1 Introduction

The electroencephalogram (EEG) signal denoising is considered a challenge due to various noises in the artifact such as eye blinking, eye motion, muscle activity, and interference with power line that can alter the original EEG signal during the recording time. Therefore, the EEG signals must be processed to achieve efficient EEG usability in order to eliminate these noises. Several techniques to reduce EEG noises such as EEG signal denoising by wavelet transform (WT) were recommended. Wavelet transform’s success depends on how best its control parameters are configured and often set experimentally. Various algorithms for optimizing configurations are used to find optimal WT parameters such as \(\beta \)-hill climbing [1]. Flower pollination algorithm (FPA) is classified as a natural-inspiring algorithm based on the flowering plants’ pollination behavior. The FPA is established by Yang in 2012 [2]. The FPA has adhered successfully for a number of optimization problems and is outperformed some selected algorithms in a previous study [3, 4]. Yang et al. [5] mooted an initial attempt to extend FPA by a random weighted sum method to deal with multi-objective engineering optimization problems (MOFPA). The MOFPA has been assessed to achieve optimum results using a few engineering optimization problems. Subsequently the same authors proposed a novel MOFPA technique [6], with several multi-objective tests functions and two bi-objective design benchmarks. Compared with other algorithms the outcomes of the proposed algorithm were highly efficient. The MOFPA has been used to tackle real world issues such as the radial distribution system [7], dynamic economic dispatch in consideration of emissions [8], transmission losses and power plant emissions, cost reduction, increased voltage stabilization [9], power loss reduction [10], and power influx problem [11].

This paper mainly aims to propose an efficient EEG signal denoising technique using a multi-objective flower pollination algorithm with wavelet transform (MOFPA-WT). The (MOFPA-WT) method is used to obtain the best parameters combination for EEG signal denoising by means of two objective functions, min(MSE) and max(SNR). The proposed method is used to combine several objectives into a composite one-objective function in accordance with a weighted sum approach. The original EEG signal is from the standard EEG dataset, Motor Movement/Imagery datasetFootnote 1 [12]. This set of data consists of 109 volunteers, and record EEG signals from 64 EEG channels based on mental tasks. The results of denoised EEG signal are assessed in terms of five measurement factors for the evaluation of the MOFPA-WT, namely SNR, SNR improvement, MSE, RMSE and PRD. It should be noted that in comparison to those methods based on FPA-MSE and FPA-SNR, the proposed method MOFPA-WT achieves efficient EEG signal denoising based on all criteria.

This paper has been arranged in the following sections. Section 2 provides an EEG background and its applications. Section 3 explains a wavelet transform and EEG wavelet-based denoising. Section 4 discusses the background of the flower pollination algorithm (FPA). Section 5 provides a description of the MOFPA-WT for tuning WT parameters. In Sect. 6 we describe the results and the discussion. Lastly, Sect. 7 outlines the conclusion and future work.

2 Electroencephalogram (EEG)

Electroencephalogram (EEG) is an electric brain activity graph recorded from the scalp. This recording shows variability in voltage resulting from ionic fluxes within the brain’s neurons [13,14,15]. EEG signals can, therefore, generate a lot information needed on brain activity. Brain EEG signals are captured by invasive or noninvasive methods [16, 17]. The major difference among these techniques is the use of electrode arrays in the brain, such as ECoG BCI for arm motion control, as the invasive method [18]. In the meantime, there are a range of techniques for recording brain activities, including EEG for electrical activity of scalp, MEGs for magnetic field fluctuations in the brain and fMRI and fNIR for changes in oxygenation level in blood resulting from neural activity [18].

H. Berger first suggested the noninvasive EEG signals could be used to collect brain activities [19]. Researchers have developed the technique of Berger for a variety of applications over several decades. In clinical applications, for example, EEG signals were used for prevention, diagnosis-detection, rehabilitation, and patient restoration. The method is also used in non-medical applications like education and self-regulation, neuromarketing and advertisements, smart environment and neuroergonomics, games and entertainment, as well as education and learning [20,21,22]. EEG signals have recently successfully been employed in safety and authentication applications as a new biometric technique [13, 20, 23, 24].

More than one artifact noises, such as eye blinking, eye movement, muscle activity and electronic device intervention, could typically corrupt the original EEG signal during the recording time [25]. Consequently, to reduce such noise, the EEG signal need to be manipulated. Various EEG noise elimination techniques like filtering and adaptive thresholding were suggested in the literature. WT was well adapted to denoise non-stationary signals, including ECG and EEG, in the last several years [1, 26,27,28,29,30,31,32,33,34].

The wavelet denoising has five parameters with different ranges of each parameter (Table 1). The efficiency of EEG signal denoising depends on the selection of the optimal combination of WT parameters. Usually, the parameters are selected based on experience or empirical evidence. In past studies, the configuration of the WT parameter is formulated to optimize MSE as its objective function [1, 32].

El-Dahshan tried to achieve an optimal configuration of wavelet denoising parameters by a genetic algorithm (GA) in [26], for ECG signals. The outcomes of its WT-based GA were better than any of those generated experimentally. Nguyen et al. in [35] suggested a genetically adaptive thresholding method for ECG signal denoising. They have used a MIT-BIH set of data [36] to test their method, which corrupts the original ECG signal with white Gaussian noise (WGN) and various SNR input noise levels. MSE and SNR were used to evaluate the performance of this procedure. WGN has effectively been eliminated from the ECG signal through their method. In order to acquire the optimum wavelet parameters, Alyasseri et al. [29] suggested that an ECG signal denoisation system based on \(\beta \)-hill climbing(\(\beta \)-hc) optimization with wavelet denoising be implemented [37]. Subsequently, the same authors used their hybrid techniques for EEG signals, and successfully obtained the optimal wavelet denoising parameters for non-stationary signals, like ECG and EEG [1, 31, 32].

The wavelet denoising has distinct advantages to denoise the non-stationary signal such as the ECG and the EEG [38]. However, there is still a problem with the current efforts when it is used for signal denoising. The major drawback for the current version is that sometimes the noise is decreased and the power of the original signal is exhausted. This is normally the case in the denoising process, as previous work has solely sought to obtain a minimum (MSE) denoising EEG signal. This is why the authors propose to develop a multi-objective function that balance the power of the denoted EEG signal by using the maximum SNR, which will reduce the mean squared error.

3 EEG denoising using wavelet transform (WT)

The wavelet transform (WT) is a powerful and common tool for the representation of time frequency domain signals. WT has been applied successfully for signal compression, function selection and other applications [29, 39]. The WT can usually be categorized in two types: the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT) [40]. Over the last few years, the WT has been widely applied in non-stationary signals like ECG and EEG, since several EEG artifact effects are proven harmful to the original EEG signal. These noises originated from eye blinking, eye movements, muscles activity, power line and EMG [34, 41,42,43]. This paper employs the DWT to denoise EEG signal. One of the methods for DWT is applied in [44] and so-called Donoho’s approach. Thus, the DWT is defined as follows [45]:

$$\begin{aligned} C(a,b)= \sum _{n \in Z} x(n)g_{j,k}(n) \end{aligned}$$
(1)

where C(a,b) denotes the wavelet dynamic coefficients, \(a=2^{-j}\), \(b=k2^{-j}\), \(j \in Z\), \(k \in Z\),Z is the set of integers; a is the size of the time scale, b is the translation, x(n) is the input EEG signal, and \(g_{j,k}(n)=2^{j/2}g(2^{j}n-k)\) is the DWT.

DWT’s task is to degrade the input signal by means of various coefficients, to correct both the high and low input signal frequencies. Figure 1 shows the denoising process of the wavelet with three decomposition levels (L=5).

Fig. 1
figure 1

EEG denoising process

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In general, the wavelet denoising process has three phases which are summarized as follows:

  • EEG signal decomposition, the original EEG signal is decomposed into five levels. The EEG signal is divided up into two parts on each level namely Approximation coefficients (cA), and Detail coefficients (cD). The cD will process using high-pass filter and cA will continue decompose for next level.

  • Thresholding where a threshold value is defined for each level according to the noise level coefficients.

  • Reconstruction, the EEG denoised signal is reconstructed using inverse discrete wavelet transform iDWT.

The wavelet denoising contains five parameters with different types of parameters (Table 1). Noise reduction efficiency depends on the choice of wavelet parameters. As shown in Fig. 1, the process of wavelet denoising has three phases. The first stage involves the decomposition of the EEG signal by DWT. In this stage, the right mother wavelet function (\(\Phi \)) is chosen to be utilized for the EEG signal decomposition task. The second WT parameter, i.e., decomposition level (L), is normally based upon the EEG signal and experience in this phase. The selection of the WT valuable parameters (one of the key objectives of this paper) has recently been carried out using FPA and GA optimizing techniques.

Fig. 2
figure 2

Soft and hard thresholding methods

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In the second phase, thresholding is applied. The wavelet has two standard thresholds (\(\beta \)), i.e., a threshold that is hard (h) and soft (s) [44, 46]. The distinction between hard and soft thresholding is shown in Fig. 2. All threshold type (soft (s) or hard (h)), the rules of selection (\(\lambda \)), and the methods of rescalation (\(\rho \)) are to be chosen. These threshold mechanisms must be implemented as the choice will affect the overall performance of denoising. The thresholding value is usually based on the amplitude of noise (\(\sigma \)) [26]. The various parameters for thresholding selection rule and rescaling methods are shown in Tables 2 and 3. Finally, Equation (2) is used for the thresholding rules.

$$\begin{aligned} s(n)= x(n)+ \sigma e(n) \end{aligned}$$
(2)

where x(n) is the original EEG signal, e is the noise, \(\sigma \) is the amplitude of the noise, and n is sample number. The wavelet parameters (\(\beta \), \(\lambda \), and \(\rho \)) must be separately applied for each wavelet coefficient (cA and cD) level. In the last phase, the denoised EEG signal is reconstructed by iDWT.

Table 1 WT Parameters of signal denoising ranges
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Table 2 Selection options of wavelet thresholding
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Table 3 Rescaling approaches of the wavelet thresholding
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4 Flower pollination algorithm

According to [47], meta-heuristic algorithms can be categorized as evolutionary algorithms [48, 49], swarm intelligence [50,51,52,53,54], and trajectory algorithm [29, 37, 55, 56].

FPA is a successful swarm-based intelligence based on the pollination behavior of flowering plants. FPA was introduced by Yang in 2012 [2] and has been successfully applied for many optimization problems [3]. The operators (rules) of FPA are summarized as follows:

  • Operator (1): Global pollination encompasses biotic and cross-pollination where pollinators carry pollens based on Levy flights.

  • Operator (2): Local pollination necessitates abiotic and self-pollination.

  • Operator (3): The probability of reproduction can be seen as the flower constancy which is commensurate to the similarity of any two flowers.

  • Operator (4): The switch probability \({p}\in [0,1]\) can be controlled between global and local pollination. Due to certain external factors, e.g., wind, local pollination will play a major part in the overall pollination activity.

These rules are described in further detail in the following subsection.

4.1 Global search of FPA (biotic)

As previously indicated, pollens of the flowers are transferred to long distances through this type of pollination, by bees, bats, birds, etc. This ensures that the most fit element is pollinated and reproduced. We can thus mathematically represent the first and third rules of the FPA as follows:

$$\begin{aligned} x_i^{itr+1}= x_i^{itr} + L_{dis}(g^*-x_i^{itr}) \end{aligned}$$
(3)

where \(x_i^{itr+1}\) the pollen i or solution vector \(x_i\) at iteration itr, and \(g^*\) is the current best solution found among all solutions at the current iteration. The parameter \(L_{dis}\) is the strength of the pollination, which essentially is a step size. Since insects may move over a long distance with various distance steps, we can use a Levy flight to mimic this characteristic efficiently [2, 3, 57]. That is, we can use a Levy flight to imitate this feature effectively

$$\begin{aligned} L_{dis}\sim \frac{\lambda \Gamma (\lambda )sin(\pi \lambda /2)}{\pi }\frac{1}{s^{1+\lambda }}, (s>> s\_0 > 0) \end{aligned}$$
(4)

\(\Gamma (\lambda )\) denotes the standard gamma function, and this distribution is valid for large steps \(s~> ~\)0. In all our simulations below, we have used \(\lambda \) =1.5.

4.2 Local search of FPA (abiotic)

There is no pollinator in this type of pollination, where it centered on the wind and diffusion to shift the pollen. The local pollination (Operator 2) and flower constancy (Operator 3) can be represented as follows:

$$\begin{aligned} x_i^{itr+1}=x_i^{itr} + \epsilon (x_j^{itr}-x_k^{itr}) \end{aligned}$$
(5)

where \(x_j^{itr}\) and \(x_k^{itr}\) are pollens from the different flowers of the same plant type. This essentially mimic the flower constancy in a limited neighborhood. Mathematically, if \(x_j^{itr}\) and \(x_k^{itr}\) come from the same species or selected from the same population, this become a local random walk if we draw \(\epsilon \) from a uniform distribution in [0,1].

4.3 Switch probability in FPA

This is the key feature of the FPA algorithm, because the performance of the FPA will effect according to the p value, where its value will determine which path will follow either global or local pollination. According to [2], the value of p can be used \(p= 0.8\) to achieve the best results for most applications.

5 Proposed MOFPA with wavelet transform (WT) for denoising electroencephalogram (EEG) signals

This section provides a discussion of the proposed MOFPA with WT (MOFPA-WT) to solve EEG signal denoising problem (see Algorithm 1). WT parameter setting is crucial to yield efficient denoising results. Therefore, wavelet denoising parameters setting is formulated as an optimization problem. In our previous attempt, the optimal parameter setting for wavelet denoising were found by FPA, in which mean squared error (MSE) was used as an objective function [1, 58]. Aside from MSE, signal-to-noise (SNR) can also evaluate the quality of any solution. SNR is another important criterion for evaluation that should be considered.

Yang et al. in [6] claimed that any problem must be initially solved using single-objective function before adopting multi-objective functions. Therefore, EEG signal denoising problem was adopted using a single-objective function in [1, 58]. In the present study, the proposed method MOFPA-WT runs through four phases, where the result of each phase is an input to the succeeding one. Figure 3 shows the flowchart of these phases and thoroughly described as follows:

Fig. 3
figure 3

Proposed method

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figure a
Phase I: :

Initialization of EEG signals and WT parameters. This phase involves three steps. First, the input EEG signal x(n) is read from the source. The WT denoising approach was developed based on an original EEG signal being corrupted by three noises, namely white Gaussian noise (WGN), power line noise (PLN), and electromyogram (EMG) estimation [26, 59, 60] the corrupted formulas of which are shown in Eqs. (6), (8), and (10), respectively. These types of noises simulate the noises that will corrupt the original EEG signal during the recording time, such as eye blinking, eye movement, and electro signal distortion. Figure 4 shows the original EEG signal and noisy EEG signals.

$$\begin{aligned} \mathrm{Noise}_\mathrm{PLN}=x(n)+N_\mathrm{PLN}(n) \end{aligned}$$
(6)

where \(N_\mathrm{PLN}(n)\) is

$$\begin{aligned} N_\mathrm{PLN}(n)= A*sin(2*\pi *f*t) \end{aligned}$$
(7)
$$\begin{aligned} \mathrm{Noise}_\mathrm{EMG}=x(n)+N_\mathrm{EMG}(n) \end{aligned}$$
(8)

where \(N_\mathrm{EMG}(n)\) is defined as follows:

$$\begin{aligned} N_\mathrm{EMG}(t)= A*rand(n),~~~~ rand \in [0,1] \end{aligned}$$
(9)
$$\begin{aligned} \mathrm{Noise}_\mathrm{WGN}=x(n)+N_\mathrm{WGN}(n) \end{aligned}$$
(10)

where \(N_\mathrm{WGN}(n)\) is

$$\begin{aligned} N_\mathrm{WGN}(t)= x(n)+ \sigma e(n) \end{aligned}$$
(11)

where A = 0.15 uV, f = 50 Hz, e is the noise, and \(\sigma \) is the amplitude of the noise in this work \(\sigma \)=15 \(\mu \)V. \(N_{PLN}\), \(N_{EMG}\), and \(N_{WGN}\) signals are added to the original EEG signal x(n) to simulate the actual noises.

Second, WT denoising parameters are initialized (i.e., \(\Phi \), L, \(\beta \), \(\lambda \), and \(\rho \)) which are shown in Table 1, as well as the parameters of MOFPA are also initialized as shown in Table 5. Finally, the input EEG signals are evaluated in terms of SNR by Eq. (14), PRD by Eq. (18), MSE by Eq. (13), and RMSE by Eq. (17) to record the results of EEG signals before and after the denoising process.

Phase II: :

Tuning WT parameters using MOFPA. This phase is the core work of this paper. The performance of WT depends on its initial parameters. Therefore, MOFPA is used to find the optimal parameters configuration. Initially, the solution of WT parameters configuration is represented as a vector \(Sol=(s_1,s_2, \ldots \,s_D)\), where D is the total number of parameters used for WT, which is normally equal to 5. \(s_1\) represents the value of the mother wavelet function (MWF) parameter \(\Phi \), \(s_2\) denotes the value of the decomposition level parameter L, \(s_3\) refers to the thresholding method \(\beta \), \(s_4\) represents the value of the thresholding selection rule parameter \(\lambda \), and \(s_5\) represents the re-scaling approach \(\rho \); the possible ranges for these parameters are selected from Table 1.

Fig. 4
figure 4

Original EEG signal corrupted using PLN, EMG, and WGN

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Figure 5 shows an example of selection of the optimal solution of WT parameters for denoising EEG signals using MOFPA. The final result of this phase is an optimized solution \(Sol'_{opt}=(s'_1,s'_2, \ldots \,s'_D)\), which will be passed onto the next phase.

The process of tuning WT parameters by using MOFPA is summarized as follows.

  1. 1.

    Initialize a set of solutions from the possible ranges of WT parameters using MOFPA. Moreover, initialize the switch probability in FPA P.

  2. 2.

    Calculate the two objective functions (i.e., min(MSE), Max(SNR)) for all solutions and determine the current best solution (g*) to be used for the global pollination later. For the local pollination, the solution will be selected randomly from the population.

  3. 3.

    Generate the rnd value and compare it with the switch probability in FPA P to manipulate the current solution (\(X_{i}\)) with global or local pollination to create the new solution (\(X'_{i}\)).

  4. 4.

    Evaluate the new solution (\(X'_{i}\)); if an improvement is observed, replace it with the current solution (\(X_{i}\)) and proceed to the next solution (\(X_{i+1}\)). Repeat Steps 3 and 4 for all solution. The proposed MOFPA-WT evaluates the solution using the multi-objective function, which is formulated in Eq. (12). This method is applied using two objective functions, namely min(MSE) and max(SNR) to achieve the best combination of WT parameters for EEG signal denoising.

    $$\begin{aligned} F_{MOFPAWT}= & {} (W_{1}*\mathbf{min} (MSE))+(W_{2}*\mathbf{max} (SNR)) \end{aligned}$$
    (12)
    $$\begin{aligned} MSE= & {} \frac{1}{N} \sum _{n=1}^N [x(n)- {\widehat{x}}(n) ]^2 \end{aligned}$$
    (13)
    $$\begin{aligned} SNR_{out}= & {} 10 \log _{10} \big \{ \frac{ \sum _{n=1}^N [x(n)]^2}{\sum _{n=1}^N [x(n)- {\widehat{x}}(n) ]^2} \big \}, \end{aligned}$$
    (14)

    where x(n) denotes the original EEG signal, and \({\widehat{x}}\)(n) is the denoised EEG signal obtained by tuning the wavelet parameters using MOFPA.

  5. 5.

    Update the current best solution (g*).

  6. 6.

    Repeat Steps 3–5 based on \(Max_{itr}\)

  7. 7.

    The Pareto optimal set contains the set of solution with the best value for two objective functions (i.e., min(MSE), max(SNR)) or at least one objective function. Finally, the Pareto front solution will be selected from the Pareto optimal set (best solution in the Pareto optimal set).

Phase III::

EEG denoising using WT based on \(Sol'_{opt}\). As mentioned in Sect. 3, the denoising process of WT involves three main steps, as shown in Fig. 1 and described as follows.

  • EEG signal decomposition using DWT. In this step, DWT is applied to decompose the noise of the input EEG signals x(n). The first two \(Sol'_{opt}\) parameters, namely MWF \(\rho \) and decomposition level L, should be used in the decomposition process. Figure 9a shows the DWT procedure for five levels, where the noisy EEG signal is divided at each level into cA and cD. The latter is processed using a high-pass filter, whereas the former is processed using a low-pass filter and is decomposed for the next level.

    The EEG signal is convolved using the high- and low-pass filters, while the block (\(\downarrow \)2), which is represented by the downsampling operator, is used to keep the even index elements of the EEG signal. The EEG signals are separated into cA and cD based on their frequency and amplitude.

  • Thresholding. This step is applied based on the noise level of the coefficients. In this step, the last three wavelet parameters, namely thresholding type (\(\beta \)), thresholding selection rules (\(\lambda \)), and re-scaling methods (\(\rho \)), must be selected from \(Sol'_{opt}\).

  • Reconstruction of the denoising EEG signal by iDWT. The value of the original EEG signals \({\widehat{X}}\) is estimated by applying iDWT on \({\widehat{X}}\) as follows:

    $$\begin{aligned} {z}[n]={i}DWT[{\widehat{X}}] \end{aligned}$$
    (15)

    The reconstruction convolves the EEG signals using upsampling (\(\uparrow \)2), which involves the insertion of zeros at the even index elements of the EEG signals. Figure 9c shows the iDWT procedure for five levels.

Phase IV::

EEG Denoising Evaluation. The final phase is evaluating the EEG outcome of WT. The performance of MOFPA-WT is evaluated according to five criteria, namely SNR formulated in Eq. (14), SNR improvement Eq. (16), MSE formulated in Eq. (13), RMSE formulated in Eq. (17), and PRD formulated in Eq. (18).

$$\begin{aligned} SNR_{imp}= & {} 10 \log _{10} \big \{ \frac{ \sum _{n=1}^N [\delta (n) - x(n)]^2}{\sum _{n=1}^N [x(n)- {\widehat{x}}(n) ]^2} \big \} \end{aligned}$$
(16)
$$\begin{aligned} RMSE= & {} \sqrt{ \frac{1}{N} \sum _{n=1}^N [x(n)- {\widehat{x}}(n) ]^2} \end{aligned}$$
(17)
$$\begin{aligned} PRD= & {} 100* \sqrt{ \frac{\sum _{n=1}^N [x(n)- {\widehat{x}}(n) ]^2}{ \sum _{n=1}^N [x(n)]^2} } \end{aligned}$$
(18)

where x(n) denotes the original EEG signal, \({\widehat{x}}(n)\) is the denoised EEG signal obtained by tuning the wavelet parameters through MOFPA, and N is the number of samples.

Fig. 5
figure 5

Mechanism of MOFPA for selecting the optimal WT parameters for denoising EEG signals

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6 Results and discussions

The EEG dataset is discussed in Sect. 6.1, the Pareto front evaluation describes in Sect. 6.2, an example to illustrative the behavior of MOFPA-WT in denoising EEG signals is explained in Sect. 6.3, the evaluation of the proposed method is presented in Sect. 6.4, comparative evaluation of MOFPA-WT with state-of-the-arts are given in Sect. 6.5, and finally comparative evaluation the proposed method with other multi-objective techniques 6.6.

To select the best parameter settings for the proposed method (MOFPA-WT), a sensitivity analysis has been conducted for the switch probability p and population size PoP parameters. The study shows the effect of both parameters using different values such that \(p=\{0.0,0.2,0.5,0.8,1.0\}\) and \(PoP=\{10,20,50\}\). The analysis study involves all measures which are the MSE, SNR, PRD, RSME, \(SNR\_imp\) as long as the computation time. As shown from the results recorded in Table 4, the best performance of the (MOFPA-WT) achieved using p=0.8, and PoP=20.

Table 4 Sensitivity analysis of MOFPA-WT to its p and population size PoP parameters for PLN noises for motor imaging EEG dataset
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The proposed MOFPA-WT is implementing using MATLAB R2017a on a LENOVO Ideapad 310, Intel Core i7, RAM 12G. Table 5 shows the parameter setting of MOFPA-WT method for denoising EEG signals.

Table 5 Parameters setting for MOFPA
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In general, the parameter values are selected according to experiments and literature. The switch probability p= 0.8 is selected because it is when the MOFPA achieves the best value for most applications [5, 6]. This is also proved after experimental study shown in the previous section. For the number of iterations, the experiments reveal that no significant improvement is observed for the WT parameters after 100 iterations (see Figs. 13, 14, and 15).

6.1 Datasets

A standard EEG signal dataset, i.e., motor/imagery, is tested on MOFPA-WTFootnote 2 [12]. EEG signals from 109 healthy subjects are collated with a brain-computer interface software BCI2000 system [61]. The EEG signals are recorded utilizing 64 fibers (EEG networks), where every antenna is decrypted in a discrete EDF file. Every volunteer conducts multiple motory/imagery tasks that are principally utilized in special disciplines, those of neurological rehabilitation and brain-computer interface applications. In general, these chores comprise of envisaging or simulating a particular action, such as opening and closing the eyelids. The EEG signals are recorded from each volunteer by asking them to perform four tasks depending on the position of the target on the screen. When the target is displayed on the right or right side of the screen, the volunteer must open and close his/her fist, which corresponds with the target position on the screen. The volunteer must open and close his/her fists or feet, if this target appears at the top or bottom of the screen.

6.2 Pareto front evaluation

The first evaluation criterion of MOFPA-WT is the Pareto front. According to [5, 62], the Pareto front (PF) can be defined as the set of non-dominated solutions:

$$\begin{aligned} PF = \{ {\mathbf {x}} \in {{\mathcal {F}}} | \not \exists {\mathbf {x}}' \in \mathcal{F}: {\mathbf {f}}({\mathbf {x}}') \prec {\mathbf {f}}({\mathbf {x}}) \}, \end{aligned}$$
(19)
Fig. 6
figure 6

Comparison results of Pareto front and MOFPA-WT for PLN noises

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where \({{\mathcal {F}}}\) is the feasible search space formed by all feasible solution vector \({\mathbf {x}}\). A solution \({\mathbf {x}}\) is said to dominate another solution \({\mathbf {x}}'\) if and only if \(x_i \le x_i'\) for \(\forall i \in \{1, 2, ..., D\}\) in the D-dimensional space. This dominance relationship can be written compactly as

$$\begin{aligned} {\mathbf {x}} \prec {\mathbf {x}}'. \end{aligned}$$
(20)

In addition, \({\mathbf {f}}=(f_1, f_2)\) corresponds to the two objectives shown in Fig. 6, where \(f_1\) and \(f_2\) refers to mean squared error (MSE) (Eq. 13) and signal-to-noise ratio (SNR) (Eq. 14), respectively.

Fig. 7
figure 7

Comparison results of Pareto front and MOFPA-WT for EMG noises

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Fig. 8
figure 8

Comparison results of Pareto front and MOFPA-WT for WGN noises

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Figures 6, 7, and 8 show the Pareto fronts and MOFPA-WT for the three different EEG noise which are PLN, ENG, and WGN, respectively. As aforementioned, we have two objectives functions: \(f_1\) represents the MSE and \(f_2\) refers to SNR. The blue points in these figures represent the Pareto fronts which are a set of non-dominated solutions to the denoising process. The plotting curve shows that there are impacted trade-offs between \(f_1\) and \(f_2\). While the red points represent the selected solutions achieved by the MOFPA-WT. Note that the selected solution is chosen based on domination criterion wherein the “min MSE” and “max SNR” are improved. The sub-figures in Figs. 6 and 7 represent the magnified portion of the original figure to show the performance of the MOFPA-WT for EEG signal denoising in specific point.

To elaborate the discussion about the performance of the proposed MOFPA-WT for EEG signal denoising, the effects of PLN and EMG noise shown in Figs. 6 and 7 are emphasized in which the denoising performance has highly influenced where it is always close to the top-left corner. This means that the MOFPA-WT is always providing higher signal-to-noise ratios. On the other hand, in Fig. 8, the MOFPA-WT obtained a sparse values and distributed along Pareto fronts approx. This can be noticed by the results achieved for WGN noise because the input EEG signal has been corrupted with a large amount of noise. For that, the proposed MOFPA-WT tried to select the dominant solutions where these solutions should have “min MSE” and “max SNR”.

6.3 Illustrative example

To explain further the behavior of MOFPA-WT in denoising EEG signals, EEG signal x(n) is supposed to have been denoised using MOFPA-WT. As mentioned in the above section and in accordance with the example in Fig. 5, MOFPA-WT involves four phases. In Phase I (initialization), the input EEG signal x(n) is corrupted with three noises types of noise, namely WGN, PLN, and EMG, as shown in Eqs. (6), (8), and (10), respectively. The SNR, MSE, RMSE, and PRD of x(n) are computed for the noised EEG signal x(n), as shown in Eqs. (14), (13), (17), and (18), respectively. Meanwhile, the MOFPA parameters are initialized as follows: switch probability (p)= 0.8, maximum number of iterations (\(Max_{itr}\))= 100, population size= 20, and dimension of the search variables (d)= 5.

In Phase II (tuning wavelet denoising parameters by MOFPA), MOFPA finds the optimal solution of the WT parameters \(Sol'_{opt}\) to address the EEG signal denoising problem. For example, let the solutions in the MOFPA population be given as shown in Fig. 5. Table 6 shows how the MOFPA iteration loops to improve the WT parameters until \(Sol'_{opt}\), which contains the optimal parameter values to be used by WT, is obtained. In Iteration 1, MOFPA determines \(Sol'_{opt}(1)\) from the initial population, where \(f(MOFPAWT)(1)= 22.2975\). In Iteration 2, \( Sol'_{opt} (2) \) with \(f(x(2))=98.3952\) is generated. Notably, the \( Sol'_{opt}\) dose not replace the previous \(Sol'_{opt})(1)\) because the current solution is worse than the previous solution. MOFPA continues to look for an optimal solution. In Iteration 8, MOFPA produces \(Sol'_{opt}(8)\) with \(f(MOFPAWT (8)=0.0075\). The best solution \(Sol'_{opt}\) will replace the previous \(Sol'_{opt}(1)\). According to Table 6, the next improvement of \(Sol'_{opt}\) is achieved at Iteration 14 where \(f(MOFPAWT)(14)=0.0075\) and the best solution \(Sol'_{opt}\) will be replaced by \(Sol'_{opt}(8)\). Thereafter, nothing changes until the last iteration is reached. The optimal parameter setting for WT is given by \(Sol'_{opt}(100)\). The \(Sol'_{opt}\) solution obtained by MOFPA in this example is shown in Fig. 7. \(Sol'_{opt}\) solution will be passed onto the next phase.

Table 6 Some recorded iterations using MOFPA to achieve optimal WT parameters
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Table 7 \(Sol'_\mathrm{opt}\) WT parameters for EEG denoising
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Phase III (EEG denoising using WT with \(Sol'_{opt}\)) involves three steps, namely EEG signal decomposition using DWT, thresholding and reconstruction. Figure 9 shows the EEG denoising process. In the decomposition step, the noisy EEG signal is decomposed using MWF \(\rho \) = bior1.3 and decomposition level L= 5 while the EEG signal is divided into five levels based on the value of L. Each level is filtered separately using the wavelet function \(\rho \) = bior1.3 to obtain a smooth EEG signal. Fig. 9 shows the decomposition process. In Thresholding, the threshold value \(\delta \) is defined based on the noisy coefficients of each level, and each cD is processed using \(Sol'_{opt}\) parameters. The \(Sol'_{opt}\) for thresholding has the following parameters: thresholding type (\(\beta \)) = soft, selection rules (\(\lambda \)) = rigrsure, and rescaling methods (\(\rho \)) = sln. The denoised EEG signal is then reconstructed using iDWT. Figure 9c shows the reconstruction of an EEG signal with five decomposition levels.

Fig. 9
figure 9

EEG denoising process

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Suppose there is an EEG signal and we need to denoise it using wavelet denoising method with the following parameters MWF \(\rho \) = bior1.3, decomposition level L= 5, thresholding type (\(\beta \)) = soft, selection rules (\(\lambda \)) = rigrsure, and rescaling methods (\(\rho \)) = sln. Therefore, in the decomposition phase, the EEG signal is decomposed using MWF \(\rho \) = bior1.3 and decomposition level L= 5 while the EEG signal is divided into five levels based on the value of L. Each level is filtered separately using the wavelet function \(\rho \) = bior1.3 to obtain a smooth EEG signal. Figure 10 shows the decomposition process. In Thresholding, the threshold value \(\delta \) is defined based on the noisy coefficients of each level, and each cD is processed using wavelet denoising parameters. As mentioned above, the wavelet denoising for thresholding has the following parameters: thresholding type (\(\beta \)) = soft, selection rules (\(\lambda \)) = rigrsure, and rescaling methods (\(\rho \)) = sln. Figure 11 shows the application of thresholding for denoising EEG signals with five decomposition levels (i.e., d1, d2, d3, d4, and d5) based on the value of wavelet denoising parameters. The dotted blue line in Fig. 11 represents the thresholding value \(\delta \) for each level. The denoised EEG signal is then reconstructed using iDWT.

Fig. 10
figure 10

EEG Decomposition process for five levels

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Fig. 11
figure 11

Thresholding and Reconstruction steps

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6.4 Evaluation of multi-objective functions in MOFPA-WT

The second experiment is conducted to show the effect of the Multi-objective functions on the performance of MOFPA-WT. Three versions of FPA-based WT are designed to evaluate the proposed multi-objective function in MOFPA-WT. The first is the proposed MOFPA-WT, which combines MSE and SNR as a multi-objective function, to find the optimal parameter setting for WT. The second is FPA with only MSE to find the optimal parameter settings for WT (i.e., FPA-MSE-WT). The third is FPA with only SNR to find the optimal parameter setting for WT (i.e., FPA-SNR-WT).

Five measures, namely SNR, SNR improvement, MSE, RMSE, and PRD, are used to evaluate the EEG signal quality, which is denoised by using the proposed MOFPA-WT. Table 8 shows the optimal WT parameters for EEG signals denoising achieved by MOFPA-WT, FPA-SNR-WT, and FPA-MSE-WT, and Table 9 shows the performance of these three methods according to PLN, EMG, and WGN noises for EEG motor movement/imaging dataset. The quality of the final outcomes of the denoised EEG signal using MOFPA-WT, FPA-SNR-WT, and FPA-MSE-WT are compared. Figure 12 shows the evaluation results of the denoising of EEG signals according to the five measures. MOFPA-WT achieves the best results for EEG signal denoising with different noises compared with FPA-SNR-WT and FPA-MSE-WT. Conversely, FPA-SNR-WT and FPA-MSE-WT shows deformation of the EEG signal in some cases, such as with EMG and WGN noises. For the PLN, the proposed MOFPA-WT achieves the highest SNR (54.3575) and SNR improvement (0.0555); meanwhile, where the other two techniques cannot preserve or improve the SNR value, in which FPA-MSE-WT and FPA-SNR-WT achieve -0.0397 and 0, respectively. For the EMG noise, the proposed method also achieves the highest SNR improvement (0.0116), whereas the SNR improvement for FPA-MSE-WT and FPA-SNR-WT is -0.0240 and 0, respectively. For high amount of noise, such as in WGN, MOFPA-WT obtains an SNR improvement of 1.1947, whereas those of FPA-SNR-WT and FPA-MSE-WT are 0 and 0.6180, respectively.

Table 8 Optimal WT parameters obtained by FPA-SNR-WT, MOFPA-WT, FPA-MSE-WT for PLN, EMG, and WGN noises for motor imaging EEG dataset
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Table 9 Performance of MOFPA in denoising EEG signals according to PLN, EMG, and WGN for EEG motor imaging dataset
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Also, the results of MOFPA-WT are evaluated using Wilcoxon signed-rank statistical test [63] to determine the significance of MOFPA-WT for EEG signal denoising compared with the other methods. Table 10 shows the pair-wise comparison between the MOFPA-WT and other methods (FPA-MSE-WT and FPA-SNR-WT). The proposed MOFPA-WT shows significant results for all EEG datasets used.

Table 10 Wilcoxon signed-rank test evaluation
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Fig. 12
figure 12

Comparison results of FPA-SNR, MOFPA-WT, and FPA-MSE for EMG, PLN, and WGN noises

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The MOFPA-WT is tested using different noise types, namely PLN, EMG, and WGN. Figs. 13, 14, and 15 show the efficient performance of MOFPA-WT, especially with PLN and EMG noises, where achieving the optimal solution at around 5 and 13 iterations, respectively, is possible; however, more than 90 iterations are needed to obtain the optimal WT parameters for WGN noise.

Fig. 13
figure 13

Convergence results of PLN

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Fig. 14
figure 14

Convergence results of EMG

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Fig. 15
figure 15

Convergence results of WGN

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6.5 Comparative evaluation MOFPA-WT with other multi-Objective techniques

Initially, the performance of the MOFPA-WT is compared with two popular multi-objective methods such as NSGA-II [64] and MOPSO [65]. The power line noise (PLN) was used to evaluate the performance of the proposed method (MOFPA-WT) with selected popular techniques. The parameters of NSGA-II used in the experiments are population size \(PoP = 50\), \(\mathrm{crossover} = 0.5\), \(\mathrm{mutation} = 0.5\), \(itr = 100\). Also, the parameters of MOPSO used in the experiments are population size \(PoP = 50\), accelerate constant \(c1 = 2\), \(c2 = 2\), maximum and minimum velocity \(vmax = 0.9\), \(vmin = 0.1\), and total number of iteration \(itr = 100\). These values are carefully selected such as used in [64] and [65]. Compared with NSGA-II [64] and MOPSO [65], MOFPA-WT produces better results and Table (11) shows comparative results using 5 measures, MSE, RMSE, PRD, SNR and \(SNR\_imp\). Moreover, MOFPA-WT, MOGA-WT and MOPSO-WT convergence rates are presented in which the suggested method (MOFPA-WT) provides faster archives than MOGA-WT and MOPSO-WT, for the best solution. Figure 16 shows the comparison results of MOFPA-WT with NSGA-II [64] and MOPSO [65] using PLN noises.

Table 11 Comparative evaluation MOFPA-WT with NSGA-II (MOGA-WT) and MOPSO (MOPSO-WT)
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Fig. 16
figure 16

Comparison results of MOFPA-WT with other multi-objective techniques using PLN noises

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Table 11 summarized the results of 100 repetitions. To further evaluate the results of MOFPA-WT compared with NSGA-II [64] and MOPSO [65]. T-test is conducted based on mean to determine the significance of MOFPAWT for EEG signal denoising. Table 12 shows the pair-wise comparison between the MOFPA-WT and other methods (NSGA-II-WT and MOPSO-WT). The proposed method (MOFPA-WT) shows significant results compared with both methods.

Table 12 T-test evaluation
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6.6 Comparing the MOFPA-WT with state-of-the-art methods

The MOFPA-WT is compared to four advanced EEG signal denoising proposed by Al-Qazzaz et al. [28], Kumari et al. [13], Bhatnagar et al. [66], and Al-Salman et al. [67]. The comparison is carried out on the basis of standard data sets, i.e., motor/imaging [12], with PLN and EMG corrupt in their original EEG signal [59, 60]. Five criteria, i.e., MSE, RMSE, \(SNR_{imp}\), and PRD, are used to assess the final outcome. Table 13 describes the performance of MOFPA-WT and these methods. Apparently, the proposed approach (MOFPA-WT) achieves the best results in comparison with other methods for every comparative measures.

Table 13 Comparing the proposed MOFPA-WT method with state-of-the-art methods for EEG signals denoising with different noises
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7 Conclusion and future work

A novel technique for EEG signal denoising problem based on MOFPA-WT is proposed in this study. The most successful denoising technique in the signal processing domain is wavelet transform (WT). The success of WT is based on the configurations of its five parameters (i.e., (i) MWF \(\Phi \), (ii) decomposition level L, (iii) thresholding function \(\beta \), (iv) threshold selection rule \(\lambda \), and (v) threshold re-scaling method \(\rho \)). The configuration of the WT parameter is critical and accomplished based on user experience. In this study, the problem is modeled as a multi-objective function based on minimum mean squared error (MSE) and maximum signal-to-noise-ratio (SNR). The optimization method called FPA is used with the multi-objective function to set the optimal WT parameter, thereby improving the EEG signal denoising results.

MOFPA-WT is evaluated using a standard EEG dataset, namely the EEG motor movement/imagery dataset. This dataset contains data of 109 volunteers, and EEG signals are captured from 64 EEG channels based on different mental tasks. These EEG signals are corrupted using three different noises, namely PLN, EMG, and WGN [26, 59, 60]. Five evaluation criteria are used, namely SNR, SNR improvement, MSE, RMSE, and PRD. To show the effect of the multi-objective aspects on the performance of the proposed MOFPA-WT, the behavior of FPA for EEG signal denoising is tested using FPA with MSE, FPA with SNR, and FPA with MSE and SNR. The experiments show that MOFPA-WT can provide more subjective results than the other two. The convergence rate and Pareto front are also studied for the proposed MOFPA-WT.

For future work, one of the possible works is to apply independent component analysis (ICA) [68], e.g., before apply the proposed method (MOFPA-WT) to achieve better results for EEG signal denoising. Also, MOFPA-WT will be applied for more challenging signal problem instances, such as person authentication or early detection of epilepsy based on EEG signal. Furthermore, the real-world applications are required to show the efficiency of MOFPA-WT.