# A matching pursuit-based signal complexity measure for the analysis of newborn EEG

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## Abstract

This paper presents a new relative measure of signal complexity, referred to here as *relative structural complexity* (RSC), which is based on the matching pursuit (MP) decomposition. By relative, we refer to the fact that this new measure is highly dependent on the decomposition dictionary used by MP. The structural part of the definition points to the fact that this new measure is related to the structure, or composition, of the signal under analysis. After a formal definition, the proposed RSC measure is used in the analysis of newborn electroencephalogram (EEG). To do this, firstly, a time–frequency decomposition dictionary is specifically designed to compactly represent the newborn EEG seizure state using MP. We then show, through the analysis of synthetic and real newborn EEG data, that the relative structural complexity measure can indicate changes in EEG structure as it transitions between the two EEG states; namely seizure and background (non-seizure).

### Keywords

Matching pursuit Relative structural complexity Coherent dictionary Time–frequency Newborn EEG## 1 Introduction

The electroencephalogram (EEG) is an important tool in the study of the central nervous system (CNS), particularly in the newborn where it provides high prognostic and diagnostic capabilities [20]. In some cases, the newborn EEG is the only indicator of CNS pathologies, with electrographic seizure events being the most significant indicator of CNS dysfunction [22].

The EEG of newborn patients suffering seizure events can be broadly classified into two main states; namely, background and seizure. The newborn EEG background signal is a complex waveform which, in the first instance, appears to be some form of noisy signal [20]. In fact, it was recently shown that the newborn EEG background can be modelled as a nonstationary stochastic 1/*f*^{γ} process [27]. Newborn EEG seizure, on the other hand, is generally characterized by periods of rhythmic spiking or repetitive sharp waves [18], whose patterns are highly variable, with complex and varied morphology and cover a variety of frequencies. These dynamical changes in structure and frequency infer that the newborn EEG is highly nonstationary. Time–frequency (TF) signal analysis techniques, which have been shown to provide informative representations of signal nonstationarities [5], are highly suitable for the newborn EEG background and seizure states.

Quadratic time–frequency distributions (QTFDs), such as the Wigner–Ville, Choi–Williams and Modified B distributions [4], are very useful for the visualization of nonstationary signals in the TF domain. A comprehensive analysis and characterization of the newborn EEG using QTFDs was previously undertaken by the present authors in [6, 7, 8, 9, 27]. The TF analysis revealed significant differences in the TF structure of newborn EEG background and seizure signals, with significant changes in structure occurring as the newborn EEG progresses from the background to seizure state, and vice versa.

Although QTFDs have proved extremely useful in the analysis of the nonstationary newborn EEG, they are not designed for signal parameterization and usually require sophisticated image processing techniques for feature extraction [23, 26, 32]. For this reason, much attention has recently been directed to the use of atomic decomposition techniques, such as the matching pursuit (MP) algorithm [21], using *overcomplete* TF *dictionaries* for TF representation and parameterization [16].

An objective of atomic decomposition techniques is to generate *sparse* (compact), yet highly informative, signal representations [13], given a particular overcomplete dictionary. The development and theoretical study of algorithms which attempt to find the sparsest representation has been the focus of many researchers recently [13, 14, 15, 17, 30, 31]. However, compact and informative representations can also be achieved through the use of a highly coherent TF dictionary [21].

In this paper, we aim to first develop a new *relative* measure of signal complexity, referred to as relative structural complexity (RSC), which is based on MP decomposition using a coherent TF dictionary. Since RSC is dependent on the use of a coherent dictionary, the second aim is to develop a TF dictionary, which is coherent with some particular newborn EEG signal structures. Finally, we apply the RSC measure, using the designed coherent dictionary, to the analysis of newborn EEG, showing that RSC is an indicator of structural change in the EEG as it transitions between the background and seizure states.

## 2 Signal processing methods

### 2.1 Time–frequency signal analysis

In the following subsections, we give brief introductions to the two TF methods mentioned in the introduction; namely, QTFDs and MP atomic decomposition. It was stated in the previous section that QTFDs are extremely useful for visualization of nonstationary signals in the TF domain but lack the compact representation of atomic decomposition techniques such as MP. Therefore, in this paper, QTFDs are used only for visualization purposes while the MP atomic decomposition technique is used in the development of the RSC measure.

#### 2.1.1 Quadratic time–frequency distributions

Quadratic TF distributions are commonly used methods for obtaining joint TF representations of nonstationary signals. The fundamental QTFD is the Wigner–Ville distribution (WVD), from which all other QTFDs can be obtained by a TF averaging or smoothing [4].

*s*(

*t*), is defined as

*K*

_{z}(

*t*,τ) is the instantaneous autocorrelation function (IAF) given by

*z*(

*t*) is the analytic associate of

*s*(

*t*) \(({\rm i.e.,}\ {z(t) = s(t) + {\frac{j} {\pi}}\int_{-\infty}^{\infty}{\frac{s(\tau)} {t-\tau}}\,{\rm d}\tau})\) [5].

The WVD satisfies a number of desirable mathematical properties. However, application of the WVD is limited by interference terms, occurring as a result of the bilinear transformation. These interferences occur in the case of nonlinear frequency modulated signals and multicomponent signals, (see [4] for details).

#### 2.1.2 Matching pursuit algorithm

*N*, \({s \in {\mathbb{R}}^{N}},\) we consider the problem of representing

*s*as a linear combination of elements from a large, overcomplete, dictionary, \({\varvec{\Upphi}} \in {\mathbb{R}}^{N \times M},\) of

*M*waveforms, where

*M*>

*N*. The individual waveforms of \({\varvec{\Upphi}}\) are referred to as atoms, denoted by \({\phi_i \in {\mathbb{R}}^N, i = 0,1,\ldots,M-1}.\) The problem of representing

*s*using the dictionary \({\varvec{\Upphi}}\) can be formulated simply as

_{i}are the atom coefficients. Methods for solving this problem are referred to as atomic decomposition techniques.

*P*− 1)

^{th}iteration. The signal approximation and the residual are updated through the following iterative process:

*R*

^{0}=

*s*. For \({k = 1,\ldots,P},\)

_{i}||

_{2}= 1. It can be easily shown that the orthogonality of

*R*

^{k + 1}and φ

_{i}

^{k}in each iteration ensures conservation of energy; that is

### 2.2 Relative structural complexity measure

In many real signal processing applications, such as machine condition monitoring and newborn EEG seizure detection, the recorded signal undergoes a change in structure, as the underlying process generating the signal undergoes some type of changes. An analytical method for identifying changes in signal structure is therefore highly desirable. The proposed RSC measure, defined in this subsection, is a method for analyzing changes in signal structure.

#### 2.2.1 Definition of relative structural complexity

^{k}≥ η

_{D}, where η

_{D}is the desired approximation accuracy.

In [2], it was reported that the number of Gabor atoms (modulated, scaled and translated Gaussian functions), chosen in a MP approximation, was related to the complexity
^{1} of a signal. This was demonstrated with synthetic signals exhibiting limit cycle and chaotic behaviour constructed from the Duffing equation.

In [21], it was stated that “a matching pursuit decomposition in a given dictionary defines a system of interpretation for signals.” Signal components, which are interpreted well with a given dictionary, are referred to as coherent structures, and these structures are indicated by strong correlation with some dictionary atoms. The more coherent a signal is with a dictionary, the larger the correlations between dictionary atoms and the signal residues [21]. From this, we infer that the more coherent a signal is with a given dictionary, the fewer MP iterations required to achieve the desired level of approximation accuracy.

Considering the findings of [ 2, 21], it seems natural to introduce a new MP-based complexity measure referred to as RSC, which gives a quantitative indication of the complexity in interpreting a signal given a decomposition dictionary.

**Definition** [RSC]: Given a decomposition dictionary \({\bf \Phi} \in {\mathbb R^{N\times M}},\) we define the RSC of a signal \({\bf s} \in {\mathbb R}^N\) as the minimum number of atoms needed by the MP decomposition to approximate the signal to a desired level of accuracy, as defined by (10).

As it can be seen from this definition, the RSC is very dependent on both the chosen decomposition dictionary and the desired accuracy of approximation. This dependence justifies the word “relative” in our definition. This complexity measure has the advantageous ability to be adapted to the signal to be approximated, given that some a priori information about the signal is available.

#### 2.2.2 Illustration of RSC using synthetic signals

- 1.
Gabor dictionary—this dictionary consists of translated, scaled and modulated versions of a Gaussian window [21].

- 2.
Wavelet Packet dictionary—this dictionary consists of approximately

*N*log_{2}(*N*) waveforms [13] and is simply a family of orthonormal wavelet bases [21]. The dictionary used in this paper was built from a Daubechies 10 quadrature mirror filter. - 3.
Cosine Packet dictionary—this dictionary also consists of approximately

*N*log_{2}(*N*) waveforms, including the standard orthonormal Fourier basis and a variety of rectangular windowed sinusoids of various widths and locations [13] (n.b. this dictionary is similar to the real Gabor dictionary).

In this experiment, a number of synthetic signals, which have varying levels of coherency with the chosen decomposition dictionary were created. To do this, two different TF dictionaries were selected. One of the dictionaries was chosen to be the decomposition dictionary, \({\varvec{\Upphi}}_D.\) The second dictionary was used only in the *composition* of the synthetic signals (i.e., not used for signal decomposition), and is referred to as the alternative dictionary, denoted by \({\varvec{\Upphi}}_A.\)

Synthetic signals of length *N* were constructed using *P* randomly selected atoms of which *P*–*L* were selected from \({\varvec{\Upphi}}_D\) and *L* from \({\varvec{\Upphi}}_A.\) The number of atoms, *L*, from \({\varvec{\Upphi}}_A\) was increased from 0 to *P*, resulting in synthetic signals with decreasing levels of coherency with the decomposition dictionary \({\varvec{\Upphi}}_D.\)

In the case where *L* is small, we would expect the signal to be highly coherent with \({\varvec{\Upphi}}_D\) and hence produce a low RSC. For the case where *L* is close to *P*, only a small proportion of \({\varvec{\Upphi}}_D\) atoms would be used to construct the signal. For this signal, we would expect low coherency with \({\varvec{\Upphi}}_D\) which translates to a large RSC.

*N*= 512,

*P*= 30, \({\varvec{\Upphi}}_D\) = Wavelet Packet dictionary and η

_{D}= 13 dB were chosen. It can be seen from Fig. 1 that two alternative dictionaries, namely Cosine Packet dictionary and Gabor dictionary, were used to illustrate RSC. This figure confirms our expectation in that the RSC of the signal increases as the number of alternative atoms,

*L*, used in the construction of the synthetic signals increases. This also confirms that the coherency between the structures of a signal and the decomposition dictionary, when using MP, can be quantified by the RSC measure. This finding suggests that a suitably chosen decomposition dictionary will allow us to produce a compact signal representation if information on signal structure is available a priori. This result will be exploited in the following subsection.

This experiment was repeated with *P* = 50 and 70, and with \({\varvec{\Upphi}}_D\) : Cosine Packet dictionary and Gabor dictionary with similar results as shown in Fig. 1.

A noteworthy remark related to Fig. 1 is that, the Cosine Packet atoms are slightly more coherent with the Wavelet Packet atoms than the Gabor atoms. This is indicated by a slower rate of rise in the RSC measure when using the Cosine Packet dictionary as the alternative compared to the Gabor dictionary.

#### 2.2.3 Coherent newborn EEG time–frequency dictionary design

*burst–suppression*. In the time domain, this pattern is characterized by a burst of high voltage activity lasting 1–10 s followed by a period of quiescence or inactivity [1]. An example of the TF representation of a burst–suppression pattern, using the modified B distribution, is shown in Fig. 2a. It can be seen that the burst of high energy masks all other patterns. The second class of pattern found in the TF representation of the newborn EEG background is the EEG activity lacking a clearly recognizable TF pattern [6, 7, 8, 9]. In this background state, a dominant TF component, following a specific instantaneous frequency law, does not exist, as shown in the example Fig. 2b.

Our previous investigation of the newborn EEG seizure state using TF analysis methods showed that the TF patterns of newborn EEG seizures can be generally characterized by piecewise linear frequency modulated (LFM) components with slowly varying amplitudes [6, 7, 8, 9]. The analysis also showed that the newborn EEG seizure is quite often multicomponent. Both these characteristics can be seen in the example TF representations, Fig. 2c and d, of two newborn EEG seizure epochs.

The piecewise LFM components characterizing newborn EEG seizure suggests that the design of a decomposition dictionary, coherent with newborn EEG seizure structures, is feasible. A more detailed analysis of the TF structures of EEG seizures [27], from a significantly larger database than the one used in [6, 7, 8, 9], allowed for the specific characterization of the newborn EEG TF structures. It was found in [27] that the LFM components of the seizures had a starting frequency of ξ_{b} ∈ [0.425,6.875] Hz and LFM slopes of ξ_{s} ∈ [−0.06,0.06] Hz/s. It was concluded from these observations that a TF dictionary coherent with the newborn EEG seizure state must include LFM atoms covering the observed ranges for ξ_{b} and ξ_{s}. It was also decided that only LFM atoms would be included in the dictionary and not piecewise LFM atoms, as this would cause a combinatorial explosion in the number of atoms in the dictionary, resulting in unrealistic processing times.

*F*

_{s}is the sampling frequency, θ ∈ [0,2π) is the starting phase and

*A*is a scalar value chosen such that ||ϕ

_{i}

^{LFM}||

_{2}= 1. For the analysis of newborn EEG, a sampling frequency of

*F*

_{s}= 20 Hz was chosen as approximately 95% of the newborn EEG power is found in frequencies less than 8 Hz [28].

Since the described set of LFM atoms do not, in general, form a complete dictionary, we construct an overcomplete coherent dictionary by combining the set of LFM atoms with an overcomplete Gabor dictionary [24, 25]. This dictionary is used for the RSC analysis of real and synthetic newborn EEG data.

#### 2.2.4 Illustration of coherent newborn EEG dictionary

To illustrate the efficiency of the proposed dictionary in representing the newborn seizure state, a MP decomposition of a 12.8 s newborn EEG seizure was performed. By choosing a short enough epoch length, the piecewise LFM structures of newborn EEG seizure can easily be approximated by LFM structures.

## 3 Newborn EEG data

### 3.1 Synthetic newborn EEG data

^{2}For the initial investigation of the RSC measure applied to newborn EEG, we created a number of realistic synthetic newborn EEG signals using the simulation methods of [27]. The synthetic newborn EEG signals are composed of four time periods; namely, preictal, seizure onset, seizure and postictal. The preictal and postictal periods were created using the newborn EEG background model described in [27], each lasting 60 s. The seizure period was created using the seizure model of [27] and lasted for 50 s. The seizure onset period was constructed using a combination of synthetic background and seizure signals. The seizure onset period was synthesized with a gradual increase in signal to background ratio (SBR), given by

*s*

_{sz}is the seizure signal and

*s*

_{bk}is the background signal. The seizure onset was synthesized to start 24 s before the full-fledged seizure. The SBR was changed every 4 s with the following successive values [-2.5,0,2.5,4,5,6] dB. Fifty synthetic newborn EEG signals of this form were created for the synthetic RSC analysis in Sect. 4.1.

### 3.2 Real newborn EEG data

The EEG from six neonates exhibiting EEG seizure periods, as marked by a pediatric neurologist from the Royal Children’s Hospital, Brisbane, Australia, were analyzed using the RSC measure. The EEG data was recorded at the Royal Brisbane and Women’s Hospital, using the MEDELEC Profile System. The raw EEG was bandpass filtered with cutoff frequencies at 0.5 and 70 Hz, and was sampled at 256 Hz. For the RSC analysis, the EEG data were digitally bandpassed filtered with cutoff frequencies at 0.5 and 10 Hz, before being downsampled to 20 Hz.

## 4 Application of RSC to newborn EEG

### 4.1 RSC analysis of synthetic newborn EEG

Synthetic newborn EEG signals were used in the initial investigation for two main reasons. Using a mathematical model, we can generate a signal that captures the main characteristics of the newborn EEG without the added complexity introduced by artefacts. Also, synthetic signals allow us to specify the exact time locations of the transitions between the two EEG states; a matter which is subjective and reviewer-dependent for the case of real newborn EEG [33]. These two characteristics of the synthetic signals allow us to assess the strengths and the weaknesses of the new complexity measure before being applied to real newborn EEG.

The RSC was implemented using a sliding, rectangular window, of length *N* = 256 samples. The overlap between successive windows was set at 60 samples (i.e., 3 s). The stopping criterion for the MP decomposition was chosen to be \({\eta_D \geq 13\,\hbox {dB}}.\)

Mean and standard deviation of relative structural complexity measures for the preitcal, seizure and postictal periods

| Preitcal | Onset | Seizure | Postictal |
---|---|---|---|---|

Mean | 32.4 | 26.5 | 11 | 31.8 |

Standard deviation | 5.35 | 5.58 | 3.52 | 5.85 |

### 4.2 RSC analysis of real newborn EEG

As in the case of the synthetic newborn EEG, the RSC was implemented using a sliding, rectangular window, of length *N* = 256 samples and a window shift of 60 samples. The MP stopping criterion was set at \({\eta_D \geq 13\,\hbox {dB}}.\)

The example in Fig. 5 exhibits similar results to the synthetic EEG data analyzed in Sect. 4.1. However, in Fig. 5a, it can be seen that there is a significant and unexpected drop in the RSC for the period between 70 and 85 s (preictal state). This sharp and unexpected drop is caused by *short-time and high-amplitude* artefacts [12, 29], as marked in Fig. 5b. This type of transient artefact often represents a large percentage of epoch energy. These signal structures also correlate highly with some small-scaled Gabor atoms causing low RSC measures for the epochs in which they are contained.

Relative structural complexity for selected nonseizure and seizure periods in the newborn electroencephalogram

Patient number | Nonseizure | Seizure |
---|---|---|

1 | 54 | 11 |

2 | 51 | 11 |

3 | 46 | 13 |

4 | 52 | 22 |

5 | 51 | 17 |

6 | 47 | 27 |

## 5 Discussion

When a system undergoes an internal change in its state, (e.g., brain seizing, power-line affected by disturbances, knock in internal combustion engine, etc.), the signals emanating from it, as measured by external sensors, usually reflect this change through a change in their structure. This enables non-invasive methods for detecting this change in state of the system.

In this paper, we presented a new method for analyzing changes in signal structure using a relative measure of signal complexity based on the MP-decomposition technique. This new measure is strongly dependent on the nature of the decomposition TF dictionary and the desired level of accuracy in the signal approximation. This new signal complexity measure can also be taken as a measure of the coherency between the signal structure and the decomposition dictionary.

Compact signal approximations using MP decomposition can be achieved by choosing a decomposition dictionary that is highly coherent with the signal under analysis. This prompted the development of a newborn EEG specific TF dictionary that was used to analyze newborn EEG.

To build a TF dictionary that is coherent with newborn EEG seizures, it was shown that the specific TF patterns in newborn EEG seizure could be translated into a set of TF atoms. This set of atoms, which was a set of LFM atoms, was combined with an overcomplete Gabor dictionary to form the new overcomplete TF dictionary. Using this dictionary, the RSC measure was applied to synthetic and real newborn EEG. The analysis indicated a significant fall in the RSC, as the newborn EEG transitioned between background and seizure periods.

Due to clear differences in RSC observed between seizure and nonseizure newborn EEG, the next logical step would be to incorporate the RSC measure into an online automatic newborn EEG seizure detection algorithm. However, there are a few limitations that need to be overcome before such an automatic detection algorithm becomes practical.

The major limitation in using the RSC as a stand-alone feature for automatic newborn EEG seizure detection is the difficulty in determining a threshold level to distinguish seizure from nonseizure. In our analysis of real newborn EEG, it was observed that the level of RSC for nonseizure and seizure varied significantly between patients. This result suggests that a patient-specific method of threshold determination, possibly through neural network training, would be required. Artefacts, particularly short-time and high-amplitude artefacts, severely limit the application of RSC. It was shown in Fig. 5 that a large amplitude artefact caused a significant and undesirable drop in the RSC. Therefore, artefacts may introduce numerous false detections. The drop in RSC for this type of artefact is due to the high coherency between artefact structures and some small-scaled Gabor atoms. The use of an appropriate artefact removal technique may alleviate the effects of artefacts. Another possibility worth investigating is the development of a new TF dictionary, which does not include atoms that are coherent with any artefacts or background structures, while still coherent with the newborn EEG seizure state.

## 6 Conclusion

Relative structural complexity, introduced in this paper, is a new MP-based measure of signal complexity, where the relativity is associated with the chosen decomposition dictionary. Development of a new TF dictionary, which is highly coherent with newborn EEG seizure structures has provided us with a method for analyzing the changes in newborn EEG signal structures as it evolves from the background state to the seizure state. The proposed methodology for analyzing changes in signal structure using the RSC measure is generic and may be used in many other signal detection problems. This method, however, is limited by the necessary a priori information for suitable dictionary selection and/or design. This limitation can be overcome via in-depth analysis of the embedded structures of the signals under analysis, as was demonstrated for the case of newborn EEG.

## Footnotes

- 1.
The definition of complexity here is related to the complexity of the phase space representation (level of chaotic behaviour) of the signal, often used in nonlinear time series analysis [19].

- 2.
The MATLAB code used to create the synthetic newborn EEG background and EEG seizure used in this paper is based on the models described in [27]. The code is freely available from http://www.som.uq.edu/research/sprcg.

## Notes

### Acknowledgments

The authors gratefully acknowledge Prof. Paul Colditz for organizing the acquisition of the real newborn EEG data and Dr. Chris Burke and Jane Richmond for their expertise in newborn EEG reading.

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