# Compressive Sampling of EEG Signals with Finite Rate of Innovation

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## Abstract

Analyses of electroencephalographic signals and subsequent diagnoses can only be done effectively on long term recordings that preserve the signals' morphologies. Currently, electroencephalographic signals are obtained at Nyquist rate or higher, thus introducing redundancies. Existing compression methods remove these redundancies, thereby achieving compression. We propose an alternative compression scheme based on a sampling theory developed for signals with a finite rate of innovation (FRI) which compresses electroencephalographic signals during acquisition. We model the signals as FRI signals and then sample them at their rate of innovation. The signals are thus effectively represented by a small set of Fourier coefficients corresponding to the signals' rate of innovation. Using the FRI theory, original signals can be reconstructed using this set of coefficients. Seventy-two hours of electroencephalographic recording are tested and results based on metrices used in compression literature and morphological similarities of electroencephalographic signals are presented. The proposed method achieves results comparable to that of wavelet compression methods, achieving low reconstruction errors while preserving the morphologiies of the signals. More importantly, it introduces a new framework to acquire electroencephalographic signals at their rate of innovation, thus entailing a less costly low-rate sampling device that does not waste precious computational resources.

### Keywords

Power Spectral Density Original Signal Compression Technique Compression Method Reconstructed Signal## 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, Open image in new window -hour monitoring of a multiple-channel EEG is needed for epilepsy patients. Traditionally, the EEG has been bandlimited to the frequency range between Open image in new window and Open image in new window Hz; thus a minimum Nyquist sampling rate of Open image in new window Hz is needed. At the quantization level of 16 bit/sample, a Open image in new window -channel EEG for a Open image in new window -hour period would amount to Open image in new window 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 Open image in new window sets of normalised EEG signals comprising Open image in new window hours were used for the study. The data is further divided into Open image in new window seconds epochs for processing. All Open image in new window patients experienced similar seizure types at similar locations on the brain. From this dataset, Open image in new window epochs of Open image in new window 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 Open image in new window channels, Open image in new window Hz sampling rate, and a Open image in new window -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 Open image in new window epochs and the epochs will be referenced as Open image in new window where Open image in new window represents the patient number and Open image in new window 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

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

#### 2.2.2. Nonuniform Splines

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

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

#### 2.2.3. Noisy Case

where Open image in new window and Open image in new window are the power spectral densities of white and Open image in new window noise, respectively and Open image in new window .

### 2.3. Spline-Based FRI Models with Additive Noise

where Open image in new window is the nonuniform spline component, and Open image in new window is the noise component. We consider the cases of nonuniform linear spline, nonuniform quadratic spline, and lastly the nonuniform cubic spline (where Open image in new window respectively in Section 2.2.2) with additive white noise and Open image in new window 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].

where Open image in new window and Open image in new window represent the numbers of bits required for the original and compressed signals, respectively.

where Open image in new window and Open image in new window are the sampled values of the original and reconstructed signals.

where Open image in new window and Open image in new window are the mean values of the original and reconstructed signals, respectively.

By plotting Open image in new window , 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

Comparison of different approximation models for Open image in new window epochs of EEG signals.

Evaluation metric | Linear spline model | Quadratic spline model | Cubic spline model |
---|---|---|---|

Average CC (%) | |||

Average RMSE | |||

Average PRD (%) | |||

Average MAXERR |

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

The assumed values of *2K,* the respective compression ratio CR and *C.*

EEG epoch | No. of coefficients Open image in new window | Compression ratio Open image in new window (%) | |
---|---|---|---|

P1_001 | 694 | 72.85 | 3.68 |

P1_002 | 716 | 72.03 | 3.61 |

P1_003 | 852 | 66.71 | 3.00 |

P1_004 | 640 | 75.00 | 4.00 |

P1_005 | 702 | 72.61 | 3.65 |

P1_006 | 880 | 65.63 | 2.91 |

P1_007 | 788 | 69.18 | 3.24 |

P1_008 | 784 | 69.34 | 3.26 |

P1_009 | 720 | 71.88 | 3.56 |

P1_010 | 742 | 70.98 | 3.45 |

P1_001 | 474 | 81.48 | 5.40 |

P2_002 | 582 | 77.27 | 4.40 |

P2_003 | 635 | 75.20 | 4.03 |

P2_004 | 706 | 72.42 | 3.63 |

P2_005 | 742 | 71.02 | 3.45 |

P2_006 | 490 | 80.86 | 5.23 |

P2_007 | 548 | 78.55 | 4.66 |

P2_008 | 610 | 76.17 | 4.20 |

P2_009 | 544 | 78.71 | 4.70 |

P2_010 | 888 | 65.31 | 2.89 |

P3_001 | 860 | 66.41 | 2.98 |

P3_002 | 836 | 67.34 | 3.06 |

P3_003 | 470 | 81.64 | 5.45 |

P3_004 | 682 | 73.36 | 3.75 |

P3_005 | 624 | 75.59 | 4.10 |

P3_006 | 490 | 80.86 | 5.23 |

P3_007 | 631 | 75.35 | 4.06 |

P3_008 | 755 | 70.51 | 3.39 |

P3_009 | 654 | 74.45 | 3.91 |

P3_010 | 820 | 67.97 | 3.12 |

With these assumptions, we represent the EEG signals as Open image in new window pieces of nonuniform linear splines embedded in Open image in new window noise and the corresponding rate of innovation will be Open image in new window with Open image in new window seconds.

### 3.1. Our Method

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

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 ( Open image in new window ), which is the ratio of mean square error of the reconstructed signals to the range of amplitudes of the signals.

Performance of our method: Open image in new window , Open image in new window , Open image in new window and Open image in new window .

EEG | Cross correlation Open image in new window (%) | Root mean square error Open image in new window | Percent root difference Open image in new window (%) | Maximum error Open image in new window |
---|---|---|---|---|

P1_001 | 94.05 | 0.0067 | 39.35 | 0.0235 |

P1_002 | 94.98 | 0.0082 | 32.37 | 0.0247 |

P1_003 | 91.87 | 0.0051 | 30.43 | 0.0212 |

P1_004 | 95.95 | 0.0024 | 34.90 | 0.0187 |

P1_005 | 93.52 | 0.0122 | 40.05 | 0.0204 |

P1_006 | 89.71 | 0.0066 | 51.37 | 0.0231 |

P1_007 | 92.42 | 0.0183 | 49.37 | 0.0213 |

P1_008 | 94.20 | 0.0091 | 35.43 | 0.0220 |

P1_009 | 94.20 | 0.0081 | 38.54 | 0.0186 |

P1_010 | 97.56 | 0.0039 | 23.05 | 0.0070 |

P2_001 | 95.08 | 0.0099 | 39.35 | 0.0102 |

P2_002 | 94.05 | 0.0055 | 39.78 | 0.0151 |

P2_003 | 93.98 | 0.0070 | 31.09 | 0.0211 |

P2_004 | 96.51 | 0.0069 | 31.52 | 0.0098 |

P2_005 | 95.36 | 0.0100 | 34.85 | 0.0178 |

P2_006 | 95.64 | 0.0063 | 30.83 | 0.0123 |

P2_007 | 92.28 | 0.0007 | 34.31 | 0.0210 |

P2_008 | 95.07 | 0.0008 | 36.31 | 0.0090 |

P2_009 | 97.26 | 0.0046 | 27.45 | 0.0012 |

P2_010 | 92.88 | 0.0096 | 43.52 | 0.0244 |

P3_001 | 95.25 | 0.0052 | 30.19 | 0.0231 |

P3_002 | 92.02 | 0.0031 | 32.11 | 0.0235 |

P3_003 | 90.03 | 0.0011 | 33.17 | 0.0209 |

P3_004 | 94.08 | 0.0032 | 33.27 | 0.0125 |

P3_005 | 93.44 | 0.0024 | 32.75 | 0.0102 |

P3_006 | 94.69 | 0.0011 | 37.07 | 0.0196 |

P3_007 | 92.07 | 0.0019 | 32.70 | 0.0180 |

P3_008 | 91.63 | 0.0012 | 28.33 | 0.0219 |

P3_009 | 90.20 | 0.0039 | 36.37 | 0.0243 |

P3_010 | 96.27 | 0.0025 | 29.95 | 0.0023 |

The Open image in new window is selected as the primary evaluation metric and our results are generated with the best Open image in new window achievable for each EEG signal. As shown in Table 3, consistently high Open image in new window ranging from Open image in new window to Open image in new window 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 Open image in new window ranging from Open image in new window to Open image in new window . 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 Open image in new window varies, the rate of innovation Open image in new window varies accordingly and leads to an increased or decreased number of spectral coefficients. Thus Open image in new window varies inversely as Open image in new window . Table 3 also tabulates the distortions arising from our method. We obtained low Open image in new window and Open image in new window , implying that our method recovers signals with some distortion. Furthermore, a Open image in new window between Open image in new window and Open image in new window is obtained, suggesting that the distortions of the reconstructed signals are very small.

Performance of our method on the Open image in new window EEG datasets.

EEG | Patient Open image in new window | Patient Open image in new window | Patient Open image in new window | Average |
---|---|---|---|---|

Mean no. of coefficients | 941 | 982 | 973 | 965 |

Mean Open image in new window (%) | 63.24 | 61.53 | 61.98 | 62.3 |

Mean Open image in new window (%) | 94.30 | 95.32 | 95.63 | 95.08 |

Mean Open image in new window (%) | 34.89 | 32.72 | 33.29 | 33.63 |

0.0080 | 0.0061 | 0.0025 | 0.0055 | |

0.0201 | 0.0302 | 0.0171 | 0.0225 |

- (i)
Compute the DFT to obtain the Fourier series coefficients Open image in new window : Open image in new window

- (ii)
Denoise: Open image in new window

- (iii)
Differentiate the denoised signal?:? Open image in new window

- (iv)
Solve a Toeplitz system of equation of size Open image in new window by Open image in new window to get Open image in new window

- (v)
Find the roots of Open image in new window by factorization, to get Open image in new window ?:? Open image in new window

- (vi)
Solve a Vandermonde system of equation of size Open image in new window by Open image in new window to get Open image in new window

- (vii)
Integrate the Open image in new window Dirac pulses- Open image in new window

Hence effectively, the computational costs involved is Open image in new window . For our dataset, the average time required to sample and reconstruct a Open image in new window -second epoch is Open image in new window seconds on an Intel Core2 Duo 2.50 GHz system with Open image in new window 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 Open image in new window ratios are obtained for some epochs, leading to a moderate compression ratio. Furthermore, it is observed that Open image in new window 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 Open image in new window 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 Open image in new window values based on certain evaluation metrics such as Open image in new window or Open image in new window

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.

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