Sparse electrocardiogram signals recovery
based on solving a row echelon-like form
of system
ISSN 1751-8849
Received on 14th January 2015
Revised on 10th August 2015
Accepted on 12th August 2015
doi: 10.1049/iet-syb.2015.0002
www.ietdl.org
Pingmei Cai
1
, Guinan Wang
1
, Shiwei Yu
1
, Hongjuan Zhang
1
✉
, Shuxue Ding
2
, Zikai Wu
3
1
Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
2
School of Computer Science and Engineering, The University of Aizu, Tsuruga, Ikki-Machi, Aizu-Wakamatsu City, Fukushima 965-8580,
Japan
3
Business School, University of Shanghai for Science and Technology, Shanghai 200093, People’s Republic of China
✉ E-mail: zhanghongjuan@shu.edu.cn
Abstract: The study of biology and medicine in a noise environment is an evolving direction in biological data analysis.
Amon g t hese studies, analysis of electrocardiogram (ECG) signals in a noise enviro nment is a challenging direction in
personalized medicine. Due to its periodic characteristic, ECG signal can be roughly regarded as s parse biomedical
signals. This study proposes a two-stage recovery algorithm for sparse biomedical signals in time domain. In the first
stage, the concentration subspaces a re found in advance. Then by exploiting these subspaces, the mixing matrix is
estimated accurately. In the second stage, based on the number of active sources at each time point, the time points
are divided into different layers. Next, by constructing some transformation matrices, these time points form a row
echelon-like system. After that, the sources at each la yer can be solved out explicitly by corresponding matrix
operations. It is noting that all these operations are conducted under a weak sparse condition that the number of active
sources is less than the number of observations. Experimental results show that the proposed method has a better
performance for sparse ECG signal recovery problem.
1 Introduction
Biological data mining in biology and medicine are often challenged
by computational problems such as curse of dimensionality, noise,
redundancy, and so on. An obvious example which is challenged
by noise is the recordings and analysis of electrocardiogram
(ECG) or foetal ECG (FECG) data, which plays an important role
in modern medicine for diagnosis of cardiac disease. High-quality
ECG signals can be utilised by physicians for identification of
physiological and pathological phenomena [1, 2]. By measuring
the indicators such as heart beat rates, morphology and dynamic
behaviours, researchers can diagnose the physical condition of a
patient or foetus. Among these parameters, the heart beat rates is
the main index for this purpose. However, in real situations, ECG,
especially FECG recordings are often corrupted by all kinds of
noises. As for FECG signal recording, which contains valuable
information about the health condition of the foetus, is always
disturbed by other noises, such as maternal ECG (MECG) with
extreme high amplitude, stomach activity, thermal noise, noise
from electrode–skin contact. Then to obtain clean signal of ECG
(FECG) from the mixture recordings is a crucial challenge in
practical applications.
In addition, one should pay attention to one thing, that is, ECG
signals always have significant periodicity characteristic; thus we
can roughly regard this kind of signals as sparse biomedical
signals. Therefore, sparse signal recovery algorithms can be used
to obtain them clearly, and related works have proved the validity
of this approach recently. For example, Zhang et al. [3, 4] have
utilised the block sparse feature of FECG to solve the FECG
telemonitoring problem, and the experiments in their works show
that the raw recordings have been reconstructed with high quality.
Inspired by these, we want to utilise a new sparse signal recovery
algorithm to achieve better recovery results of the ECG recordings.
The method of sparse signal recovery has been widely used in the
field of signal processing recently such as image denosing [5, 6],
image separation [7], biomedical signal processing [8], radar
detection [9] etc. Besides, classifying genomic and proteomic data
is very important to predict diseases in a very early stage and to
investigate signalling pathways. However, this poses many
computationally challenging problems such as curse of
dimensionality, noise, redundancy, and so on. Therefore, the
principle of sparse signal recovery has been applied to analyse
high-dimensional biological data within the frameworks of
clustering, classification, and dimension reduction approaches [10,
11]. The basic model of sparse signal recovery can be described as
x = As + v, (1)
where x [ R
M×1
is the observed signal and A [ R
M×N
denotes the
dictionary, that is, the mixing matrix. The vector s [ R
N×1
is the
source signal which is to be recovered, and its non-zero elements
should be below a threshold to ensure the global optimal solution.
Besides, v is a noise vector which is usually Gaussian distributed.
Noting that s is sparse, that is, most samples in it are nearly zero,
and just a few per cent take significant values [12, 13]. In this
case, model (1) can be solved by dictionary learning approach,
which usually includes two steps. Specifically, the first step is for
learning dictionary A. After the dictionary is learned, we come to
the second stage to recover the sparse source signals based on the
dictionary obtained from the first stage. Many dictionary learning
algorithms can be utilised to deal with the first step such
as Focuss-based dictionary learning algorithm, iterative least
squares-based dictionary learning algorithm [14], K -singular value
decomposition (SVD) algorithm [15], K-means clustering
algorithm [16] etc. However, both K-SVD algorithm and K-means
clustering algorithm converge to a local minimum, which may
produce counterintuitive results sometimes [16]. Therefore, we
would better employ other more appropriate algorithm to estimate
the dictionary of biological signal recovery problem. After
dictionary is learned, Step two can be worked out by many
algorithms [16], for example, orthogonal matching pursuit
algorithm [17], Focuss algorithm [18], iterative weighted algorithm
[19] etc.
IET Systems Biology
Research Article
IET Syst. Biol., pp. 1–7
1
&
The Institution of Engineering and Technology 2015
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