1. Introduction

The electroencephalogram (EEG) is a recording of the brain's neural activities. Since its discovery by Berger [1], many research activities have focussed on how to automatically extract useful information about the brain's conditions based on the distinct characteristics of these electrical signals. Valuable information about the human brain conveyed by the EEG is used in various studies like the nervous system, sleep disorders, epilepsy, and dementia [2]. These applications require acquisition, storage, and automatic processing of EEG during an extended period of time. For example, -hour monitoring of a multiple-channel EEG is needed for epilepsy patients. Traditionally, the EEG has been bandlimited to the frequency range between and  Hz; thus a minimum Nyquist sampling rate of  Hz is needed. At the quantization level of 16 bit/sample, a -channel EEG for a -hour period would amount to megabytes. Hence, to efficiently store and transmit a huge amount of data, effective compression techniques are desired. While lossy techniques yield higher compression, because of reliability considerations, lossy data compression techniques are not used as the morphology of the signals which are not always well retained. Excellent surveys of the performance of lossless and lossy EEG compression techniques can be found in [3] to [4]. Antoniol and Tonella presented and discussed several classical lossless EEG signal compression methods such as Huffman coding, predictive compression, and transform compression [3]. In [5], Memon et al. discussed lossless compression techniques ranging from simple dictionary searches to sophisticated context modeling. A long-term EEG compression method using features obtained from the signals' power spectral density was proposed in [6] while multi-channel EEG signals were compressed by exploiting the intercorrelation among the EEG channels through the Karhunen-Loeve transform in [7]. Nielsen et al. proposed a signal-dependent wavelet compression scheme that adapted optimal wavelets to biomedical signals for compression [8]. A near-lossless compression method described in [9] compressed EEG signals using neural network predictors followed by nonuniform quantization. More recently, a new compression method based on the construction process of the classified signature and envelop vector sets of the EEG signals [4].

The techniques presented above operated on EEG signals obtained at or above Nyquist rate. This acquisition process leads to a collection of huge amounts of irrelevant data, only to be discarded during the compression stage of the signals. Furthermore, transients, which are common in EEG signals, are not bandlimited. Hence, Shannon's sampling theory cannot be applied to sampling EEG signals. Over the last few years, advancements in signal processing and data acquisition introduced a new sampling theory known as compressive sampling or compressed sensing [10]. Aviyente proposed a compressed sensing framework for EEG compression by exploiting the sparsity of EEG signals in a Gabor frame [11]. This method, however, does not operate on the analog EEG signals directly. Compressive sampling, on the other hand, asserts that its acquisition system directly translates analog signals into compressed digital form so that one can recover super-resolved signals from a few measurements [10]. Similarly, we propose to approach the problem of compressing EEG signals at source. In order to address the nonbandlimitedness of the EEG signals, our compression method will be based on the theory of sampling signals with finite rate of innovation (FRI) [12]. This theory has recently been investigated for a compression technique for electrocardiogram (ECG) signals [13] and neonatal EEG seizure signals [14] as well as for EEG seizure source localisation [15].

Our paper is organised as follows. In Section 2, a description of the EEG data, a review on sampling signals with finite rate of innovation, and an FRI model of EEG signals are presented. A scheme for compressively sampling EEG signals with finite rate of innovation will be described in Section 3. Results and discussions will be presented in Section 4 and finally, a conclusion will summarise our findings and provide directions for our future work.

2. EEG Data Description and the FRI Model

2.1. EEG Data Description

A total of sets of normalised EEG signals comprising hours were used for the study. The data is further divided into seconds epochs for processing. All patients experienced similar seizure types at similar locations on the brain. From this dataset, epochs of seconds duration were selected for establishing a finite rate of innovation model of EEG signals while the rest of the data were used to evaluate our compression scheme. The EEG data were acquired using a Neurofile NT digital video EEG system with channels,  Hz sampling rate, and a -bit analogue-to-digital converter. Notch or bandpass filters have not been applied. More details of the database can be found in [16]. In our experiments, these EEG signals are assumed to be the source signals. For each patient, there will be epochs and the epochs will be referenced as where represents the patient number and represents the epoch number.

2.2. Review of Sampling Signals with Finite Rate of Innovation

Consider classes of parametric signals with a finite number of degrees of freedom per unit of time, which is defined as the rate of innovation (e.g., streams of Dirac pulses, nonuniform splines, and piecewise polynomials). It is shown in [12] that although these signals are not bandlimited, they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel, and then perfectly reconstructed by solving systems of linear equations.

2.2.1. Periodic Stream of Dirac Pulses

Consider a stream of Dirac pulses periodized with period , where and for all . This signal has degrees of freedom per period, thus the rate of innovation is

(1)

By taking a continuous-time periodic sinc sampling kernel with bandwidth greater than or equal to the rate of innovation given by (1), and sampling at uniform locations = where = and then the samples defined by = sufficiently represent [12].

2.2.2. Nonuniform Splines

A signal is a nonuniform spline of degree with knots at if and only if its th derivative is a stream of weighted Dirac pulses [17]. Here, the rate of innovation is

Consider a continuous-time periodic nonuniform linear spline with period , containing pieces of maximum degree . By following the sampling method described in Section 2.2.1, is uniquely defined by = = [12].

2.2.3. Noisy Case

In this section, we briefly present two types of noise signals that are added to the FRI signals. The first type of noise signal considered is the white noise, which is a zero-mean signal characterised by a flat power spectral density. The second type of noise signal is the noise whose power spectral density is inversely proportional to its frequency [18]. Accordingly, we define

(2)

where and are the power spectral densities of white and noise, respectively and .

2.3. Spline-Based FRI Models with Additive Noise

In this section, an FRI model of EEG signals is validated and presented. In particular, we model the EEG signals as

(3)

where is the nonuniform spline component, and is the noise component. We consider the cases of nonuniform linear spline, nonuniform quadratic spline, and lastly the nonuniform cubic spline (where respectively in Section 2.2.2) with additive white noise and noise. We also compare the models with the original signals based on the performance metrics described below and conclude with a suitable FRI model for EEG signals.

2.3.1. Performance Metrics

The following evaluation metrics were employed to determine our method's performance [4].

The compression ratio () is defined as a ratio between the number of bits required to represent the original signal and the compressed signal. First, we define a ratio

(4)

where and represent the numbers of bits required for the original and compressed signals, respectively.

Thus we can define a commonly used in the literature as

(5)

A metric that can be used to measure distortion is percent root difference (). This metric is commonly used for measuring the distortions in reconstructed biomedical signals such as Electrocardiographic (ECG) signals and EEG signals. For signals of length , can be defined as

(6)

where and are the sampled values of the original and reconstructed signals.

Another distortion metric is the root mean square error (). In data compression, we are interested in finding an optimal approximation for minimizing this metric as defined by the following formula:

(7)

Since the similarity between the reconstructed and original signal is crucial from the clinical point of view, the cross correlation () is used to evaluate the similarity between the original signal and its reconstruction.

(8)

where and are the mean values of the original and reconstructed signals, respectively.

In order to understand the local distortions between the original and the reconstructed signals, two metrics, the maximum error () and the peak amplitude related error () [19], will be computed. The maximum error metric is defined as

(9)

and it shows how large the error is between every sample of the original and reconstructed signals. This metric should ideally be small if both signals are similar. The is defined as

(10)

By plotting , one will be able to understand the locations and magnitudes of the errors between the original and reconstructed signals.

2.3.2. Comparison of Models

As depicted in Figure 1, a comparison between the original EEG signal and its approximations shows that the nonuniform linear spline model gives the best approximation where a of , with an of , a of and a of are achieved. An evaluation of sets of EEG signals in Table 1 shows that the nonuniform linear and cubic spline models best fitted the EEG signals. Since the results achieved with a linear spline model are very close to that of a cubic spline model, the nonuniform linear spline model is chosen as our EEG model to minimise computation costs.

Table 1 Comparison of different approximation models for epochs of EEG signals.
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Figure 1
figure 1

Comparison of different spline models of EEG. (a) Original EEG signal; (b) Linear spline model, CC = 99.22%, RMSE = 0.0020, PRD = 13.07%, MAXERR = 0.0012; (c) Quadratic spline model, CC = 96.00%, RMSE = 0.0046, PRD = 29.65%, MAXERR =0.023; (d) Cubic spline model CC = 99.13%, RMSE = 0.0021, PRD = 13.65%, MAXERR = 0.0011.

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Figure 2 illustrates the nonuniform linear spline model for different EEG signals whereby the approximations closely model the original signals with different pieces of linear splines. As such, we conclude that with our data, different EEG signals can be modelled with for each seconds EEG segment.

Figure 2
figure 2

Examples of modelling different EEG signals with K-pieces nonuniform linear splines. (a) K = 37 pieces. (b) K = 23 pieces. (c) K = 53 pieces. (d) K = 45 pieces.

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The nonuniform linear spline model makes up approximately % of the EEG signal. We observed that the residue signal (EEG signal-nonuniform linear splines) resembled a noise signal with a power spectral density The range of this noise amplitude was found to be and was estimated to be around by computing the slope of a fitted line onto the log-log plot of versus as shown in Figures 3(a) and 3(b) [20]. Thus, our EEG signals are modelled as nonuniform linear splines embedded in additive noise. In this way, the EEG signals have been cast as signals with finite rate of innovation embedded in noise and this provides a motivation to exploit the FRI framework for compressive sampling of EEG signals.

Figure 3
figure 3

(a) The average power spectral density of the residue signal compared to that of 1/f noise (b) The estimation of α from the log-log plot of average power spectral density versus frequency of the residue signal.

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3. Compressive Sampling of EEG Signals with Finite Rate of Innovation

Since the EEG signals are modelled as an FRI signal noise, we will employ methods developed in [12, 21] to acquire and reconstruct them. It is important that the rate of innovation of the EEG signals is known and in our case, it has to be estimated from the source signals. Let us assume that the number of pieces of linear splines needed to represent the EEG signals is given as shown in Table 2.

Table 2 The assumed values of 2K, the respective compression ratio CR and C.
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With these assumptions, we represent the EEG signals as pieces of nonuniform linear splines embedded in noise and the corresponding rate of innovation will be with  seconds.

3.1. Our Method

Figure 4 shows our proposed EEG signal acquisition process with finite rate of innovation. Since the value of is assumed to be known, the samples of the EEG signal are obtained based on the descriptions in Section 2.2.1. Corresponding to the representation for the EEG signals, a reconstruction method is presented in Figure 5.

Figure 4
figure 4

Our proposed scheme for sampling EEG signals with finite rate of innovation.

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Figure 5
figure 5

The FRI reconstruction block diagram for EEG signals.

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In order to perform the Cadzow's noise reduction, a rectangular () Toeplitz matrix is created from the spectral values of the source signal in the form

(11)

We then perform a singular vector decomposition of the matrix , and enforce rank on by choosing only most significant singular values. This is iterated until the ratio of the largest singular value of the to that of the is smaller than a preset threshold. Thus the denoised DFT coefficients can be extracted from [21].

Since our EEG signals are modelled as nonuniform linear splines, we perform a differentiation operation twice on the denoised signals so as to reduce them into a stream of Dirac pulses. In order to find the locations and weights of the Dirac pulses, consider a filter whose -transform has zeros at , that is, Since the CTFS of the differentiated EEG signal is a linear combination of complex exponentials , it follows that is an annihilating filter and satisfies the following condition:

(12)

where

(13)

The coefficients of the annihilating filter are found solving (12) which is equivalent to the following Toeplitz linear system of equations:

(14)

Thus the locations of the Dirac pulses are given by the roots of . Next, the weights of the Dirac pulses are given by solving the Vandermonde system of equations given by

(15)

Lastly, the stream of Dirac pulses is integrated twice to obtain the reconstructed EEG signals which correspond to the nonuniform linear spline approximation of the EEG signals.

4. Results and Discussions

In this section, we will present our results based on the performance metrics in Section 2.3.1. Comparisons to wavelet based compression techniques using discrete wavelet transform with the Daubechies and Coiflets wavelets [8] will be included in our discussions. These wavelet transforms are performed with four detailed levels and one approximation. Both the wavelet coefficients and the FRI innovation parameters are coded using Huffman coding. We also compared our results to those found in [11] in terms of normalised mean square error (), which is the ratio of mean square error of the reconstructed signals to the range of amplitudes of the signals.

We applied our method on the sets of EEG signals and the results of selected epochs are tabulated in Table 3.

Table 3 Performance of our method: , , and .
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The is selected as the primary evaluation metric and our results are generated with the best achievable for each EEG signal. As shown in Table 3, consistently high ranging from to is achieved for our dataset. This implies that there is a great similarity in the morphology between the original and reconstructed EEG signals. This result is highly desirable because such diagnostic features are extremely important and must be preserved. Our method achieves a ranging from to . This is due to the morphology of the EEG signals, where some signals need more linear splines to model them compared to the others. As varies, the rate of innovation varies accordingly and leads to an increased or decreased number of spectral coefficients. Thus varies inversely as . Table 3 also tabulates the distortions arising from our method. We obtained low and , implying that our method recovers signals with some distortion. Furthermore, a between and is obtained, suggesting that the distortions of the reconstructed signals are very small.

A typical plot of and a histogram of the errors are shown in Figures 6 and 7. Errors between original samples of the signal and the corresponding reconstructed samples are amplified and shown in the plot. The values of are generally less than of the original signal, although some values are larger than of the original signal. By comparing Figure 8 to Figure 6, the is relatively high at and the value is . However, the differences between the original and reconstructed signals cannot be distinguished morphologically. Thus alone cannot measure how well morphologies of the EEG signals are retained. The histogram of errors showed a concentration of errors in the range between to , and some outliers in the larger error bins, thus contributing to distortions. Although the results are satisfactory, we observed that the distortions arise from the estimation of the innovation parameters. Let us make a comparison of the obtained by sampling both the original EEG epochs and their nonuniform linear spline approximation (i.e., noise-free signals). If the estimation of the innovation parameters is accurate, the obtained should be the same. As shown in Figure 9, the obtained for the noise-free signal is higher than the noisy signal, due to wrong estimations of locations and weights.

Figure 6
figure 6

A plot of PARE for the signals shown in Figure 8. Errors between original and reconstructed samples are depicted as spikes in this plot.

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Figure 7
figure 7

Histogram plot of the errors obtained for an epoch.

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Figure 8
figure 8

A comparison between the original EEG signal and its reconstruction.PRD = 34.35%, CC = 98.43%, RMSE = 0.0046, and MAXERR = 0.0187.

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Figure 9
figure 9

Comparison of the 30 epochs' CC in the event of (i) sampling the original EEG epochs and (ii) sampling their nonuniform linear spline approximations (noise free signals).

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In addition, we present results for sampling three -hour recordings. Figure 10 shows how the various metrics change in a continuous EEG recording of one patient. As illustrated in Figure 10(a), the number of coefficients varies between and , thus showing that for the same patient, the innovation parameters cannot be assumed to be constant. We also noted that although varies in a large range, is quite consistently lying in the range of to as in Figure 10(b). Furthermore, the distortion in the EEG is kept very low, as depicted by Figure 10(c). Compression ratio for this recording ranges from about to , which varies faithfully with . Table 4 tabulates the mean and number of coefficients for each dataset. On the average, we achieved a of around with low and errors of around , and respectively. In addition, the achieved is around . Based on the observations of our experiments, a minimum value of and a maximum value of will maintain the morphologies of the reconstructed signals visually.

Table 4 Performance of our method on the EEG datasets.
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Figure 10
figure 10

Variation of the evaluation metrics on 343 epochs corresponding to 3430 seconds of continuous EEG from one patient. (a) The number of coefficient 2K versus epochs. (b) CC versus epochs. (c) RMSE versus epochs. (d) CR versus epochs.

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Figure 11 presents an example of the strength and uniqueness of our sampling scheme. The EEG signal's original sampling frequency is  Hz (Figure 11(a)). Our system estimated to be and modelled the signal with samples of the original signal (Figure 11(b)), which has a of . Effectively, we are sampling the original signal at  Hz. We effectively reconstructed the  Hz signal as shown in Figure 11(c) with = , = , = , = . As a comparison, we reconstructed the signal in Figure 11(b) with the traditional sinc interpolation method (Figure 11(d)) with = , = , = , = . Clearly, we are able to represent EEG signals with a low number of samples and reconstruct them with high fidelity.

Figure 11
figure 11

Comparison of reconstruction of an EEG signal acquired by our method. (a) The original signal sampled at 256 Hz. (b) The EEG signal acquired by our method at 47.5 Hz with 2K = 474. (c) The reconstructed EEG signal at 256 Hz with our method CC = 97.08%, RMSE = 0.0099, PRD = 22.43%, MAXERR = 0.0016. (d) The reconstructed EEG signal using the traditional sinc interpolation method CC = 90.11%, RMSE = 0.0147, PRD = 48.30%, MAXERR = 0.428.

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Figure 12 shows the relation between and with respectively. The is inversely proportional to the rate of innovation, as shown in the earlier discussion. In order to achieve a high has to be compromised. Similarly, as we increase , the error involved such as and will increase together, though not in a linear form. A comparison is made with traditional sinc interpolation and the performance of our method is superior since an interpolation process is unable to faithfully reconstruct signals acquired at a low-sampling rate into one of a higher sampling rate. Next we compared our results with that of compressing the EEG signals using wavelet compression methods. As illustrated, our method achieves comparable results in terms of and although the achieved by wavelet methods is slightly better. Since only indicates how much error is incurred in the reconstruction without reference to the morphology of the signals, our results do not indicate that our reconstructed signals differ largely from the original signals morphologically. Furthermore, our method entails a less costly low-rate sampling device and does not waste precious computational resources collecting extra data only to discard them subsequently.

Figure 12
figure 12

A comparison of the performance of our method with sinc interpolation, wavelet compression using Daubechies and coiflets wavelets on an EEG epoch. (a) Variation of CC with CR. (b) Variation of PRD with CR. (c) Variation of RMSE with CR. (d) Variation of ρ with CR

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Figure 13 illustrates a comparison between an original and a reconstructed EEG signal with the coefficients, we reconstructed the signal and achieved = , = , = , = and = Furthermore, the of the reconstructed signal is in the range of to as opposed to achieved by the method in [11]. A zoomed-in view in Figure 14 confirms that the morphology of the original signal is conserved in the reconstruction.

Figure 13
figure 13

A comparison between (a) an original EEG signal and (b) the reconstructed EEG signal.2K = 1184, CR = 81.48%, CC = 97.52%, RMSE = 0.0044, PRD = 24.55%, MAXERR = 0.0022.

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Figure 14
figure 14

A zoom-in view of the comparison of original and reconstructed EEGs in Figure 13.

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Lastly we will discuss about the computational costs of our scheme. With reference to Figure 5, the computational complexity can be estimated as follows.

  1. (i)

    Compute the DFT to obtain the Fourier series coefficients:

  2. (ii)

    Denoise:

  3. (iii)

    Differentiate the denoised signal?:?

  4. (iv)

    Solve a Toeplitz system of equation of size by to get

  5. (v)

    Find the roots of by factorization, to get ?:?

  6. (vi)

    Solve a Vandermonde system of equation of size by to get

  7. (vii)

    Integrate the Dirac pulses-

Hence effectively, the computational costs involved is . For our dataset, the average time required to sample and reconstruct a -second epoch is seconds on an Intel Core2 Duo 2.50 GHz system with RAM. This computational time can be improved by employing fast algorithms on dedicated digital signal processors to achieve a realtime EEG signal acquisition and display.

5. Conclusions

We proposed an approach to compress EEG signals at source based on the finite rate of innovation sampling theory. Unlike traditional compression methods which acquire many data samples and later discard redundant ones, our proposed method relies on acquiring a small set of data from the original signal based on the signal's rate of innovation and then reconstructing the signal with high resolution. Even though a small set of data is obtained, our method retains the morphologies of the EEG signals. It yielded promising results such as good cross correlation and low distortions at a low computational cost. In this way, we achieve computational savings which can be utilised in other more important signal processing stages. Moderate ratios are obtained for some epochs, leading to a moderate compression ratio. Furthermore, it is observed that changes depending on the state of the EEG, thus leading to a variable rate of innovation. Valuable information such as the occurrences of EEG abnormalities can be extracted through tracking the changes in the rate of innovation across the EEG. As such, the advantage of our compression method lies in the ability to compress EEG signals and track changes across EEG states concurrently. Although the accuracy of the estimated affects the entire scheme, as discussed in [21], it can be estimated from the rank of a Toeplitz matrix. However, more research is needed to determine the correct duration of EEG signals to yield optimal values based on certain evaluation metrics such as or

We will continue our work to minimise the local errors caused by outliers and to include adaptive rate of innovation to cater to the changing states of EEG signals. Finally we will investigate how EEG signals can be source compressed with finite rate of innovation in real time.