On the trade-off between compression efficiency and distortion of a new compression algorithm for multichannel EEG signals based on singular value decomposition


ACTA IMEKO 
ISSN: 2221-870X 
June 2022, Volume 11, Number 2, 1 - 7 

 

ACTA IMEKO | www.imeko.org June 2022 | Volume 11 | Number 2 | 1 

On the trade-off between compression efficiency and 
distortion of a new compression algorithm for multichannel 
EEG signals based on singular value decomposition 

Giuseppe Campobello1, Giovanni Gugliandolo1, Angelica Quercia2, Elisa Tatti3, Maria Felice Ghilardi3, 
Giovanni Crupi2, Angelo Quartarone2, Nicola Donato1 

1 Department of Engineering, University of Messina, Contrada di Dio, S. Agata, 98166 Messina, Italy 
2 BIOMORF Department, University of Messina, AOU "G. Martino", Via C. Valeria 1, 98125, Messina, Italy 
3 CUNY School of Medicine,  CUNY, 160 Convent Avenue, New York, NY 10031, USA 

 

 

Section: RESEARCH PAPER  

Keywords: Biomedical signal processing; electroencephalograph (EEG); EEG measurements; near-lossless compression; singular value decomposition (SVD) 

Citation: Giuseppe Campobello, Giovanni Gugliandolo, Angelica Quercia, Elisa Tatti, Maria Felice Ghilardi, Giovanni Crupi, Angelo Quartarone, Nicola Donato, 
On the trade-off between compression efficiency and distortion of a new compression algorithm for multichannel EEG signals based on singular value 
decomposition, Acta IMEKO, vol. 11, no. 2, article 30, June 2022, identifier: IMEKO-ACTA-11 (2022)-02-30 

Section Editor: Francesco Lamonaca, University of Calabria, Italy 

Received October 24, 2021; In final form February 22, 2022; Published June 2022 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Giuseppe Campobello, e-mail: gcampobello@unime.it   

 

1. INTRODUCTION 

Since its invention by the German psychiatrist Hans Berger 
almost a century ago [1], electroencephalography (EEG) has 
continuously evolved, becoming a powerful and extensively used 
method that allows measuring safely and noninvasively the 
spatiotemporal dynamics of the brain activity with a high 
temporal resolution in the range of milliseconds, which enables 
detecting rapid changes in the brain rhythms [2]. The brain 
rhythms are the periodic fluctuations of human EEG, which are 
associated with cognitive processes, physiological states, and 
neurological disorders [3]. Hence, the use of EEG can range 
from basic research to clinical applications [4]. Among the 
various applications of EEG, it is worth highlighting its recent 
application in brain computer interface (BCI) research [5], [6]. 

EEG consists of a neurophysiological measurement of the 
electrical activity generated by the brain through multiple 

electrodes placed on the scalp surface. EEG data are measured 
as the electrical potential difference between two electrodes: 
active and reference electrodes. At neurophysiological level, the 
electrical potential differences are mostly generated by the 
summation of both excitatory and inhibitory post-synaptic 
potentials in tens of thousands of cortical pyramidal neurons that 
are synchronously activated [7]. Hence, the brain sources of the 
electrical potentials recorded by EEG may be suited to an infinite 
number of configurations, thereby limiting the spatial resolution 
of scalp EEG. To overcome this drawback, several source 
localization methods have been proposed and their application 
with high-density EEG (HD-EEG) systems, such as 64-256 
electrodes, can lead to a remarkable improvement in EEG spatial 
resolution [3], [8]. 

Many applications require that EEG systems have to record 
continuously for several days or even weeks and this might easily 
yield several gigabytes (GB) of generated data, which makes 
compression algorithms necessary for efficient data handling. As 

ABSTRACT 
In this article we investigate the trade-off between the compression ratio and distortion of a recently published compression technique 
specifically devised for multichannel electroencephalograph (EEG) signals. In our previous paper, we proved that, when singular value 
decomposition (SVD) is already performed for denoising or removing unwanted artifacts, it is possible to exploit the same SVD for 
compression purpose by achieving a compression ratio in the order of 10 and a percentage root mean square distortion in the order of 
0.01 %. In this article, we successfully demonstrate how, with a negligible increase in the computational cost of the algorithm, it is 
possible to further improve the compression ratio by about 10 % by maintaining the same distortion level or, alternatively, to improve 
the compression ratio by about 50 % by still maintaining the distortion level below the 0.1 %. 

mailto:gcampobello@unime.it


 

ACTA IMEKO | www.imeko.org June 2022 | Volume 11 | Number 2 | 2 

an illustrative example, about 2.6 GB of data per day are 
generated by an EEG system recording the data from 64 
electrodes with a sampling rate of 250 Hz and a 16-bit resolution. 
It should be mentioned that intracranial EEG recordings can 
generate even terabytes (TB) of data per day [9]. Therefore, EEG 
data need to be largely compressed to efficiently manage their 
storage. Furthermore, data compression is necessary also to 
reduce both transmission rate and power consumptions when 
telemonitoring EEG via wireless [10], [11]. For instance, wireless 
wearable EEG systems for long-term recordings should operate 
under a low-power budget, due to limitation on battery lifetime, 
and then the power consumption needs to be significantly 
reduced by compressing the data before transmission [12]. 

Various EEG compression algorithms have been developed 
to minimize the number of bits needed to represent EEG data 
by exploiting inter and/or intra-channel correlations of EEG 
signals. 

EEG compression algorithms can be classified into two main 
categories: lossless and lossy compression [13], [14]. As the main 
goal of the compression algorithms is to reduce the size of the 
data, their performance is, typically, evaluated by using the 
compression ratio (CR), which is calculated as the ratio between 
the number of bits required to represent the original and 
compressed EEG data. 

Generally, lossy compression enables superior compression 
performance compared to the lossless counterpart but it cannot 
guarantee a reconstruction of the exact original data from the 
compressed version. In such a case, the percent root-mean-
square distortion (PRD) is used as indicator for assessment of 
the quality of the reconstructed signal, which is affected by the 
distortion introduced by the lossy compression. Typically, 
lossless compression algorithms are preferred in clinical practice 
to avoid diagnostic errors, since important medical information 
may be disregarded using lossy compression and, in addition, 
there is a lack of legislation and/or approved standards on lossy 
compression, making EEG reconstruction a more critical 
requirement than compression performance. On the other hand, 
the lossless compression approach has a limited impact on 
storage requirements for EEG applications. As a matter of fact, 
the use of the state-of-the-art lossless compression algorithms 
allows achieving typical compression ratios in the order of 2 or 3 
[15]-[21]. 

On the other hand, EEG signals are of very small amplitude, 

typically in the order of microvolts (V), and thus they can be 
easily contaminated by noise and artifacts, which should be 
filtered to highlight and/or extract the actual clinical information 
[22], [23]. To accomplish this task, digital filters and denoising 
procedures based on wavelets, principal component analysis 
(PCA) and/or independent component analysis (ICA) are often 
used [24]-[28]. 

This enables the development of near-lossless PCA/ICA-
based compression algorithms that can achieve much higher 
compression ratios than those obtained with lossless 
compression algorithms with a tolerable reconstruction 
distortion for the application of interest. Different near-lossless 
EEG compression schemes based on parallel factor 
decomposition (PARAFAC) and singular value decomposition 
(SVD) have been investigated and compared with wavelet-based 
compression techniques [29]. In most cases, PARAFAC leads 
achieving better compression performance but the maximum CR 
obtained with a PRD lower than 2 % was 4.96 [29]. A near-
lossless algorithm able to obtain a CR of 4.58 with a PRD in the 
range between 0.27 % and 7.28 %, depending on the specific 

dataset under study, has been proposed in [30]. More recently, a 
SVD-based compression scheme able to obtain 80 % data 
compression (i.e., CR = 5) with a PDR of 5 % has been reported 
in [31]. 

More recently, in [32] we proposed a near-lossless 
compression algorithm for EEG signals able to achieve a 

compression ratio in the order of 10 with a 𝑃𝑅𝐷 < 0.01 %. In 
particular, the algorithm has been specifically devised for 
achieving a very low distortion in comparison to other state-of-
the-art solutions. In this paper, we present an improved version 
of our previous algorithm and particular attention is given to 
achieve a good trade-off between compression efficiency and 
distortion. 

The rest of this paper is organized as follows. In Section II, 
we briefly review SVD and describe our original algorithm 
proposed in [32]. In Section III, we illustrate the proposed 
algorithm. In Section IV, we present the experimental results 
obtained on a real-world EEG dataset. Finally, future works and 
our conclusions are drawn in Section V. 

2. SINGULAR VALUE DECOMPOSITION 

EEG signals are easily contaminated with artifacts and noise 
and, therefore, they need to be filtered before extracting the 
actual clinical information. For this purpose, SVD-based PCA 
and ICA techniques are commonly used. 

In order to briefly review how SVD is exploited in this 

context, let us consider a high-density 𝑁-channel EEG system 
whose signals are sampled for a time interval 𝑇 and at a rate of 
𝑓𝑠 samples per second (sps). In this case, we have 𝑀 = 𝑇 ⋅ 𝑓𝑠 
samples per channel and thus an overall number of samples equal 

to 𝑁 ⋅ 𝑀. We assume that such as samples are represented by a 
𝑁 × 𝑀 matrix 𝐴. 

It is known from the SVD theory that it possible to 

decompose a matrix 𝐴 into three matrices 𝑈, 𝛴, and 𝑉, such that 
𝐴 = 𝑈𝛴𝑉 𝑇. In particular, 𝛴 is a diagonal matrix whose diagonal 
elements, i.e., 𝜎𝑖  with 𝑖 ∈ [1, . . . , 𝑁], are named singular values. 
Moreover, a rank 𝑘 approximation of 𝐴, i.e., 𝐴𝑘 = 𝑈𝑘 𝛴𝑘 𝑉𝑘

𝑇 , 

exists which minimizes the norm ||𝐴 − 𝐴𝑘 || and that can be 
obtained by considering the submatrices 𝑈𝑘  and 𝑉𝑘 , given by the 
first 𝑘 columns of 𝑈 and 𝑉, respectively, and the leading principal 
minor of order 𝑘 of 𝛴, i.e., 𝛴𝑘 , containing the first 𝑘 < 𝑁 
singular values. 

In the specific context of EEG, the desired rank 𝑘, and thus 
the number of singular values exploited for approximation, is 
chosen by clinicians, or other EEG experts, in order to reduce 
the effect of undesired artifacts and noise by keeping unaltered 
the clinical information. 

In this case the actual clinical information is contained in 𝐴𝑘 
and, with the aim of reducing storage resources that are needed 

to store EEG samples, it is mandatory to encode the matrix 𝐴𝑘 
in the most efficient manner. 

In [32] authors proposed a solution for the above problem by 
deriving the near-lossless compression algorithm, as reported in 
Figure 1. 

The basic idea of the algorithm is to decompose the matrix 

𝐴𝑘  into two matrices, 𝑋𝑘  and 𝑌𝑘 , such that 𝐴𝑘 = 𝑋𝑘 𝑌𝑘 . In 
particular, the matrices 𝑋𝑘  and 𝑌𝑘  can be obtained, as shown in 

step 2, by first evaluating the matrix 𝑆 = 𝛴1/2 and then 
considering the first 𝑘 columns of the matrix 𝑈𝑆 and the first 𝑘 
rows of the matrix 𝑆𝑉 𝑇 , i.e., 𝑋𝑘 = (𝑈𝑆)[: , 1 ∶ 𝑘] and 𝑌𝑘 =



 

ACTA IMEKO | www.imeko.org June 2022 | Volume 11 | Number 2 | 3 

(𝑆𝑉 𝑇 )[1 ∶ 𝑘, ∶] in Matlab-like notation. Successively (see step 3), 
maximum absolute values of the matrices 𝑋𝑘  and 𝑌𝑘 , i.e., 𝑚𝑋 =
max(|𝑋𝑘 |) and 𝑚𝑌 = max(|𝑌𝑘 |), are evaluated. Such values are 
used in the last step, i.e., step 4, to transform the floating-point 

matrices 𝑋𝑘  and 𝑌𝑘  into two integer matrices, �̃�𝑘  and �̃�𝑘 , on the 
basis of the following equations: 

�̃�𝑘 = round(𝑚𝑌 ⋅ 𝑋𝑘 ) 

�̃�𝑘 = round(𝑚𝑋 ⋅ 𝑌𝑘 ) .
 (1) 

Note that the round() operator in above equations is the 
usual rounding operator, i.e., it rounds a floating point number 
to the nearest integer number. 

It is worth observing that actual dimensions of the matrices 

𝐴𝑘, �̃�𝑘 , and �̃�𝑘  are, 𝑁 × 𝑀, 𝑁 × 𝑘, and 𝑘 × 𝑀, respectively. 

Thus, the number of elements in �̃�𝑘  and �̃�𝑘 is lower than the 
number of EEG samples in the matrix 𝐴𝑘. 

Therefore, the matrices �̃�𝑘  and �̃�𝑘  can be considered as an 
alternative but compressed representation of the matrix 𝐴𝑘. 

In particular, the expected compression ratio can be derived 
as follows. 

Let us indicate with 𝑤 the number of bits used to represent 
each EEG sample in 𝐴𝑘. Considering the actual dimensions of 
the matrices 𝐴𝑘, the overall number of bits needed for 
representing the matrix 𝐴𝑘 is 𝐵𝑜 = 𝑤 ⋅ 𝑁 ⋅ 𝑀. 

In the same way, if we suppose that 𝑤 + 𝑎 is the maximum 

number of bits needed to represent the elements of �̃�𝑘  and �̃�𝑘 , 
the overall number of bits needed for representing the 

compressed matrices �̃�𝑘  and �̃�𝑘  is at most 𝐵𝑐 = (𝑤 + 𝑎) ⋅ (𝑁 +
𝑀) ⋅ 𝑘 and therefore the compression ratio can be evaluated as 

𝐶𝑅 =
𝐵𝑜
𝐵𝑐

=
𝑤 ⋅ 𝑀 ⋅ 𝑁

(𝑤 + 𝑎) ⋅ (𝑁 + 𝑀) ⋅ 𝑘
 . (2) 

In particular, when 𝑤 + 𝑎 ≈ 𝑤 and 𝑀 >> 𝑁, the expected 
compression ratio of the proposed algorithm can be 

approximated as 𝐶𝑅 ≈ 𝑁/𝑘. Therefore, a considerable 
compression can be achieved when 𝑁 >> 𝑘, i.e., in the case of 
high-density EEG systems with correlated signals. 

For instance, in the case 𝑁 = 256 and 𝑘 = 15 we have 𝐶𝑅 ≈
17 so that each GB of EEG data can be compressed and thus 
stored in less than 60 MB. 

In their paper, authors proved that, given the matrices �̃�𝑘  and 

�̃�𝑘  and the scale factor 𝑠 = 𝑚𝑋 ⋅ 𝑚𝑌 , an effective approximation 

�̃�𝑘 of the matrix 𝐴𝑘 is given by the following equation 

�̃�𝑘 = round (
�̃�𝑘  �̃�𝑘

𝑠
) . (3) 

Basically, the above relation provides the reconstruction 
equation needed for decompression. 

Experimental results reported in [32] have shown that the 

maximum absolute error 𝑀𝐴𝐸 = |𝐴𝑘 − �̃�𝑘 | introduced by the 

above approximation is bounded by 𝑀𝐴𝐸 ≤ 2, that is a 
negligible error in comparison to the actual range of the original 

EEG samples, i.e. [−2𝑤−1, +2𝑤−1 − 1]. 

3. PROPOSED ALGORITHM 

In this section, we slightly modify the previous algorithm with 
the aim of: 

1) improving the compression ratio; 
2) parameterizing the algorithm. 
In particular, we derived a new version of the algorithm able 

to achieve different trade-offs between compression efficiency 
and distortion. 

Basically, the new algorithm exploits the fact that consecutive 

values in the matrix �̃�𝑘  are highly correlated. Therefore, a further 
reduction in the number of bits, and thus an increasing in the 
compression ratio, can be obtained by encoding the differences 

between consecutive values in �̃�𝑘 instead of the matrix �̃�𝑘 itself. 
More precisely, let us introduce the matrix 

�̃�𝑌𝑘 = [(�̃�𝑘 [1, : ])
𝑇 ; diff(�̃�𝑘

𝑇 )] , (4) 

where diff() returns the matrix of differences along the first 

dimension. It is worth observing that the matrix �̃�𝑘 can be exactly 

recovered from �̃�𝑌𝑘  as 

�̃�𝑘 = cumsum(�̃�𝑌𝑘 )
𝑇  , (5) 

where cumsum() is the cumulative sum of elements along the 
first dimension. 

Therefore, no further losses are introduced if, instead of the 

matrix �̃�𝑘, the matrix of differences �̃�𝑌𝑘  is stored or transmitted. 
On the basis of the previous observation, a new compression 

algorithm for EEG has been derived and can be summarized as 
shown in Figure 2. Note that, in comparison to the previous 
algorithm, we introduced a new step (see step 5), highlighted in 
bold for the sake of readability. 

Reconstruction, i.e., decompression, can be easily achieved by 

obtaining �̃�𝑘  with (5) and thus using again (3) to recover �̃�𝑘. 

 

Figure 1. Illustration of the compression algorithm proposed in [32].  

 

Figure 2. Illustration of the proposed compression algorithm.  

• INPUTS: an integer number 𝑘 < 𝑁 and a matrix 𝐴, formed 
by 𝑁 × 𝑀 EEG samples; 

• OUTPUTS: integer matrices �̃�𝑘  and �̃�𝑘 and the scale factor 
𝑠 = 𝑚𝑋 ⋅ 𝑚𝑌. 

• ALGORITHM: 

1) Use SVD to decompose 𝐴 as 𝐴 = 𝑈𝛴𝑉𝑇 

2) Obtain the matrices 𝑆 = 𝛴1/2, 
𝑋𝑘 = (𝑈𝑆)[: ,1: 𝑘] and 𝑌𝑘 = (𝑆𝑉

𝑇 )[1: 𝑘, : ] 
3) Evaluate 𝑚𝑋 = max(|𝑋𝑘 |), 𝑚𝑌 = max(|𝑌𝑘 |) 
4) Calculate 𝑠 = 𝑚𝑋 ⋅ 𝑚𝑌, 

�̃�𝑘 = round(𝑚𝑌 ⋅ 𝑋𝑘 ) and �̃�𝑘 = round(𝑚𝑋 ⋅ 𝑌𝑘 ) 

• INPUTS: an integer number 𝑘 < 𝑁, the matrix 𝐴 formed by 
𝑁 × 𝑀 EEG samples and a scale factor 𝐹; 

• OUTPUTS: integer matrices �̃�𝑘  and 𝐷�̃�𝑘 and the scale factor 
𝑠 = 𝑚𝑋 ⋅ 𝑚𝑌. 

• ALGORITHM: 

1) Use SVD to decompose 𝐴 as 𝐴 = 𝑈𝛴𝑉𝑇 

2) Obtain the matrices 𝑆 = 𝛴1/2, 
𝑋𝑘 = (𝑈𝑆)[: ,1: 𝑘] and 𝑌𝑘 = (𝑆𝑉

𝑇 )[1: 𝑘, : ] 
3) 𝑚𝑋 = max(|𝑋𝑘 |)/𝐹, 𝑚𝑌 = max(|𝑌𝑘 |)/𝐹 
4) Calculate 𝑠 = 𝑚𝑋 ⋅ 𝑚𝑌, 

�̃�𝑘 = round(𝑚𝑌 ⋅ 𝑋𝑘 ) and �̃�𝑘 = round(𝑚𝑋 ⋅ 𝑌𝑘 ) 
5) Calculate the matrix 

�̃�𝑌𝑘 = [(�̃�𝑘 [1, : ])
𝑇 ; diff(�̃�𝑘

𝑇 )] 



 

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Note that in the new algorithm we introduced a new input 

parameter, i.e., the scale factor 𝐹, which can be used to achieve 
different tradeoffs between compression efficiency and 

distortion. In particular, the factor 𝐹 is exploited for reducing 
𝑚𝑋 and 𝑚𝑌 (see step 3 in Figure 2). 

It is worth nothing that 𝑚𝑋 and 𝑚𝑌 in the new and previous 
algorithms assume the same values when 𝐹 = 1. 

Therefore, intuitively and as confirmed in experimental 
results reported in the next section, we have no difference in the 

distortion achieved by the two algorithms when 𝐹 = 1. 
Instead, by choosing a value of 𝐹 greater than 1, it is possible 

to achieve a greater compression ratio. 
This can be easily justified by observing that, according to (1), 

by reducing 𝑚𝑋 and 𝑚𝑌 we further reduce the dynamic range of 

the elements in the matrices �̃�𝑘  and �̃�𝑘  and thus the number of 
bits that are needed for their representation. 

Obviously, a greater compression ratio is obtained at the cost 
of a greater distortion. 

Nevertheless, experimental results reported in the next 
section show that the proposed algorithm improves the 
compression ratio by about 10 % by achieving the same 
distortion level of our previous algorithm, i.e., 0.01 %. Moreover, 
a substantial increase in the compression ratio, up to 50 %, can 
be achieved by still maintaining the distortion level below the 
0.1 %. 

  

4. MEASUREMENT-BASED RESULTS 

The proposed compression algorithm is applied to a dataset 
containing real EEG signals, which have been preprocessed by 
EEG experts to denoise and remove artifacts. The EEG dataset 
under study has been provided by CUNY School of Medicine 
(New York, NY, USA). This dataset refers to awake EEG 
recordings from the research work published in [25]. In this 
study, Tatti et al. have investigated the role of beta oscillations 
(13.5-25 Hz) in the sensorimotor system in a group of healthy 
individuals. In this experiment, participants were asked to 
perform planar reaching movements (mov test). Mov test required 
the participants to reach a target, located at different distances 
and directions, that appeared on a screen in non-repeating and 
unpredictable order at 3 seconds interval. Participants were asked 
to make reaching movements by moving a cursor on a digitizing 
tablet with their right hand to targets appearing on the screen. 
The total testing time was approximately five to six minutes for 
each EEG recording (96 targets). 

Each mov test was measured with a 256-channel high-density 
EEG system (HydroCel Geodesic Sensor Net, HCGSN, 
produced by Electrical Geodesic Inc., Eugene, OR, USA), 
amplified using a Net Amp 300 amplifier, and sampled at 250 Hz 
with 16-bit resolution using the Net Station software (version 
5.0). EEG was noninvasively recorded using scalp electrodes and 

electrode-skin impedances were kept lower than 50 k. The 
EEGLAB toolbox (v13.6.5b) for MATLAB (v.2016b) was used 
for off-line preprocessing of the gathered EEG data [33], [34]. 
The signal of each recording was first filtered using a finite 
impulse response (FIR) bandpass filter with a passband that 
extends from 1 Hz to 80 Hz and notch filtered at 60 Hz. Then, 
each recording was segmented in 4-seconds epochs and visually 
examined to remove sporadic artifacts and channels with poor 
signal quality. Moreover, ICA with PCA-based dimension 
reduction (max 108 components) was employed to identify 

stereotypical artifacts (e.g., ocular, muscle, and 
electrocardiographic artifacts). Only ICA components with 
specific activity patterns and component maps characteristic of 
artefactual activity were removed. Electrodes with poor signal 
quality were reconstructed with spherical spline interpolation 
procedures, whereas those located on the cheeks and neck were 
excluded, resulting in 180 signals. 

After the preprocessing, all signals were re-referenced to their 
initial average values and processed EEG data were exported in 
the European Data Format (EDF) [35] by means of the 
EEGLAB toolbox. 

In particular, with the aim of evaluating the performance of 
the proposed algorithm, six EDF files, which are related to three 
subjects (labelled with the Subject Numbers SN_M2, SN_M4 
and SN_M5) and two sets of mov tests for each subject 
(“alltrials_1” and “alltrials_4”), have been tested. 

Data range, number of samples, and a few other information 
of the above mentioned EDF files are reported in Table 1. It is 
worth observing that samples are represented with 16-bit integer 

numbers, i.e., 𝑤 = 16, and that the overall number of samples 
exploited for tests is more than 80 ⋅ 106. 

In order to apply the proposed algorithm, each EDF file has 
been read and related data have been processed in blocks of 

𝑁 × 𝑀 samples, where 𝑁 has been chosen equal to 180, i.e., 𝑁 
coincides with the number of EEG channels remained after the 

preprecessing phase, and 𝑀 has been fixed equal to 1,000. So 
that each block of samples represents 4 seconds of data recorded 
by the multichannel EEG systems. 

The proposed compression algorithm, i.e., the algorithm in 
Figure 2, has been applied to each block and the average 
compression ratio have been evaluated according to the relation: 

𝐶𝑅𝐹 =
1

𝐿
∑

𝐵𝑜,𝑖
𝐵𝑐,𝑖

𝐿

𝑖=1

 , (6) 

where 𝐿 represents the number of blocks processed, 𝐵𝑜,𝑖  is the 

number of bits that are needed to represent the 𝑖-th block before 
compression, and 𝐵𝑐,𝑖 is the number of bits that are needed to 
represent the same block but after compression. Note that we 

use the subscript 𝐹 to highlight the scale factor used for 
compression, e.g., 𝐶𝑅2 is the compression ratio achieved when 
𝐹 = 2. When the scale factor is not expressly stated we assume 
𝐹 = 1. 

Subsequently, each compressed block has been reconstructed 

according to (5) and (3). Finally, distortion metrics, i.e., 𝑃𝑅𝐷 and 
𝑀𝐴𝐸, have been evaluated on the whole EDF file using the 
following equations: 

𝑃𝑅𝐷 = 100 ⋅ √
∑ ∑ (𝑀⋅𝐿𝑗=1

𝑁
𝑖=1 𝑎𝑖𝑗 − �̃�𝑖𝑗 )

2

∑ ∑ 𝑎𝑖𝑗
2𝑀⋅𝐿

𝑗=1
𝑁
𝑖=1

 (7) 

𝑀𝐴𝐸 = max
𝑖,𝑗

|𝑎𝑖𝑗 − �̃�𝑖𝑗 | , (8) 

where �̃�𝑖𝑗  are the integer values obtained after reconstruction 

and 𝑎𝑖𝑗  are the original samples. 
In our experiments, we further evaluated the compression 

efficiency of the proposed algorithm with respect to the near-
lossless compression algorithm proposed in [32]. In particular, 

compression efficiency (𝐶𝐸) is here defined as: 



 

ACTA IMEKO | www.imeko.org June 2022 | Volume 11 | Number 2 | 5 

𝐶𝐸 = 100 ⋅
𝐶𝑅𝐹 − 𝐶𝑅0

𝐶𝑅0
 , (9) 

where 𝐶𝑅0 is the compression ratio obtained with the algorithm 
proposed in [32], i.e., the algorithm reported in Figure 1. 

Similarly, we use 𝑃𝑅𝐷0 and 𝑀𝐴𝐸0 for referring related distortion 
metrics. 

Compression results achieved with the proposed 

compression algorithm by setting the scale factor 𝐹 = 1 are 
shown in Table 2. 

More precisely, for each compressed file, we reported the 

number of singular values exploited for compression (𝑘), the 
compression ratio (𝐶𝑅1), values of the distortion metrics (𝑃𝑅𝐷 
and 𝑀𝐴𝐸), and the compression efficiency (𝐶𝐸) of the proposed 
algorithm and, in brackets, corresponding compression results 
obtained with the algorithm proposed in [32], evaluated on the 
same files and considering the same number of singular values. 

As it is possible to observe, the compression ratio achieved 

by the proposed algorithm when 𝐹 = 1 is near 𝑁/𝑘, which 
confirms our analytical results reported in Section III. 

Note that the PRD is less than 0.01 % for all EDF files tested. 

In particular, PRD values obtained with 𝐹 = 1 are the same 
values obtained in [32]. The same consideration can be extended 
to the MAE. This confirms that the two algorithms in Figure 2 
and Figure 1 have the same performance in terms of distortion 

when 𝐹 = 1. 
However, by observing the results on compression efficiency 

(see the last column of Table 2), it is possible to conclude that, 
in comparison to the algorithm proposed in [32], the new one 
proposed in this paper is able to improve the compression ratio 

in a range between 7 % and 9 %. Moreover, the scale factor 𝐹 
introduced in the new algorithm provides the possibility to 

achieve even higher compression ratios, obviously at the cost of 
a greater distortion. 

We investigated the trade-off between compression efficiency 
and distortion of the proposed algorithm by considering 

different values of the scale factor 𝐹 within the range [1, 16]. 
In particular, we reported in Table 3 compression ratios 

(𝐶𝑅𝐹), distortion metrics (𝑃𝑅𝐷 and 𝑀𝐴𝐸), and compression 
efficiency (𝐶𝐸) corresponding to 𝐹 ∈ {1,2,4,8,16}. 

As can be observed in Table 3, by increasing 𝐹 it is possible 
to improve the compression ratio and thus the compression 

efficiency. In particular, by fixing 𝐹 = 16, the proposed 
algorithm is able to improve the compression ratio by about the 

50 % by maintaining the 𝑃𝑅𝐷 below the 0.1 % threshold (in fact 
the 𝑃𝑅𝐷 is at most equal to 0.081 % for all the files tested). 

It is also worth noting that the MAE obtained in our 

experimental results is approximatively equal to 2 𝐹. 
Finally, we evaluated the distribution of absolute errors in 

recovered signals. 
In Figure 3 we reported the cumulative distribution function 

(CDF) of the absolute errors, i.e., the probability 𝑃(|𝑒𝑟𝑟| ≤ 𝑥) 

Table 1. EDF files used as dataset. 

File name Channels Duration (s) Number of samples Physical Range (μV) Data range 

SN_M2_alltrials_1 180 344 15 480 000 [-35.500, +30.314] [-32768, 32767] 

SN_M2_alltrials_4 180 340 15 300 000 [-48.550, +38.539] [-32768, 32767] 

SN_M4_alltrials_1 180 328 14 760 000 [-34.532, +34.756] [-32768, 32767] 

SN_M4_alltrials_4 180 264 11 880 000 [-41.100, +40.673] [-32768, 32767] 

SN_M5_alltrials_1 180 324 14 580 000 [-41.463, +38.867] [-32768, 32767] 

SN_M5_alltrials_4 180 308 13 860 000 [-41.347, +46.929] [-32768, 32767] 

Table 2. Compression ratio (CR), percent root-mean-square distortion (PRD), 
maximum absolute error (MAE), and compression efficiency (CE) of the 
proposed algorithm when F = 1 (values of CR0 and PRD0 are reported in 
brackets). 

File name k CR1 PRD (%) MAE CE (%) 

SN_M2_alltrials_1 20 8.6 
(7.9) 

0.0065 
(0.0065) 

2 (2) 8.6 

SN_M2_alltrials_4 20 8.6 
(7.9) 

0.0063 
(0.0063) 

2 (2) 8.9 

SN_M4_alltrials_1 12 14.6 
(13.6) 

0.0075 
(0.0075) 

2 (2) 7.4 

SN_M4_alltrials_4 12 14.4 
(13.4) 

0.0069 
(0.0069) 

2 (2) 7.5 

SN_M5_alltrials_1 13 13.4 
(12.5) 

0.0071 
(0.0071) 

2 (2) 7.2 

SN_M5_alltrials_4 13 13.2 
(12.3) 

0.0069 
(0.0069) 

2 (2) 7.3 

Table 3. Compression results achieved with the proposed algorithm for 
different values of the scale factor F. 

File name F CRF PRD (%) MAE CE (%) 

SN_M2_alltrials_1 1 8.58 0.006 2 8.6 
 2 9.23 0.010 4 16.8 
 4 9.98 0.018 8 26.3 
 8 10.87 0.035 16 37.6 
 16 11.94 0.069 32 51.1 

SN_M2_alltrials_4 1 8.60 0.006 2 8.9 
 2 9.25 0.010 4 17.1 
 4 10.01 0.017 8 26.7 
 8 10.91 0.033 17 38.1 
 16 11.98 0.066 31 51.6 

SN_M4_alltrials_1 1 14.55 0.007 2 7.4 
 2 15.68 0.012 5 15.7 
 4 16.99 0.021 7 25.4 
 8 18.53 0.041 15 36.8 
 16 20.39 0.081 34 50.5 

SN_M4_alltrials_4 1 14.43 0.007 2 7.5 
 2 15.53 0.011 4 15.7 
 4 16.81 0.019 8 25.3 
 8 18.33 0.036 16 36.6 
 16 20.14 0.072 31 50.1 

SN_M5_alltrials_1 1 13.42 0.007 2 7.2 
 2 14.45 0.011 5 15.4 
 4 15.66 0.020 8 25.1 
 8 17.08 0.039 20 36.4 
 16 18.79 0.077 34 50.1 

SN_M5_alltrials_4 1 13.16 0.007 2 7.3 
 2 14.16 0.010 4 15.5 
 4 15.31 0.019 9 24.9 
 8 16.67 0.037 16 36.0 
 16 18.30 0.073 35 49.3 



 

ACTA IMEKO | www.imeko.org June 2022 | Volume 11 | Number 2 | 6 

that the absolute error (|𝑒𝑟𝑟|) is lower than a threshold 𝑥, 
achieved for different values of 𝐹. 

As can be observed in Figure 3, the percentage of samples 

with an absolute error lower than 𝐹 after reconstruction is near 
to 100 % for all the EDF files in the dataset. Note that the 

vertical lines in Figure 3 represent the condition 𝑃(|𝑒𝑟𝑟| ≤ 𝐹)). 
Therefore, we can state that the scale factor 𝐹, which is 

needed as input in the proposed algorithm, can be fixed 

according to the desired MAE, i.e., for a given value of 𝐹, the 
MAE obtained after reconstruction will be, with high probability, 

within the range [𝐹, 2𝐹]. 

5. CONCLUSIONS 

In this paper, we developed and validated an improved 
version of a recently proposed near-lossless compression 
algorithm for multichannel EEG signals. The algorithm exploits 
the fact that SVD is usually performed on EEG signals for 
artifacts removal or denoising tasks. Experimental results, 
reported in this paper, show that the developed algorithm is able 
to achieve a compression ratio proportional to the number of 
EEG channels with a root-mean-square distortion less than 
0.01 %. Moreover, with proper settings of input parameters, the 
compression ratio can be further improved up to 50 % by 
maintaining the distortion level below the 0.1 %. Moreover, the 
algorithm allows the desired maximum absolute error to be fixed 
a priori. It should be highlighted that, although an EEG dataset 

has been considered as a case study, the proposed compression 
algorithm can be quite straightforwardly applied to different 
types of dataset. In a future work, we will further investigate 
performance of the proposed algorithm considering more 
extended datasets and other types of signals. 

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Figure 3. Cumulative distribution function (CDF) of the absolute errors obtained after reconstruction with the proposed near-lossless algorithm for different 
scale factors (F). Vertical lines represent the condition P(|err| ≤ F). 

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