https://orcid.org/0000-0002-7420-8977
Figures
Abstract
A greedy pursuit strategy which finds a common basis for approximating a set of similar signals is proposed. The strategy extends the Optimized Orthogonal Matching Pursuit approach to selecting the subspace containing the approximation of all the signals in the set. The method, called Simultaneous Optimized Orthogonal Matching Pursuit, is stepwise optimal in the sense of minimizing at each iteration the mean square error norm of the signals in the set. When applied to compression of electrocardiograms, significant gains over other transformation based compression techniques are demonstrated on the MIT-BIH Arrhythmia dataset.
Citation: Rebollo-Neira L (2025) Simultaneous optimized orthogonal matching pursuit with application to ECG compression. PLoS One 20(6): e0325555. https://doi.org/10.1371/journal.pone.0325555
Editor: Nattapol Aunsri, Mae Fah Luang University, THAILAND
Received: June 6, 2024; Accepted: May 13, 2025; Published: June 11, 2025
Copyright: © 2025 Laura Rebollo-Neira. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data are available Aston Repository: https://doi.org/10.17036/researchdata.aston.ac.uk.00000655
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Important signals in everyday life such as natural images, audio, and electrocardiogram records, are in general highly compressible. This implies that the original signal, available as a large set of numerical values, can be transformed into a set of much smaller cardinality or a set containing a large proportion of zero values. The transformation, which should not compromise the informational content of the data, is frequently called sparse representation. Traditional methods for sparse representation of signals are realized by applying an orthogonal transformation and disregarding the least relevant points in the transformed domain. Subsequently the signal is recovered by means of the inverse transformation. However, alternative transformations, which are not orthogonal but adapted to a signal at hand, have been shown to render high level of sparsity. Such transformations aim at representing a signal as a superposition of elements, which are called ‘atoms’ and are selected from a large set called ‘dictionary’. The superposition is said to be sparse if it involves a number of atoms much smaller than the number of numerical values representing the original signal.
Given a dictionary, the problem of finding the sparsest approximation of a signal, up to some acceptable error, is an NP-hard problem [1]. In practice it is addressed by tractable methodologies known as Pursuit Strategies. Such methodologies can be grouped for the most part in two broad categories. Namely, Basis Pursuit and Greedy Pursuit Strategies. The Basis Pursuit (BP) approach endeavors to obtain a tractable sparse solution by minimization of the 1-norm [2]. Greedy algorithms seek for a sparse solution by stepwise selection of dictionary’s atoms. When dealing with real data the latter are in general more convenient. From the seminal Matching Pursuit (MP) [3] and Orthogonal Matching Pursuit (OMP) [4] methods, a number of Greedy Pursuit Strategies have been developed to improve the process of sparsely representing single signals [5–16]. Due to complexity issues and memory requirements, most of these techniques are to be applied by segmenting the signal and approximating each segment independently of the others. Nonetheless, when the segments bear similarity with each other, for some applications it is convenient to look for the dictionary’s atoms suitable to represent all the segments simultaneously. The greedy Pursuit Strategy which has been dedicated to simultaneously approximate a set of signal is based on OMP [4] and has been termed Simultaneous Orthogonal Matching Pursuit (SOMP) [17]. Since in this work we extend the Optimized Orthogonal Matching Pursuit method [6] to simultaneously approximate a set of signals, we term the new approach Simultaneous Optimized Orthogonal Matching Pursuit (SOOMP).
The difference between SOMP and the SOOMP approach introduced in this work is equivalent to the difference between OMP and OOMP methods, both for approximating single signals. OOMP is stepwise optimal in the sense of minimizing at each iteration the norm of the residual error. Whilst OMP minimizes the norm of the error only with respect to the coefficients of the atomic superposition, OOMP minimizes the norm of the error with respect to those coefficients and the selection of a new atom. In the case of multiple signals SOOMP is designed to minimize the mean value of the error norm squared. An additional advantage arises from the proposed implementation. Based on adaptive biorthogonalization, the SOOMP method produces at each iteration the common dual basis to the basis of selected atoms. This allows to calculate the coefficients of the representation of each signal in the set simply by computation of inner products. We implement the previous SOMP method in an equivalent manner and compare the two approaches for the approximation of stereo music, by selecting atoms from a highly coherent trigonometric dictionary. The practical relevance of the SOOMP approach is further illustrated by using it for compression of electrocardiogram (ECG) records.
An ECG signal represents a sequence of heartbeats which, if properly segmented and aligned, are suitable to be simultaneously approximated. This property is shown to benefit compression. Reliable comparison with other compression techniques is made possible by recuse to an adaptive quantization procedure that facilitates to reconstruct the whole ECG record at the required quality. The compression results are shown to significantly improve upon results produced by different transformation based approaches.
The paper is organized as follows: Sect 2 introduces the problem and the mathematical notation. Sect 3 establishes the proposed SOOMP approach for simultaneous approximation of a set of similar signals. Sect 4 compares the SOOMP and SOMP approaches for the simultaneous approximation of stereo music. Sect 5 applies the proposed SOOMP method for compressing digital ECG records and produces reliable comparisons with previously reported results. The conclusions are presented in Sect 6.
2 Mathematical introduction of the problem
In order to pose in mathematical terms the problem to be addressed we need to introduce the notation used throughout the paper as well as some preliminary background.
The sets of real, integer, and natural numbers are indicated by , and
, respectively. Boldface letters are used to indicate Euclidean vectors or matrices whilst standard mathematical fonts indicate components, e.g.,
is a vector of components
and
is a matrix of elements
which when not leaving room for ambiguity will also be represented as
. A set of Q signals of equal length N, to be simultaneously approximated in a common subspace, is represented as a set of vectors
. The inner product is indicated as
, e.g. for
and
The 2-norm induced by the inner product is denoted as , e.g. for
A set of M vectors
such that and N < M, is called a redundant dictionary for
and its elements are called atoms.
In our context a signal is assumed to be well approximated by an element, say
, belonging to a finite dimensional subspace
. This assumption implies that, within a tolerance
much larger than the numerical errors in the calculations,
is accepted to be a good approximation of
if
Examples of signals fulfilling this definition are, amongst others, audio signals, and electrocardiograms. These are all signals with acceptable approximations which, without affecting their informational content, do not necessarily produce a highly accurate point-wise reproduction of the signals. These type of signals are suitable for lossy compression.
Since this work concerns approximation of similar signals we need to make an assumption on the signals that will be considered. We say that a finite set of Q signals are similar if they can be well approximated in a subspace
of dimension k, with k significantly smaller than N. This is equivalent to assuming that there exists a common basis
for
such that each signal
is approximated as
The quality of the approximated set will be assessed in mean value
where with
.
3 Strategy for simultaneous approximation of a set of signals
Given a set of similar signals and a dictionary, the aim is to simultaneously approximate all the signals in the set
within a common subspace
. In other words, each signal
is to be approximated as a k-term atomic superposition
where the atoms in (1) are selected from the given dictionary according to the criterion of optimality that will be established by Proposition 1 in the next subsection. Let us suppose for the moment that these atoms are known. Assigning a weight
to the signal
, with
, the coefficients
in (1) are required to minimize the mean value of the square norm of the errors in the approximation of the set of signals, i.e.
Since the above minimization is equivalent to finding the components
of each vector
such that
Accordingly, the minimization with respect to the coefficients in (1) can be implemented by adaptive biorthogonalization [18], as proposed within the OOMP algorithm for a single signal [6],
with vectors calculated as will be described in the next section.
The selection of the atoms in the decomposition (1) such that
poses an intractable problem (for a dictionary of M atoms there are possibilities to be checked). We address the selection in a tractable manner by extending the OOMP strategy to simultaneously approximate a set of similar signals. The extended strategy is refereed to as SOOMP (Simultaneous OOMP).
3.1. SOOMP algorithm
The algorithm is initialized by setting: ,
,
and k = 0. The first atom is selected as the one corresponding to the index
such that
This first atom is used to assign , calculate
and iterate as prescribed below.
- (1) Upgrade the set
, increase
, and select the index of a new atom for the approximation as
(6)
- (2) Compute the corresponding new vector
as
(7)
including for numerical accuracy the re-orthogonalization step:(8)
- (3) Upgrade vectors
as
(9)
- (4) Update
as
(10)
- (5) If the stopping criterion is met finish the iterations. Otherwise repeat steps (1)–(5).
Note: given a tolerance , as stopping criterion we set:
or
depending on convenience for the particular application.
Once the iterations have finished calculate the coefficients for the decomposition (1) as
For calculate the final approximation of each signal
as
Remark 1. The set of vectors as given in (9) fulfills that
where is the orthogonal projector of
onto
. Please find the proof in [6], or as a particular case of the more general proof in [18].
Proposition 1. The recursive selection of the indices , as proposed in (6), is stepwise optimal. It minimizes, at each iteration, the mean of the square distance between the set of signals
and their corresponding approximations
.
Proof: For k = 0 it is clear that selected as in (5) minimizes the mean of the square distance
as given by
Let us assume that the indices selected as proposed in (6) minimize, in the specified stepwise sense, the mean square distance
We shall prove by induction that if the atoms are fixed, at iteration k + 1 the atom
selected as in (6) minimizes
. The proof stems from the fact that at iteration k the approximation
of each signal
is the orthogonal projection of
onto the subspace
(c.f. Remark 1).
Consider that is augmented by one element, say
, so that
, where
indicates direct sum. The orthogonal projection of each signal
onto
can be expressed as
Thus
Since is optimized and fixed at iteration k, it is true that at iteration k + 1 the index of the atom which minimizes
fulfils
The proof is concluded using the self-adjoint properties of to write:
Moreover, since all atoms are normalized and the set is an orthonormal basis for
we have
so that
which shows the equivalence between (13) and (6).
Corollary 1. The selection criterion (6) guaranties that, if for the right hand side of (6) is not zero, the selected elements
are linearly independent.
Proof: For k = 1 the single atom is linearly independent. Let us assume that the first k selected atoms
are linearly independent and prove that then the newly selected atom
is also linearly independent. The proof is achieved by contradiction. Indeed, if
is linearly dependent then
, for some scalers
, so that
and from (14) we gather that the right hand side of (6) is zero. This contradiction leads to conclude that the selected elements
by criterion (6) are linearly independent.
Remark 2. If the dictionary is complete or over-complete, i.e.,
, then by selecting N atoms the method can reconstruct the exact signals
Otherwise if
the selected atoms provide the orthogonal projections
, where
. It should be stressed, though, that with a suitable dictionary all the signals
are expected to be well approximated in a subspace
with k significantly smaller than N. Otherwise the representation would not qualify to be sparse.
Remark 3. The complexity of the SOOMP algorithm, at each iteration, is O(NMQ). For equal weights the difference between the complexity of the SOOMP selection criterion (13) and the SOMP one [17] is the denominator in the right hand side of (13). This introduces extra computations of complexity O(NM) so that the order of complexity of both algorithms is equivalent. However, as will be illustrated in the next section, due to the reduction in the number of selected atoms, and hence the number of iterations, the approximation may run faster if SOOMP rather than SOMP is used.
4 Numerical example
We illustrate here the algorithm’s implementation by simultaneously approximating stereophonic music. This type of music, commonly called stereo music, is usually produced by using two independent audio channels and
. The example is dedicated to showing the sparsity obtained with the proposed SOOMP method and the precursor SOMP one, when simultaneously approximating both channels using a highly coherent trigonometric dictionary.
Since the signals structure varies with time, approximations with trigonometric dictionaries are carried out on a partition of the signal. In this example the partition consists of disjoint segments, called frames, which are assumed to be all of the same size L. The signal representation is realized by independent approximation of each frame.
As shown in [15,16], for music representation the combination of a Redundant Discrete Cosine Dictionary and a Redundant Discrete Sine Dictionary
, defined below, renders higher sparsity than when using pure
or
dictionaries of the same redundancy as the combination
.
where and
are normalization factors.
The signal representation is realized by independent approximation of each frame. The atoms are selected for simultaneously representing both channels
and
in the ith frame, for
, with I the integral part of N/L. In the numerical simulations the number of atoms in each dictionary
and
is 2L so that the total number of atoms in
is 4L.
As a metric of approximation quality we use the standard Signal to Noise Ratio () calculate as
where are the approximations of the channels
. Since the frames are disjoint the approximation of each channel
is obtained by the concatenation of the approximations
of the corresponding frames i.e.,
, where
indicates the concatenation operation. The numbers
of atoms for approximating each i-frame are decided to meet the condition:
On defining
the parameters in (15) are determined as
where is fixed to the same value for each frame in oder to achieve the expected approximation quality.
The metric of sparsity is considered to account for the number of elements in the approximation of the whole signals. For this we define the Sparsity Ratio (SR) as
where is the total number of atoms in the signal representation and N the number of samples in each of the channels. Thus, a large value of
indicates a high level of sparsity.
The numerical example is realized using four stereo clips of melodic music: 1. Classic Orchestra. 2. Classic Guitar. 3. Chopin Piano. 4. Piazzolla Tango. All four clips are of the same length, N = 256000 samples in each channel (5.8 secs). The approximation is carried out on frames of length L = 1024 and for qualities corresponding to dB,
dB, and
dB. The improvements in the values of
obtained with SOOMP, with respect to those obtained with SOMP, are noticeable from the comparison of the 3rd and 4th columns in Table 1 for all four clips and all approximation qualities. The 5th column displays the gain over SOMP yielded by SOOMP. The 6th and 8th columns show the approximation times (the values are the average of five independent runs with MATLAB in a Laptop Core i7-1165G7). The 7th and 9th columns give the corresponding standard deviations.
The left graphs in Fig 1 show 2000 samples in channel 1 of the original chips as well as the corresponding approximation up to 25 dB. The right graphs have the same description but correspond to the 2000 samples in channel 2. As illustrated by the graphs in Fig 1, SNR = 25 dB produces already very good pointwise approximation of the signals.
The graphs on the left correspond to one of the channels and the graphs on the right to the other.
5 Application to compression of ECG records
A digital ECG signal represents a sequence of heartbeats. In a typical record each heartbeat is characterized by a combination of three graphical deflections, known as QRS complex, and two lateral and less visually noticeable P and T waves. A short segment of a typical ECG record is illustrated in Fig 2.
In order to simultaneously approximate all the beats in a record we need to segment and align the beats to meet the requirement of being similar. The procedure is discussed in the next subsection.
5.1. Segmentation and alignment of heartbeats
The QRS complex is segmented once the central R peak is detected. This can be effectively done by the Pan Tompkins method [19]. In our numerical examples we use the off-the-shelf MATLAB implementation of this algorithm [20]. Since the distance between peaks in a record is not uniform, the length of the segmented beats should be passed to the decoder. The segmented peaks are placed in arrays of equal length L by padding with zeros. Fig 3 illustrates the resulting configuration with 80 heartbeats. Fig 4 shows the two dimensional image of the segmented and aligned heartbeats corresponding to records 111 and 100 in the MIT-BIH Arrhythmia database [21].
For simultaneously approximating heartbeats we use a wavelet dictionary. Given a partition of the interval [c, d] the dictionary is constructed as follows [22,23].
with
and
where indicates the restriction of the function
to the interval [c, d]. Different families of wavelet basis and dictionaries for approximation of heartbeats have been compared in [22], where the Cohen–Daubechies–Feauveau family was singled out as the most effective one. We have confirmed the same outcome for simultaneous approximations and adopted the Cohen–Daubechies–Feauveau
dictionary of redundancy approximately two introduced in [22].
The prototype functions and
are plotted in the left and right graphs of Fig 5 respectively. The MATLAB codes for producing numerically both functions and building the dictionary (16) are described in [23]. The codes have been made available in [25] together with of the complete MATLAB software for reproducing the numerical examples in this work.
The segmented and aligned heartbeats are simultaneously approximated using the SOOMP approach by assigning the same weight to each heartbeat, i.e. . In this case the algorithm stops at iteration k if
with
indicating the mean of
and
a fixed value of the metric of quality
, which is defined by
where is the whole ECG record,
is the reconstructed record from the approximated heartbeats and
is the mean of
.
Given a required value of , at the approximation step of the processing the parameter
is fixed as 0.8
, in order to achieve the target value
at the quantization step described in Subsect 5.2.
Even if the approximation is realized to achieve the required by the whole record, it is interesting to calculate the quality metric for each heartbeat in the array. To this end we defined
where is the approximation of the beat
and
its mean value. The values of
for the simultaneous approximation of records 111 and 100 are shown in the left and right graphs of Fig 6 respectively.
The total produced by the reconstruction of the whole records is, in both cases,
. The mean value
for record 111 is 9.0 with std=4 while for record 100 the mean
is 9.4 with std=1.4. However, as noticeable in the figures, for some qs the
values are much higher than for others. This is a consequence of the irregularities of the beats, which can be perceived in the left image of Fig 4.
The top left graph of Fig 7 depicts a heartbeat in records 111 and its approximation. This heartbeat yields . As shown by the red line the figure, such high
value is produced by a smooth version of the noisy signal. On the contrary, for regular heartbeats the
values are close to the
of the whole record. The right graph of the figure shows the approximation of one of those beats. The left bottom graph is the approximation of one of the few beats in record 100 which yields the outlier value
. The right bottom graph shows the approximation of one of the other beats.
The right graph corresponds to a beat yielding close to the mean value
. The bottom graphs have the same description but the heartbeats are from record 100.
By the simultaneous approximation of the aligned heartbeats these are transformed into a reduced set of numbers which allow to reconstruct the approximated heartbeats. This set consists of a) the k indices corresponding to the common atoms in the decomposition of the heartbeats (c.f. (1)) b) the different coefficients
in the decomposition of each heartbeat (c.f. (1)). These coefficients can be placed in a two dimensional array
as illustrated in Fig 8. The top image on the left in this figure shows the magnitude of the array
arising from the approximation up to
of record. 111. The bottom left image corresponds to the approximation up to
of record 100.
The bottom left image depicts the magnitude of entries in the arrays corresponding to the approximation of records 100. The images on the right are the magnitude of the entries in the arrays
(top) and
(bottom) arising by applying the discrete cosine transform on the columns of the arrays represented by the images on the left.
It is clear from the location of the brightest pixels in the left images of Fig 8 that the coefficients of largest magnitude are concentrated in vertical lines. This suggests that, to favor compression for storing these values, it is convenient to apply an orthogonal transformation to map the coefficients in the vertical direction to smaller values which eventually might be quantized to zero. Consequently, by applying the discrete cosine transform on each column of , we create the transformed array
with the following entries
where indicates the one dimensional discrete cosine transform operating on the n-th column of array
. The transformed points corresponding to the left images in Fig 8 are represented in the right images of this figure. The introduction of this step to decorrelate the vertical entries in the array
is key to boost the performance of the adopted encoding strategy described below. The notorious change of intensity in the images on the right of Fig 8 indicates that after quantization some of the entries of the transformed arrays will be mapped to zero. Within the encoding strategy described in the next section, this effect enhances compression.
5.2. Encoding
At the encoding step the array
is expressed as a vector
of
components, adopting the column-major order. The encoding of this vector follows the procedure outline in [26]. The components of
are converted to integer numbers by a mid-tread uniform quantizer as follows:
where indicates the largest integer smaller or equal to x and
is the quantization parameter. For comparison with results in other publications in the numerical examples the quantization parameter
is set to produce the required quality of the reconstructed signal.
The absolute value of the elements (23) are placed in a smaller vector, say
, after the elimination of zeros. The signs are encoded separately in a vector
using a binary alphabet: 1 for + and 0 for –.
Assuming that the nonzero values in (23) occur at the positions , these indices are re-ordered in ascending order
, i.e.
. This induces new order in the coefficients,
and in the corresponding signs
. Defining
−
the array
stores the indices
with unique recovery.
Finally the vectors , as well as the length of the heartbeats
, are compressed using adaptive Huffman coding implemented by the off-the-shelf MATLAB function Huff06 [27]. The additional numbers which have to be passed to the decoder are:
- (i) The indices
of the selected dictionary’s atoms forming the common basis.
- (ii) The quantization parameter
.
- (iii) The mean value of the 1D ECG record (if not previously subtracted).
- (iv) The number of rows and columns of
, i.e. Q and k.
5.3. 1D ECG signal recovery
At the decoding stage, after reverting Huffman coding, the locations of the nonzero entries in the transformed array after quantization are readily obtained. This allows the recovery of the array
as follows.
- (i) Set
and
.
- (ii) Reshape the vector
to produce a 2D array
of size
. The array
is recovered from the
one by inverting the
transformation (c.f. (22)).
- (iii) Each row of the recovered array
gives the coefficients in the decomposition (1) of the approximated heartbeats, i.e.
- (iv) Finally the reconstructed beats
are assembled in a 1D record using the distance between heartbeats that was stored in the vector
.
The achieved compression ratio , which is defined as
depends on the required quality of the recovered signal. In the numerical examples the quality of the recovered records is assessed by the as defined in (20). It is pertinent to stress the importance of adopting this normalized metric for comparison of reconstruction quality. The subtraction of
avoids dependence on the signal baseline.
6 Numerical tests
For the numerical test we use the MIT-BIH Arrhythmia database [21]. Each of the records is of 30 min length, consisting of N = 650000 11-bit samples at a frequency of 360 Hz.
For comparison purposes we compress the subset of records reported in [28,29], and [30] and reproduce the values of in those publications. This is achieved as follows: The SOOMP method is applied to approximate the set of heartbeats in each record up to 80% the target PRDN. The quantization parameter
is automatically fixed, by a bisection algorithm, in order to reproduce the target
for the whole record within two decimal places.
The first, second and third columns of Table 2 reproduce the results published in [28]. The comparison is relevant because the approach [28] is also based on approximation of heartbeats using a dictionary. The techniques are very different though. Whilst our dictionary does not have to be stored because it is numerically generated, the dictionary in [28] is part of the ECG record to be compressed. Moreover, the method for finding the sparse representation is different and so is the procedure to store the parameters that should be passed to the decoder.
Our compression results are shown in the forth column of Table 2. These results demonstrate a significant gain in for the same recovery quality. For further comparison we apply the fast compression algorithm [26], which does not require peak segmentation or Huffman coding. This method has been already shown to improve the average
for the 48 records in the MIT-BIH Arrhythmia dataset with respect to the results in [31,32], and [33], for a broad rage of average qualities. For comparison with [28] in Table 2 the compression is realized to reproduce the
listed in the second column for each record.
The first, second and third columns of Table 3 reproduce the results published in [29], which are achieved with an approach based on the Singular Value Decomposition (SVD).
Our compression ratios () are shown in the the forth column of Table 3. The fifth column shows the
produced by the fast compression algorithm [26].
The first, second and third columns of Table 4 reproduce the results published in [30], which are also obtained with a Singular Value Decomposition based approach. Our are shown in the forth column of this table. The fifth column shows the
produced by the fast compression algorithm [26].
Note: The MATLAB software for reproducing the tables is available on http://www.nonlinear-approx.info/examples/node017.html
7 Conclusions
The Optimized Orthogonal Matching Pursuit approach has been extended with the purpose of selecting a common basis for the simultaneous approximation of a set of similar signals. The extended approach, termed Simultaneous Optimized Orthogonal Matching Pursuit, minimizes at each iteration the mean value square error norm of the joint approximation. The algorithm’s implementation was demonstrated by approximating stereophonic music using a highly coherent trigonometric dictionary. The applicability of the method to ECG compression was illustrated on records taken from the MIT-BIH Arrhythmia database. The particular records were selected for comparison purposes as in [28,29], and [30]. The simultaneous approximations of aligned heartbeats was used for compressing a whole record. The adopted compression strategy was shown to improve upon compression results achieved by other methods for the same reconstruction quality. The comparison was made possible by means of an iterative quantization procedure which delivers the required quality.
While the proposed approach involves detection and alignment of R-peaks, it is the approximation step which introduces the highest computational cost. In order to address this matter as a line of future work, it would be interesting to investigate the possibility of selecting a suitable subspace from a whole data set (instead of a subspace for each single record as is done here). Certainly approximating new records using previously selected atoms would significantly speed the compression procedure. It is still to be discerned if the SOOMP approach could pick out a common low dimension subspace to approximate, up to a given quality, any new ECG record. We feel confident that the results presented in this work will motivate further research in the topic.
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