1 Introduction

The automatic classification of movement-related EEG signals has received increasing attention in the field of brain-computer interfaces (BCIs). BCIs are a way to establish a direct connection between the human brain and external devices [1], and noninvasive EEG-based BCIs can be used for the rehabilitation of brain nerves [2]. EEG-based BCIs include P300, motor imagery (MI), and other types. In most cases, the P300 paradigm requires the subject's eyes to always look at the visual stimuli appearing on the screen, and at the same time, the EEG signal of this process is acquired. The P300 signal generally appears 300–400 ms after the event and is a positive potential signal [3]. The latency and amplitude of the P300 signal indirectly reflect the emotional and psychological changes of the subject when facing a stimulus or reflecting a potential intent [4]. The advantage of the P300 signal is that the subject can generate it without training because it is one of the internal signals in the brain [5]. However, when performing the analysis, it is necessary to overlay and average the signals many times to observe the changes in the EEG signal caused by the stimulation event. Furthermore, during the process of acquiring signals, the subject’s eyes will be easily fatigued, and the practical effect is possibly reduced [6]. Motor imagery refers to a process of thinking that uses the brain to imagine the movement of its limbs, but there is actually no movement [7]. This thinking process is spontaneous by the brain and does not require external stimulation [8].

BCIs based on motor imagery have recently received increasing attention. To obtain a more effective EEG MI BCI, there are generally two optimization strategies: one is to maintain the high separability of the MI EEG features during feature extraction, and the other is to optimize the classifier based on the specific needs. In contrast, the use of the former as an optimization strategy has attracted increasing attention in recent years. In machine learning, many methods exist for EEG feature extraction, which can be divided into two categories according to the purpose of the processing: temporal filtering and spatial filtering. The temporal filter focuses on the temporal aspects of EEG signals, and its advantage lies in its low computational complexity while retaining an acceptable discrimination performance among simple MI tasks [9]. Spatial filtering reduces the influence of spatial noise. Typical spatial filters include common spatial pattern (CSP) and common average reference (CAR) filters [10]. These spatial filter methods are helpful to solve the problem of MI binary classification. In 1999, G. Pfurtscheller and F.H. Lopes da Silva [11] confirmed that when preparing and performing an exercise with one hand, the amplitude of the α band (8–12 Hz) and β band (13–30 Hz) in the contralateral sensorimotor cortical EEG signals decreased. This phenomenon is called event-related desynchronization (ERD), which is the reduced amplitude of activated cortical EEG signals. At the same time, the amplitude of the α band (8–12 Hz) and β band (13–30 Hz) in the cerebral cortex signaled on the same side as the exercising hand increased; this signal is called event-related synchronization (ERS). This outcome means that the corresponding cortex increased in amplitude in the resting state [12]. Some studies have also indicated that motor imagery tasks are associated with attenuation or increase in localized brain rhythm activity called ERD/ERS [13].

One of the most popular and efficient methods to extract ERD/ERS-related features is the CSP, which is widely used for motor imagery BCI designs [14, 15]. CSP aims to find an optimal spatial filter by maximizing the filtered variance of one class and minimizing the filtered variance of the other class simultaneously [16]. Generally, the improvement of CSP can be divided into two categories according to the direction: one is to optimize the objective function, and the other is to optimize the channel quality of EEG signals. Some studies adopt the second strategy to improve CSP; for instance, Dong et al. [17] established the CSP-CIM to improve the robustness of CSP with respect to outliers. The correntropy-induced metric (CIM) can saturate when two vectors are far apart from each other in the input space, and this property makes it appropriate for learning problems with outliers in the data and helps to build a robust objective function [17]. Of course, there are also studies that adopt the first strategy to improve CSP. Y. Park and W. Chung [18, 19] considered a local CSP generated from individual channels and their neighbors (termed “local regions” [20]) rather than a global CSP generated from all channels. To overcome some of the shortcomings of the global CSP approach, a local temporal CSP (LTCSP) was developed in [16]. Recently, Z. Yu et al. [21] proposed introducing a weight function based on the phase-locking value (PLV) [22] into the framework of LTCSP. B. Chakraborty et al. [23] also considered using the phase information and amplitude information of the EEG signal to model the CSP. The novelty of this method is that the formula for calculating phase information and amplitude information is completed with the help of complex space. A sparse CSP algorithm that incorporates the sparse technique and iterative search into the CSP was proposed by Fu et al. [24]. Geirnaert et al. [25] proposed a new attention decoding method by using the filter-bank common spatial pattern filters (FB-CSP). Compared to the traditional auditory attention decoding (AAD) algorithms, FB-CSP does not require access to clean source envelopes. More recently, an integrated framework, termed common time–frequency-spatial patterns (CTFSP), was proposed to extract sparse CSP features from multiple frequency bands in multiple time windows [26].

A new optimization method that can increase the difference between different classes of motor imagery tasks is desirable. Recent studies have indicated that the use of the Euler representation for feature extraction has greatly contributed to the improvement of image classification accuracy. By combining these studies [27,28,29,30], some advantages of the Euler representation have been confirmed as follows. (a) The Euler representation obtains a larger margin, which is helpful for image classification by using Euler representation to measure the distance between pixels in each image. (b) Moreover, compared with the measurement of images of the same category, the Euler representation can obtain a larger margin when measuring the distance between images of different categories. In other words, it can obtain a better separability between different class images.

Inspired by these previous research results, it is also possible to utilize the Euler representation for the extraction of EEG signal features. Therefore, in our proposed method, we apply the Euler representation to discriminate the characteristics of EEG data between the different MI tasks. In this paper, we present a novel feature extraction method for MI classification, which is called e-CSP. The method uses the Euler representation as a kernel function in the input of the CSP. Each sample value is mapped to a complex space by the formula of the Euler representation, and then the CSP is performed in Euler space. We use linear discriminant analysis (LDA) to observe the classification performance of our proposed method. Higher classification accuracy indicates higher class separability. A higher separability can be considered a better motor imagery discrimination performance [31]. The rest of the paper is organized as follows. Section 2 introduces the conventional CSP algorithm and our proposed method. The experiment and dataset description is presented in Sect. 3. Computational results are analyzed in Sect. 4. Finally, the conclusions are drawn in Sect. 5.

2 Methods

2.1 Common spatial pattern

The input EEG signals can be defined as a matrix of [N*C*T], where C is the number of EEG channels, N represents the number of time samples per trial, and T is the number of trials. It is assumed here that the data matrix of each trial is zero mean, i.e., each class of data is the center matrix. The sample covariance matrices of the two classes are computed as

$${R}_{1}=\frac{{X}_{1}{X}_{1}^{T}}{tr({X}_{1}{X}_{1}^{T})},{R}_{2}=\frac{{X}_{2}{X}_{2}^{T}}{tr({X}_{2}{X}_{2}^{T})},$$
(1)

where X1 and X2 are sample data from two classes, while R1 and R2 are each of the respective covariance matrices. The final spatial covariance \(\overline{{R }_{1}}\) and \(\overline{{R }_{2}}\) are computed by averaging over the trials under each class. Then, the total covariance matrix R can be formed as

$$R=\overline{{R }_{1}}+\overline{{R }_{2}}.$$
(2)

Due to the symmetry of the covariance matrix R, the eigenvalue decomposition is given by

$$R=UD{U}^{T},$$

where D is the diagonal matrix formed by the corresponding eigenvalues, and U is the eigenvector matrix of the matrix. Since the eigenvalues are arranged in ascending order [32, 33], descending order is performed first. Then, the whitening value matrix of R is computed as

$$P=\sqrt{{D}^{-1}}{U}^{T}$$
(3)

and transforms the mean covariance matrices \(\overline{{R }_{1}}\) and \(\overline{{R }_{2}}\) as

$${S}_{1}=P\overline{{R }_{1}}{P}^{T},{S}_{2}=P\overline{{R }_{2}}{P}^{T}.$$

Note that, by principal component decomposition, S1 and S2 can be formulated as

$${S}_{1}={B}_{1}{\Lambda }_{1}{{B}_{1}}^{T},{S}_{2}={B}_{2}{\Lambda }_{2}{{B}_{2}}^{T},$$

where B1 and B2 are the eigenvector matrices of S1 and S2, and \({\Lambda }_{1}\) and \({\Lambda }_{2}\) are associated eigenvalue matrices with diagonal entries. It can be demonstrated that the eigenvectors of the matrix and the eigenvector matrices are equal [32, 34]; at the same time, the sum of the diagonal arrays and of the two eigenvalue matrices is an identity matrix. That is

$${B}_{1}={B}_{2}=B,{\Lambda }_{1}+{\Lambda }_{2}=I\text{.}$$

Thus, the eigenvector corresponding to the largest eigenvalue allows S1 to S2 to have the smallest eigenvalue and vice versa. Finally, the projection matrix is the corresponding spatial filter W, which is defined as

$$W={B}^{T}P.$$
(4)

Using the projection matrix W, the EEG signals X1 and X2 are projected as

$$\begin{array}{ccc}{F}_{1}\text{=}{W}^{T}{X}_{1}& \mathrm{and}& {F}_{2}\text{=}{W}^{T}{X}_{2},\end{array}$$

where F1 and F2 (usually after a log transformation) are used for classification.

Combining the aforementioned analysis, we find that the spatial filter is obtained by comparing the filtered variance of one class to the filtered variance of the other class. Thus, the objective function of CSP can be written as

$$J(w)=\frac{tr({w}^{T}\overline{{R }_{1}}w)}{tr({w}^{T}\overline{{R }_{2}}w)},$$
(5)

where w is the spatial filter W obtained by the above process. \(\overline{{R }_{1}}\) and \(\overline{{R }_{2}}\) are computed by averaging over the trials under each class. When the objective function reaches a maximum, the spatial filter W is optimal.

2.2 Euler representation for CSP

2.2.1 Motivation

To solve the optimization problem of the spatial filter in the CSP objective function, we consider using the kernel trick property to capture the nonlinear similarity of features to improve the discrimination ability of CSP. Different from the commonly used kernel function, which maps data into a higher-dimensional hidden space, the Euler representation maps data into an explicit space that has the same dimensionality as that of the original data space. In our proposed Euler representation for the CSP method, the Euler representation is used as a kernel function in the input of the algorithm, and it is expected to improve the CSP. To better explain the formulation of the Euler representation for CSP, we first introduce the definition of the cosine kernel function, i.e., cosine distance, and then analyze its advantages.

Definition:

Given two arbitrary vectors \({x}_{j}\) and \({x}_{p}\in {R}^{m}\) [35], the cosine distance metric between them is.

$$d\text{(}{x}_{j}\text{,}{x}_{p}\text{)=}\sum_{c={1}}^{m}\{1-\mathrm{cos}(\alpha \pi ({x}_{j}(c)-{x}_{p}(c)))\},$$
(6)

where \({x}_{j}(c)\) is the c-th element of \({x}_{j}\).

In most feature extraction algorithms, the Euclidean distance is generally used as the measurement metric of discrimination between data. In fact, the cosine distance can also be used as a measurement metric. Two classes of motor imagery data are marked as A and B. Tables 1 and 2 list the distance between the same class and different classes of data under different measurement metrics. By comparing the ratio of the between-class distance to the within-class distance, we find that under the cosine distance metric, this ratio is always greater than that of the Euclidean distance. The results illustrate that the cosine distance is more advantageous in determining the difference between classes, which will greatly contribute to improving the accuracy of classification. Thus, the cosine distance metric, which is defined in Eq. (6), is a more effective kernel function.

Table 1 Comparison of normalized dissimilarity measures between the same class
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Table 2 Comparison of normalized dissimilarity measures between different classes
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Combining the aforementioned analysis and the comparison with the Euclidean distance, we can obtain the following two characteristics of cosine distance:

  1. (1)

    Using the cosine distance metric enlarges the distance between all sample value margins between different channels at the same time point. This enlargement demonstrates that the metric of the cosine distance can help obtain a large margin, making it advantageous for the classification of motor imagery.

  2. (2)

    Compared to Euclidean distance, the cosine distance metric can enlarge the distance between samples of different classes.

    Therefore, the cosine distance metric not only obtains a greater degree of discrimination between different channels but also increases the degree of discrimination between the different classes of motor imagery data. This outcome greatly improves classification accuracy. Combining the aforementioned analysis for the cosine distance metric, we build a robust formulation for CSP. By simple algebra, Eq. (6) can be rewritten as Eq. (7).

    $$\begin{array}{l}d\text{(}{x}_{j}\text{,}{x}_{p}\text{)=}\sum_{c={1}}^{m}\{1-\mathrm{cos}(\alpha \pi ({x}_{j}(c)-{x}_{p}(c)))\}\\ ={\Vert \frac{1}{\sqrt{2}}({e}^{i\alpha \pi {\text{x}}_{j}}-{e}^{i\alpha \pi {\text{x}}_{p}})\Vert }_{2}^{2}\\ ={\Vert {z}_{j}-{z}_{p}\Vert }_{2}^{2}\end{array}$$
    (7)

where

$${z}_{j}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{e}^{i\alpha \pi {\text{x}}_{j}(1)}\\ ...\\ {e}^{i\alpha \pi {\text{x}}_{j}(c)}\end{array}\right]=\frac{1}{\sqrt{2}}{e}^{i\alpha \pi {\text{x}}_{j}}$$
(8)

is called the Euler representation of \({x}_{j}\).

Equation (7) illustrates that the sample values of different channels at two time points obtained by the cosine distance metric are equivalent to the ℓ2-norm between the corresponding two vectors with the Euler representation. In addition, the ℓ2-norm is a better function expression of the optimization problem than the cosine distance in practical applications. Therefore, our proposed method first maps the sample values \({x}_{j}\) normalized in [0, 1] onto the complex representation \({z}_{j}\in {C}^{m}\) and then performs the complex CSP in Euler space. To more intuitively observe the contribution of the Euler representation for EEG data, we use its formulation, which is defined in Eq. (8) for EEG data. As shown in Fig. 1, the results indicate that the use of the Euler representation can obtain more consequential information. This outcome further confirms that our consideration is feasible and that the Euler representation is meaningful as an optimization strategy for CSP.

Fig. 1
figure 1

Contribution of the Euler representation for EEG data. a The traditional data; b the data using the Euler representation

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2.2.2 Euler CSP

The Euler representation is used to transform the two classes of motor imagery data, and for the EEG data of a single trial, \({Z}_{\text{n1}}\text{=}\frac{1}{\sqrt{2}}{e}^{i\alpha \pi {X}_{n1}}\), where \({X}_{n1}\) is a vector consisting of the sampled values of each channel at a time point, then the EEG data for a single trial can be denoted as \(Z=[{Z}_{n1},{Z}_{n2}...{Z}_{N}]\), and data for all trials are denoted as such.

By simple algebra [16], the objective function of CSP (Eq. (5)) gives

$$\begin{array}{l}J(w)=\frac{tr({w}^{T}\overline{{R }_{1}}w)}{tr({w}^{T}\overline{{R }_{2}}w)}\\ =\frac{tr({w}^{T}(\frac{1}{{T}_{1}}{\sum }_{i=1}^{{T}_{1}}{R}_{1}(i))w)}{tr({w}^{T}(\frac{1}{{T}_{2}}{\sum }_{i=1}^{{T}_{2}}{R}_{2}(i))w)}\\ =\frac{\frac{1}{{T}_{1}}{\sum }_{i=1}^{{T}_{1}}{\sum }_{j=1}^{C}({\gamma }_{j}^{T}{X}_{1}(i){X}_{1}{(i)}^{T}/tr({X}_{1}(i){X}_{1}{(i)}^{T}))}{\frac{1}{{T}_{2}}{\sum }_{i=1}^{{T}_{2}}{\sum }_{j=1}^{C}({\gamma }_{j}^{T}{X}_{2}(i){X}_{2}{(i)}^{T}/tr({X}_{2}(i){X}_{2}{(i)}^{T}))}\end{array},$$
(9)

where (i) denotes the i-th trial, \({\gamma }_{j}\) is the j-th column of matrix w, and T1 and T2 correspond to the numbers of trials for X1 and X2, respectively. On account of the use of the Euler representation for X1 and X2, the numerator of the objective function becomes

$$\frac{1}{{T}_{1}}{\sum }_{i=1}^{{T}_{1}}{\sum }_{j=1}^{C}\left({\gamma }_{j}^{T}{Z}_{1}\left(i\right){Z}_{1}{\left(i\right)}^{T}/tr\left({Z}_{1}\left(i\right){Z}_{1}{\left(i\right)}^{T}\right)\right),$$
(10)

and the denominator is likewise formulated. The covariance matrices of X1 and X2 are calculated by

$${G}_{1}=\frac{{Z}_{1}{Z}_{1}^{T}}{tr({Z}_{1}{Z}_{1}^{T})},{G}_{2}=\frac{{Z}_{2}{Z}_{2}^{T}}{tr({Z}_{2}{Z}_{2}^{T})},$$
(11)

where Z1 and Z2 are the Euler representations of X1 and X2, respectively. Then, the total covariance matrix G is calculated by

$$G={G}_{1}+{G}_{2}.$$
(12)

Hence, the objective function of e-CSP is rewritten as

$$J(Q)=\frac{tr({Q}^{T}\overline{{G }_{1}}Q)}{tr({Q}^{T}\overline{{G }_{2}}Q)},$$
(13)

where Q is the new projection matrix of e-CSP, \(\overline{{G }_{1}}\) is the spatial covariance matrix of the average trials for one class and \(\overline{{G }_{2}}\) is the spatial covariance matrix of the average trials for the other class.

The procedure of solving the objective function, i.e., e-CSP, is summarized in the following algorithm.

figure g

3 Data and experiment

In this section, we use two publicly available EEG data sets from BCI competitions and a new Motor Imagery dataset from Cho et al. [36] to evaluate the performance of the proposed algorithm and compare the results with the conventional CSP algorithm. In our experiments, linear discriminant analysis (LDA) and support vector machines (SVM) are used as classifiers. As shown in Fig. 2, the experiments mainly include three parts: preprocessing, feature extraction, and classification. To fully verify the performance of our proposed method, we design four experimental paradigms, which are described below.

  1. (1)

    For experiment 1, we compare the classification accuracies of the e-CSP with other optimization methods of CSP in the BCI competition III dataset IVa and BCI competition III dataset IIIa publicly available datasets under the LDA classifier.

  2. (2)

    For experiment 2, we empirically set the maximum number of projection vectors as the number of electrodes and set the parameter α in the e-CSP to 0.9, 1.2, and 0.5 in the BCI competition III dataset IVa, BCI competition III dataset IIIa, and Cho’s dataset, respectively. In addition, to obtain the accuracies of the e-CSP and the CSP, we also calculated the F1-score based on the accuracy to fully evaluate the performance of the e-CSP.

  3. (3)

    For experiment 3, we set the parameter α in the e-CSP to 1.1 in the BCI competition III dataset IVa and BCI competition III dataset IIIa. We set the parameter α in the e-CSP to 0.5 in Cho’s dataset. Then, we compare the classification accuracies of the e-CSP and CSP algorithms under the LDA and SVM classifiers.

  4. (4)

    For experiment 4, we respectively set the number of projection vectors to 30, 48, and 24 in the BCI competition III dataset IIIa, BCI competition III dataset IVa, and Cho’s dataset. We set the parameter α in the e-CSP from 0.1 to 1.9 to observe its influence on the accuracy.

Fig. 2
figure 2

The block diagram of the experiments

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All experiments are performed on the Windows 10 operating system, and the fivefold cross-validation was used for model evaluation. It includes three steps. First of all, the dataset was divided into five subsets, and then the four subsets of that were used to train the model while the rest subset was used to test and calculate the accuracy of the method; finally, this process was repeated five times such that each subset was used once as testing set. We recorded the final averaged accuracy. The rest of this section includes the description of the specific information of the dataset and the preprocessing of the data.

3.1 Data description

3.1.1 BCI competition datasets

The two publicly available datasets used to evaluate the performance of our proposed method are the BCI competition III dataset IVa and the BCI competition III dataset IIIa. The detailed information is described in Table 3.

Table 3 BCI competition datasets information
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3.1.2 Cho’s dataset

This dataset conducted a BCI experiment for motor imagery movement (MI movement) of the left and right hands with 52 subjects (19 females, mean age ± SD age = 24.8 ± 3.86 years). Each subject took part in the same experiment, and subject ID was denoted and indexed as S01, S02, …, S52. In this paper, we randomly select the data of 10 subjects to evaluate the performance of the proposed method. The detailed information is described in Table 4.

Table 4 Cho’s dataset information
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For the MI performance, some brief descriptions were provided [36]. The performance of subjects “S02” and “S23” are worse than that of the other subjects, as will be seen in the experiments.

3.2 Data preprocessing

According to the background knowledge of neurophysiology, it can be known that during imagination, the neuronal cluster discharge of the sensorimotor cortex of the brain can cause ERS or ERD [37]. The characteristic frequency of these phenomena mainly includes an α-rhythm of 8–12 Hz and a β-rhythm of 13–30 Hz. Thus, in the preprocessing phase, a fifth-order Butterworth filter was first used for 8–30 Hz bandpass filtering. As in [38], the cut-off speed will increase when the order of the filter increases. For MI signals, the selection of the 5th order filter usually leads to the optimal cut-off speed. For the signal data for feature extraction, EEG data between 0.5 and 2.5 s after the visual cue of the motor imagery task were selected because this is the best time period for the EEG to detect ERD/ERS [38].

4 Results and discussion

In this section, the results of the three experiments will be detailed in turn. The classification accuracy is used as an index to evaluate the performance of the algorithm in our study. Higher accuracy indicates higher class separability. A higher separability can be considered a better motor imagery discrimination performance. Moreover, the F1-score is also used as an index to compare the performance of the e-CSP and the CSP. The results include three parts: the performance comparison of e-CSP and CSP based on other optimization strategies, the performance comparison of e-CSP and CSP, and the impact of the parameter α in the Euler representation.

4.1 Performance comparison of the e-CSP with other methods on BCI competition datasets

To evaluate the performance of the proposed method, we also compare the classification accuracies of several other methods in two public competition datasets. We obtained the accuracies of the other three methods in the two competition datasets from previous studies [16, 21, 23]. The results are presented in Table 5, Table 6, and Fig. 3. Table 5 lists the classification accuracies of the proposed e-CSP and the other three methods in the BCI competition III dataset IVa. The e-CSP obtains higher classification accuracies for the subjects “av” and “aw” than the other three methods. The mean classification accuracy of the e-CSP is better than that of the other methods. The e-CSP classification accuracy is 22.02%, 12.92%, and 11.68% higher than those of the CSP phase + amp, LTCSP, and p-LTCSP, respectively. As seen in Table 6, the performance of the CSP phase + amp [23] is very unstable in comparison with the e-CSP. The LTCSP [16] and p-LTCSP [21] are both CSPs based on temporally local manifolds, and they both show better classification performance for subject “al” than the e-CSP. However, the mean classification accuracy of e-CSP is higher than that of the LTCSP and p-LTCSP.

Table 5 Classification accuracies of the proposed method and different methods in the BCI competition III dataset IVa. The bold content in each row of the table represents the highest accuracy achieved in that row
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Table 6 Classification accuracies of the proposed method and different methods in the BCI competition III dataset IIIa. The bold content in each row of the table represents the highest accuracy achieved in that row
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Fig. 3
figure 3

Average classification accuracies of the proposed method and different methods on the two datasets. a The BCI competition III dataset IVa, and b the BCI competition III dataset IIIa

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We also evaluate the performance of the e-CSP in the BCI competition III dataset IIIa. Table 6 shows the classification accuracies of the e-CSP, CSP phase + amp, LTCSP, and p-LTCSP. It is evident that the e-CSP also achieves the greatest mean classification accuracy. The mean classification accuracy of the e-CSP is better than that of the other methods, with accuracies that are 13.56%, 6.32%, and 6.24% higher than those of the CSP phase + amp, LTCSP, and p-LTCSP, respectively.

The classification accuracies of the four methods on the two datasets are presented in Fig. 3. By contrast, the e-CSP has relatively superior classification accuracies across the eight subjects. A T-test was performed for the e-CSP and one of the other methods one by one. The results also reveal that the accuracies of the e-CSP are significantly higher than other methods (P < 0.05). The results demonstrate that the Euler representation is indeed a meaningful optimization strategy based on CSP.

4.2 Performance comparison of e-CSP with CSP

In addition, to verify the effectiveness of the e-CSP on BCI competition datasets, we also performed experiments using Cho’s dataset. The average classification accuracies across all subjects of the conventional CSP and the proposed e-CSP with the number of projection vectors can be seen in Fig. 4 and Table 7. As shown in Fig. 4, some results are revealed in the curve of the classification accuracies across the 8 subjects.

  1. (1)

    When the number of projection vectors is small, the classification accuracy of the e-CSP is higher than that of the CSP, and when it reaches a certain threshold, the performance of the conventional CSP will be slightly higher than that of the e-CSP. For different subjects, the threshold points are different, but the overall performance trend is the same. As shown in Fig. 4(b), when the number of projection vectors is set as 40, the classification accuracy of the CSP and e-CSP is the same. When the number of projection vectors is less than 40, the classification accuracy obtained by the e-CSP is higher than that obtained by the CSP. However, when the number of projection vectors is greater than 40, the classification accuracy obtained by the e-CSP is lower than that of the CSP. The results illustrate that to achieve the same classification accuracy: the e-CSP requires fewer features than those of the CSP. In other words, the e-CSP is more effective.

  2. (2)

    In addition, principal component analysis (PCA) is used to select the features with a variance of 95% from the extracted features. During the evaluation process, it is found that the proportion of the effective features among all the features extracted by the e-CSP is always higher than that of the CSP when extracting the same number of features. This result further validates the effectiveness of the e-CSP.

  3. (3)

    As shown in Fig. 4(a), (b), and (d), although the accuracy obtained by the e-CSP is greater than 80% regardless of the number of the projection vectors, the accuracies show a decreasing trend. In previous works, the number of the pair of filters (i.e., projection vectors) was empirically set to 3 [37]. In this paper, in order to compare the impact of the number of projection vectors, we set it from the maximum value to the minimum value. The trend shown in the figure demonstrates that a few leading projection vectors contain more discriminative features.

Fig. 4
figure 4

Average classification accuracies versus the number of projection vectors in the BCI competition III dataset IVa and BCI competition III dataset IIIa. The horizontal axis in the figure represents the number of projection vectors, and the vertical axis represents the classification accuracy

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Table 7 Classification accuracies versus the number of projection vectors for each subject in the Cho’s dataset
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The above results indicate that, compared with conventional CSP, the e-CSP shows better classification performance when the number of projection vectors is relatively small. When the number of features extracted by these two algorithms is the same, the features extracted by the e-CSP contain more effective features. These results reveal that the e-CSP has better feature discrimination performance, which provides a more efficient proposal for feature extraction of EEG data. The good performance of the e-CSP can also be confirmed in the curve of classification accuracy of the left- and right-hand motor imagery tasks. In Fig. 4(g), it can be observed that under various projection vector numbers, the classification accuracy of the e-CSP is higher than that of the conventional CSP from a macro perspective. Moreover, the average accuracy of the e-CSP is significantly higher than that of CSP by performing a T-test (P < 0.05). In summary, by comparing the experimental results above, it is obvious that a substantial improvement of the CSP is obtained by the e-CSP in the majority of cases.

As shown in Table 7, some results similar to those on the two public datasets can also be observed in Cho’s dataset. Although the classification accuracy of the e-CSP is not the same when the number of projection vectors is different, the same is that the classification accuracy of the e-CSP is higher than the CSP in the majority of cases. Furthermore, this behavior becomes more obvious when the number of projection vectors is small. Some other phenomena are also deserved to be discussed. When the number of projection vectors was set as 48 and 52, the classification accuracies of subjects “S01” and “S43” obtained by the e-CSP are not greater than that by the CSP. It may be due to the impact of the parameter α. Thus, we also inspect the fluctuation of classification accuracy of the e-CSP with the change of the parameter α in the following section. In summary, the performance of Cho’s dataset has further verified the effectiveness of the e-CSP.

Based on the classification accuracy, we calculated the F1-score to further compare the performance of the two methods. F1-score is defined as the harmonic mean of precision and recall, whose range is [0, 1] [39]. The results are shown in Table 8. It is demonstrated that the F1-score of the e-CSP is higher than that of the CSP. It is worth mentioning that better accuracy does not mean a higher F1-score, but the higher F1-score represents the robustness of the method.

Table 8 Average F1-score of the CSP and the e-CSP for each subject in three datasets
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In Fig. 5, the average classification accuracy of the conventional CSP and e-CSP algorithms across all subjects under both LDA and SVM classifiers are obtained. In the experiment, the linear kernel function was set for the SVM classifier, and the LDA classifier was followed the Fisher criterion. In the majority of cases, the classification accuracy of the e-CSP is higher than that of the CSP regardless of whether LDA or SVM is used. Statistical significance of accuracy was estimated by performing a T-test with a confidence level of 0.05, which verified the result mentioned above. It is evident that the results obtained by the SVM are in exceptionally good agreement with those obtained by the LDA. This outcome proves that our proposed algorithm has no classifier dependency, and the classifier has no effect on the essence of the results.

Fig. 5
figure 5

Average classification accuracies versus the different methods under two classifiers on two datasets. a The BCI competition III dataset IVa; b the BCI competition III dataset IIIa; c the subjects S01, S02, S23, S41, and S43 in the Cho’s dataset; and d the subjects S03, S06, S14, S35, and S50 in the Cho’s dataset

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4.3 Performance comparison of e-CSP based on parameter α

The change in the average classification accuracies across all subjects of parameter α in the Euler representation of the two public datasets and Cho’s dataset is reported in Fig. 6. Tables 9 and 10, respectively, list the value of α when the e-CSP obtains the best classification accuracy. The parameter α is set from 0.1 to 1.9 with a step size of 0.1 to observe the change in classification accuracy. Similar behavior is observed in the BCI competition III dataset IVa and BCI competition III dataset III a two publicly datasets; although the values of the best classification accuracy of each subject are not the same, they are concentrated in the range of 0.8–1.3. However, some different results between the two datasets can also be noticed. As shown in Fig. 6(a), the parameter α has a small influence on the performance of e-CSP, while it is seen that in another dataset (Fig. 6(b) ), with the change in parameter α, the classification accuracy displays large fluctuations.

Fig. 6
figure 6

Average classification accuracies versus the α value of different datasets. a The BCI competition III dataset IVa; b the BCI competition III dataset IIIa; c the subjects S03, S06, S14, S35, and S50 in Cho’s dataset; and d the subjects S01, S02, S23, S41, and S43 in the Cho’s dataset. The horizontal axis in the figure represents the value of the parameter α, and the vertical axis represents the classification accuracy

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Table 9 Values of α for the best classification of the proposed method in the two datasets
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Table 10 Values of α for the best classification of the proposed method in the Cho’s dataset
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However, in Cho’s dataset, it can be observed that as the parameters change, the performance of the e-CSP does not show similar behavior as in the two public datasets. As shown in Fig. 6(c), for the subjects “S35” and “S50”, the e-CSP can achieve relatively high accuracies when the values of the parameter α are concentrated in the range of 0.1–0.5; moreover, as the parameter α increases, the accuracy decreases. As shown in Fig. 6(d), the curve of the classification accuracy for the subject “S01” shows a rising trend with the increase of the parameter α. For the most of subjects, when the parameter α is in [0.3 0.8] interval, the e-CSP overall has good performance. Thus, we set the parameter α as 0.5 in experiment 1 and experiment 2 on Cho’s dataset.

Moreover, we compared the classification accuracies of the CSP and the e-CSP in lower limb MI data. The dataset was collected by our laboratory, which was based on the MI task of the right foot and left foot. As shown in Fig. 7, the average classification accuracy of the e-CSP is greater than that of the CSP.

Fig. 7
figure 7

Average classification accuracies of the proposed method and CSP in the lower limb MI data

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5 Conclusion

A novel feature extraction method is proposed for EEG classification, which is called the e-CSP. Different from other methods proposed to optimize the objective function of the CSP, e-CSP maps the sample values into the Euler space by explicit Euler representation and then performs the complex CSP in Euler space. Compared to the conventional CSP, e-CSP uses the cosine distance metric to measure the within-class and between-class scatters in the criterion function. The effectiveness of the e-CSP is illustrated by comparing with the CSP in the BCI competition III dataset IVa, the BCI competition III dataset IIIa and the Cho’s dataset. From the experimental results, the e-CSP performs better than the CSP under both the LDA and SVM classifiers. Moreover, the e-CSP enhances EEG classification and outperforms some other recent related methods in the publicly available BCI competition datasets.