rieta, César Sánchez, and David Moratal
Abstract—The analysis of the surface electrocardiogram (ECG) is the
most extended noninvasive technique in medical diagnosis of atrial fibril-
lation (AF). In order to use the ECG as a tool for the analysis of AF, we
need to separate the atrial activity (AA) from other cardioelectric signals.
In this matter, statistical signal processing techniques, like blind source sep-
aration (BSS), are able to perform a multilead statistical analysis with the
aim to obtain the AA. Linear BSS techniques can be divided in two groups
depending on the mixing model: algorithms where instantaneous mixing of
sources is assumed, and convolutive BSS (CBSS) algorithms. In this work, a
comparison of performance between one relevant CBSS algorithm, namely
Infomax, and one of the most effective independent component analysis
(ICA) algorithms, namely FastICA, is developed. To carry out the study,
pseudoreal AF ECGs have been synthesized by adding fibrillation activity
to normal sinus rhythm. The algorithm performances are expressed by two
indexes: the signal to interference ratio (
) and the cross-correla-
tion (
) between the original and the estimated AA. Results empirically
prove that the instantaneous mixing model is the one that obtains the best
results in the AA extraction, given that the mean
obtained by the
FastICA algorithm (
) is higher than the main ob-
tained by Infomax (
). Also the obtained by FastICA
(
) is higher than the one obtained by Infomax ( ).
Index Terms—Atrial fibrillation, convolutive blind source separation,
ECG, independent component analysis.
I. INTRODUCTION
The prevalence of atrial fibrillation (AF) is less than 1% in people
under 60 years old, but it increases significantly in those over 70, ap-
proximating to 10% in those over 80 [1]. The exhaustive analysis of AF
episodes requires to separate previously the atrial activity (AA) com-
ponent from other bioelectric signals that contribute to the electrocar-
diogram (ECG) formation [2]. Some of these signals are ventricular
activity (VA), muscular activity, noise and artifacts introduced by elec-
trodes, and the powerline interference [3].
Blind Source Separation (BSS) techniques have been successfully
applied to the extraction of the AA from the ECG of AF episodes by as-
suming instantaneous and linear mixing of the bioelectric signals in the
human body [3]. The feasibility in this matter of the FastICA algorithm,
based on independent components analysis (ICA), is proved in [3] and
[4]. Nonetheless, the propagation delays of the bioelectric signal in the
Manuscript received July 26, 2006; revised October 28, 2006. This work was
supported in part by the the Generalitat Valenciana under Project GV06/299.
Asterisk indicates corresponding author.
*C. Vayá is with the Department of Innovation in Bioengineering, Castilla-la
Mancha University, Escuela Politécnica Superior de Cuenca, Camino del
Pozuelo s/n, 16071 Cuenca, Spain (e-mail: carlos.vaya@uclm.es).
J. J. Rieta and D. Moratal are with Biomedical Synergy, Polytechnic Univer-
sity of Valencia, EPSG, Carretera Nazaret Oliva s/n, 46730 Gandía, Valencia,
Spain.
C. Sánchez is with the Department of Innovation in Bioengineering,
Castilla-la Mancha University, Escuela Politécnica Superior de Cuenca,
Camino del Pozuelo s/n, 16071 Cuenca, Spain (e-mail:carlos.vaya@uclm.es).
Digital Object Identifier 10.1109/TBME.2006.889778
body and specially the movements of the heart could violate the ICA
assumptions of spatial stationarity and instantaneous mixing of phys-
ical sources [3]. On the contrary, in the convolutive model, these cir-
cumstances can be considered by using convolutive BSS (CBSS) algo-
rithms. CBSS algorithms have not been applied yet to the extraction
of the AA. Our objective is to compare the performance of one rele-
vant CBSS algorithm, namely Infomax [5], and the FastICA algorithm
[6]. The main goals of this contribution are, on the one hand, to reveal
whether CBSS algorithms are reliable to extract the AA from the ECG
of AF episodes when the typical ECG sampling rate of 1 kHz is used
and, on the other hand, to study their possible improvement in the AA
estimation with respect to the instantaneous BSS algorithms.
II. B
LIND
SOURCE SEPARATION
EXTRACTION TECHNIQUES
Time-domain-based techniques, like Average Beat Substraction [7],
have been well accepted and used in clinical applications [8] to cancel
out the QRS complex and the T wave. Nonetheless, they are only ap-
plied to the leads where atrial fibrillation is more easily distinguish-
able, e.g., V1. This means that if we apply QRST cancellation tech-
niques to different leads, we would obtain different atrial activities as
well. Consequently, they do not make use of the information included
in every lead in an unified way. On the contrary, BSS techniques make
a multilead statistical analysis by exploiting the spatial diversity that
multiple spatially separated electrodes introduce [3], [9]. When linear
mixing of sources is considered, we talk about linear BSS techniques.
Three main conditions must be fulfilled in order to apply linear BSS
techniques: 1) independence of sources; 2) nongaussianity of sources;
3) linear mixing of sources [10]. Within the context of AF episodes,
these three conditions are applicable. First, in AF episodes the bioelec-
tric sources of the heart generating AA and VA can be considered to
be statistically independent. This is justified because only a portion of
atrial waveforms traverse the atrioventricular node and provoke ven-
tricular depolarization [3], [4]. Furthermore, the physical origin of the
atrial wavefront that has been able to produce ventricular depolariza-
tion could be very variable. This uncoordinated operation of AA and
VA during an AF episode allows us to consider them as statistically
independent [4]. Second, AA has a sub-Gaussian probability distribu-
tion whereas the VA is clearly super-Gaussian [3]. Finally, the mixing
process of the bioelectric sources in the human body can be considered
to be linear [11], [12]. Therefore, linear BSS techniques can be applied
to extract the AA in AF episodes.
These are the most important linear models used in BSS:
1) Instantaneous Linear Mixing Model: Within the assumption of
linearity, the simplest situation is the instantaneous linear model. In
this case, the instantaneous mixing of the cardioelectric sources in the
human body is assumed. This can be expressed as [6]
(1)
where
is a matrix that represents the linear transformation of source
mixing,
is the column vector that contains the observations at the
th sampling time, contains the sources, and the additive
noise in every source. The BSS algorithms based on the instantaneous
linear mixing model try to find the matrix
as the pseudoinverse of
the
that obtains the best estimation of the sources [6]:
(2)
0018-9294/$25.00 © 2007 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007 1531
The global mixing-unmixing system is characterized by the matrix ,
so that
.
The FastICA algorithm [6], based on the instantaneous mixing
model, has already been applied to the extraction of the AA from
ECGs of AF episodes [3]. In FastICA the instantaneous mixing of the
sources is assumed and quite good results in the extraction of the AA
have been obtained by using this algorithm [2].
2) Convolutive Linear Mixing Model: In the convolutive model, a
more realistic case is considered. Here, weighted and, besides, delayed
contributions in the generation of the observations are taken into ac-
count [6].
The convolutive model is a more general linear mixing BSS model
where coefficients of
are substituted by finite-duration impulse re-
sponse (FIR) filters and the product operator of (1) and (2) by the con-
volution operator
. Indeed, the previous instantaneous model could
be considered as a particular case of the convolutive model where the
length of the filters is one.
The performance of several CBSS algorithms has been tested, but
only the results obtained by Infomax [5] are presented here. Infomax
belongs to an important family of algorithms based on information
theory. Mutual information (MI) is a natural measure of the depen-
dence between random signals [6]. MI is always nonnegative, and zero
if and only if the variables are statistically independent. MI can be used
as a criterion for obtaining the original sources of the BSS problem [6].
The other algorithms that have been tested are the Multi-channel
Blind Least-Mean Square (MBLMS) algorithm [13], the time-delayed
decorrelation (TDD) algorithm [14], and the Convolutive Blind Signal
Separation (CoBliSS) algorithm [15]. Results obtained by TDD are
very similar to results obtained by Infomax, and results obtained by
MBLMS and TDD are quite poorer. Consequently, only results of In-
fomax have been reported in this paper.
III. P
ERFORMANCE ESTIMATION AND
DATABASE
A. Spectral Concentration (SC)
Considering that the typical spectral morphology for AA is charac-
terized by a very pronounced peak in frequencies from 5 to 8 Hz, with
no harmonics and insignificant amplitudes above 15 Hz [16], it is pos-
sible to define a performance extraction index capable of evaluating
AA extraction quality based on the spectral concentration [9]. The SC
can be defined as
(3)
where
is the power spectral density of the AA signal,
is the fre-
quency,
is the sampling rate (1 kHz), and is the main frequency
peak of the AA. Experimental observations prove that the AA is the
estimated independent component with the highest
when the limits
of the numerator summation are those specified in (3) [8]. Therefore,
the
can be used as an indicator to select, in an automated way, the
estimated source that best matches the AA. A wider distance between
frequency limits introduces high frequency noise components in the
calculation of the
indicator so that the probability of discriminating
correctly the AA is reduced. A shorter distance between limits reduces
the AA power considered in
so that the probability of AA discrim-
ination is reduced too.
B. Performance Indexes
1) Signal-to-Interference Ratio (SIR): Considering
as the one of
the
observations with the highest contribution of AA and as the
one of the
sources that contains the AA, the SIR of
expressed in
decibels is defined as [17]
(4)
where
are the FIR filters of the mixing matrix
, which represent
the contribution of the
th source on the th observation.
stands for
the mathematical expectation. In the same way, considering that
is
the estimated source with the highest contribution of AA (i.e, the esti-
mated source with the highest
), the SIR of
is [17]
(5)
where
are the FIR filters of the global system matrix
. Finally,
the signal-to-interference ratio
is defined as
(6)
2) Cross-Correlation: The cross-correlation between the original
AA and the estimated AA is calculated as
(7)
where
and
are the original and the estimated AA, respectively.
C. ECG Database
The calculation of the previously defined indexes needs the original
sources and the mixing matrix to be known. Given that the sources that
contribute to the generation of real ECGs are unknown, we needed to
establish two environments of synthesized ECGs, so that the perfor-
mance indexes could be applied.
1) First Environment: In the first environment, 15 pairs of separated
AA and VA of AF episodes are mixed using the convolutive model.
These separated AA and VA have been previously obtained from the
V1 lead of real ECGs by using QRST cancellation [7]. The mixing is
made by random mixing matrices
. Obviously, this is not the real
mixing process of the bioelectric signals in the human body. Nonethe-
less, it can be considered as a valid and objective way of testing and
discarding those algorithms (CoBliSS and MBLMS) that were clearly
useless to extract the AA. The FIR filters length of these mixing ma-
trices (
) has been changed from 1 to 8. All ECGs are 12 s in length,
and they all were obtained at a sampling rate of 1 kHz. In the sepa-
ration process, the filter length of the separation matrix
is an ad-
justable parameter in the CBSS algorithms (
). It has been changed
in the tests from 2 to 32. The value of two is the lowest value allowed
by all the tested CBSS algorithms. The value of 32 has been chosen so
that the controlled maximum delay due to the mixing-unmixing process
equals to 40 ms in duration, accordingly to the aforementioned max-
imum value of
and the sampling rate [13].
2) Second Environment: In the second environment, AF 12-leads
ECG are synthesized by adding AA and VA of every lead. AA and VA
were previously separated from real ECGs of AF episodes by using
QRST cancelation [7]. Therefore, the synthesized ECGs are exactly the
original real recorded signals. In this way we know their AA and VA
components, so that the previously defined parameters can be applied.
The convolutive mixing is assumed to be present in the generation of
the real ECGs and, consequently, in the separated AA and VA. This
synthesis process is depicted in Fig. 1. All resulting ECG recordings
last for 8 s and are sampled at 1 KHz. This second environment com-
prises 20 synthesized ECGs. The maximum value of
has been fixed
1532 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 8, AUGUST 2007
Fig. 1. Second environment synthetic ECG generation. The AF 12-leads ECGs
are synthesized by adding the AA and the VA of every lead, previously separated
from real ECGs of AF episodes.
Fig. 2. and mean values of Infomax and FastICA algorithms in
the first environment as a function of the FIR filters length of the random mixing
matrices (
).
to 128 by considering 128 ms as a reasonable maximum propagation
delay for all the bioelectric signals in the human body [11]. The res-
piratory induced axis-shifts of all the analyzed ECG recordings have
been previously eliminated by signal filtering. All these recordings are
free from ectopic beats or other nonideal characteristics. This allows
us to develop an initial work focused in the performance of the CBSS
algorithms when regular signals of AF are considered.
Note that the presence of previous convolution in the initial AA and
VA is not an obstacle to the study of performance of CBSS algorithms.
If previous convolution is present, the CBSS algorithms would try to
invert the global linear system composed by the previous convolution
and the additional simulated convolution. On the other hand, given that
the mixing process of sources is unknown, the only way to test the
performance of extraction when no direct access to the atrial activity is
available (we are using surface recordings) is to define a set of synthetic
signals so that we can compare the signals estimated by BSS algorithms
with the original signals.
IV. R
ESULTS
A. First Environment Results
The global mean
obtained in this environment by FastICA
algorithm (
) is higher than the global mean
obtained by Infomax (
). Also the obtained by
FastICA (
) is higher than this obtained by Infomax (
).
The influence of
in the first environment is shown in Fig. 2. Both
performance indexes,
and , decrease when mixing filters
length (
) increases. Mean values of
obtained by Infomax
are several decibels lower than the respective FastICA mean values,
regardless of
. Furthermore, FastICA
mean values are always
nearer to one than those of Infomax.
Fig. 3 illustrates the influence of the separation filters length (
).
The mean values of FastICA are included only as a reference constant
Fig. 3.
and mean values of Infomax and FastICA algorithms in
the first environment as a function of the FIR filters length of the separation
matrices (
).
Fig. 4. and mean values of Infomax and FastICA algorithms in
the second environment as a function of the FIR filters length of the separation
matrices (
).
value, given that FastICA does not match the convolutive model and,
therefore, the filters length parameter cannot be chosen. Mean
and
of Infomax are slightly lower than the respective indexes of
FastICA.
decreases when
increases. Thus, the best separation
performance of Infomax is obtained when the instantaneous model is
considered in this environment.
B. Second Environment Results
Also in this environment, the global mean
obtained by Fas-
tICA algorithm (
) is several decibels higher than the one
obtained by Infomax (
). In the same way, the global mean
obtained by FastICA (
) is higher than the one obtained
by Infomax (
).
The influence of
in the second environment is shown in Fig. 4.
Note that the highest values of
and
, in the Infomax algo-
rithm, are obtained when
equals to eight.
V. C
ONCLUSION
Infomax obtained acceptable results but poorer than those of Fas-
tICA. With regard to the
parameter in the first environment, Fas-
tICA always exceeds the results obtained by the Infomax algorithm in
the case of instantaneous mixing (
). Hence the instantaneous
linear mixing model can be considered as an acceptable approximation
for the bioelectric mixing in the human body. Given that the instan-
taneous linear mixing model is the particular case of the convolutive
mixing model when
is equal to one, we can conclude that CBSS
algorithms need an improvement to reach at least the performance of
the FastICA algorithm in the case of instantaneous mixing and to ob-
tain better results in the convolutive case.
The Infomax algorithm has been easily adapted to the standard
12-lead real ECG of the second environment, reaching its best perfor-
mance when a convolutive model (
) is considered. An in-depth
analysis of the Infomax algorithm and its adjustment to the special
features of ECG signals could obtain an appropriate algorithm for the
extraction of the AA.
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Use of Genetic Algorithms to Optimize Fiber Optic Probe
Design for the Extraction of Tissue Optical Properties
Gregory M. Palmer and Nirmala Ramanujam*
Abstract—This paper outlines a framework by which the optimal illu-
mination/collection geometry can be identified for a particular biomedical
application. In this paper, this framework was used to identify the optimal
probe geometry for the accurate determination of tissue optical properties
representative of that in the ultraviolet-visible (UV-VIS) spectral range. An
optimal probe geometry was identified which consisted of a single illumina-
tion and two collection fibers, one of which is insensitive to changes in scat-
tering properties, and the other is insensitive to changes in the attenuation
coefficient. Using this probe geometry in conjunction with a neural network
algorithm, the optical properties could be extracted with root-mean-square
errors of 0.30
for the reduced scattering coefficient (tested range of
3–40
), and 0.41 for the absorption coefficient (tested range of
0–80
).
I. I
NTRODUCTION
The diffuse reflectance spectrum, which reflects the absorption and
scattering properties of a turbid medium, is sensitive to a number of im-
portant physiological indicators and thus, is a useful tool for the early
diagnosis of precancers and cancers. The illumination/collection geom-
etry is a critical aspect of tissue diffuse reflectance spectroscopy in that
it affects sensitivity to the absorption and scattering properties of the
medium, the sensing depth and the signal-to-noise ratio. Specialized
probe geometry designs have been previously shown to be useful in
characterizing tissue optical properties from diffuse reflectance spectra
[1], [2].
This paper outlines a framework by which the optimal illumina-
tion/collection geometry can be identified given a particular design ob-
jective. Here, the framework is used to identify the optimal probe ge-
ometry for the accurate determination of tissue optical properties rep-
resentative of that in the ultraviolet-visible (UV-VIS) spectral range.
The unique benefits of this approach are 1) no
a priori information is
needed about the tissue absorbers and scatterers and 2) there is no re-
quirement for complex multiple source-detector separation fiber probe
geometries.
II. M
ETHODS
A. General Optimization Methodology
The optimization methodology proceeds in the following manner.
First, a population of fiber probe geometries was randomly initialized.
Next, diffuse reflectance measurements were simulated for each of
these probe geometries for a wide range of tissue optical properties
using Monte Carlo modeling. The training data set for each of the
probe geometries was used to optimize a neural network algorithm
to take the diffuse reflectance as an input, and output the optical
properties. The optimized neural network algorithm was applied to an
Manuscript received November 9, 2005; revised November 15, 2006. This
work was supported in part by University of Wisconsin through Radiological
Sciences Training Grant 5T32CA009206-27, sponsored by the National Insti-
tutes of Health, and in part by the National Institutes of Health (NIH) under
Grant 1R01CA100559-01A1. Asterisk indicates corresponding author.
G. M. Palmer is with the Department of Radiation Oncology, Duke Univer-
sity, Durham, NC 27710 USA (e-mail: greg.palmer@duke.edu).
*N. Ramanujam is with the Department of Biomedical Engineering, Duke
University, Durham, NC 27703 USA. (e-mail: nimmi@duke.edu).
Digital Object Identifier 10.1109/TBME.2006.889779
0018-9294/$25.00 © 2007 IEEE
