INTRODUCTION
Wavelets - designing from signal patterns In this par t- II, cer tain records are illustrated with annotations that lead to the concept of identifying and comparisons of spike-wave patterns using wavelets. The usefulness of nonlinear wavelets is brought forth and some nonlinear wavelets are illustrated. Next, a technique has been developed of picking a wave pattern from the EEG itself and forming a wavelet of it. When we use this wavelet, though crude, and analyse the patient’s lengthy record, we might localize regions which are the preliminaries for the development of the large amplitude abnormal spike and wave patterns.
When we know that a signal junk comprises of a collection of signal patter ns characteristic of the originator of the signal, then it is better to use a wavelet that mimics the basic
Material Science Research India Vol. 5(2), 347-358 (2008)
Electro encephalography measurements
using wavelet based signal processing (Part 2)
S. ANANTHI, V. VIDYADEVI and K. PADMANABHAN
Central Instrumentation Research and Department, University of Madras, A.C. Tech Campus, Chennai - 25 (India).
(Received: July 20, 2008; Accepted: August 29, 2008)
ABSTRACT
In Part –II of this paper, some nonlinear oscillation based wavelets are introduced as useful for EEG analysis. Next, it is shown how a new wavelet based on the spike and wave activity picked up from a patient’s own record could be designed. With such a designed wavelet, analysis made on the normal e.e.g. record itself is able to indicate any lurking low-amplitude activity characteristic of that pathology. While photic stimulation would induce that activity in the patient and elicit a really significant large amplitude spike and wave, it is better avoided. With this method, since such an activity can be inferred in the normal (un-stimulated) record itself by the picked wavelet analysis, it provides for a safer diagnostic procedure.
Key words: Van der Pol wavelet, picking a wavelet from signal, design of wavelets.
oscillatory pattern of it. Naturally originating signals from multiple entities might have signals of a complex nature, both in their frequency distribution and their wave pattern variations. Non linear oscillators of various kinds are known, such as the Van der Pol, for instance, which are due to the effects of saturation, hysterisis and similar nonlinear effects. The available wavelets may not sufficiently characterize the signals generated by such effects. In such a situation, trying out with available wavelets could only give some initial information about the signal and its time course ; it may be a poor fit.
Typical illustrations
Initially, let us illustrate the analysis of some record segments using available wavelets and compare between them.
Patient Name: X1, Record Number-2682, Age14 (Fp2- F4,F4- C4, T4-T6 T6 O2)
Fig. 1: (a) Signal (b) Fourier spectrum (c) Wavelet plot at the max. scale
Note: In this signal, we have D = 0.5 – 4 Hz -More D activity and q activity.
From the Fig.1, the scale 21 plot of DB10 wavelet ( Maximum scale plot) indicates two events : 1. t = 120 to 200
A peak with slow tail, a second peak and a doublet peak.
This pattern repeats again between t = 440 to 510
2. t = 350 to 420. A biphasic twin-complex with a centre high amplitude peak is noted. This is repeated again with less amplitude between t = 570 to 640. This indicates two asynchronous abnormal neural foci are eliciting these complexes
This record shows predominant wave patterns at low frequency (<5Hz). At t =260, there is a large spiky biphasic wave ( Fig.2). This is followed by 2 mono-phasic waves of good amplitude. Are all these events synchronous?
Observing the DB10 maximum amplitude scale plot, the second wave is nonlinear, which shows clearly in DB 10 as a broad non-linear wave and the bior.3.7 shows the transition at t=320.
So, it is noted that only the more nonlinear wavelets are able to bring out the transitions of consecutive wave spikes.
The record shows the sudden (low frequency) wave spikes (Fig.3). The Fourier spectrum is second from top. It is required to find if these two are spikes arising synchronously from a single site. This information is not found from the record itself.
The nonlinear wavelet DB 10 and Bior5.5 indicate that these two waves are differential events. The transition between 2 spike-wave complexes is clear here.
CASE-2
Patient Details: X2, Record- 2690, Age 45
Fig. 2: Signal and several wavelet maximal scale time scale plots CASE 3:
R no: 2697 age 19
Fig. 3: Records of low frequency spikes analysed with various wavelets
Mildly this transition is visible. No transition visible with (Morlet, Mex.hat). wavelets.
Db6 (frequency-6.Hz, scale-14) wavelet decomposed plot ( Fig.4c). At t=90 and t=280, the
pattern passes through a null. There are up to 8 wave groups noticeable. The first group is between t=15 and 80. The second group, also consisting of similar waves, is between 95 and 160. These are similar groups.
Fig. 4: (a) Signal Segment (b) Fourier Spectrum (c) DB10 max.line wavelet spectrum
There is another different wave group between t=165 to 210. This is a nonlinear oscillation. There is a theorem which states: If Ψ is a wavelet and Φ is a bounded function which is integrable, then the convolution Ψ*Φ is also a wavelet. Taking two such functions, one a wavelet and another the van der pol wave pattern, we get a true wavelet.
Such a wavelet is described in fig.5 below. One such wavelet is constructed from the DB6 wavelet with the Van der Pol oscillatory wave. Such wavelets are not crude wavelets, (like the Morlet). These possess the necessary properties of wavelets. It is possible to find a scaling function for
Fig. 5:
(c) Convolved wavelet
Fig. 6: More wavelet max. scale plots for the signal of fig. 4 a.
it. The first part of this figure below matches with this nonlinear wavegroup. Another group occurs between 220 and 265. This also has a combined Van der pol oscillation. Lastly there are two similar groups from 300 to 335, 335 to 370. These are better linear oscillations, and the transition between them is visible only in the DB 10 wavelet.
While the nonlinear wavelets indicate the transitions (eg., t=90) of the different wave groups distinctly, the smoothing wavelets below (Morlet and Mex. Hat) show as if there is a continuous chain of wave oscillations. These are not useful to show the nonlinear patterns evinced by the DB10 wavelet. The db10 pattern and the biorthogonal 5.5 show the pattern of this wavelet of fig.5 in their figures.
Fig. 7
Thus, with the smoothing wavelets, (Mex. Hat somewhat better), the different nonlinear wave
groups in the signal are not noticeable. However these serve to indicate the groups of waves.
CASE-5:
Patient Name: X5
While the smoothing wavelets show four wave clusters, with the central one large and two on left and one on right in this record of 800 data samples, the DB10 and Biorth.5.5 wavelets ( Fig.9) show the first and last as arising out of one neural focus. The central wave cluster looks similar to a Morlet wavelet itself. The record itself shows this as a nonlinear oscillation, but even the nonlinearising wavelets show them as near sinusoidal oscillations.
Thus this central wave group is not likely to be nonlinear in origin though it appears like that in the record. In ictal records, the high amplitude oscillations are nonlinear and they also show up as non linear in the wavelet patterns. One diagnostic inference was that in this inter ictal record, the neural focus is still generating within its linear region and only if it transgresses this, the patient is likely to get into a seizure.
Fig. 9
CASE-6
Patient name: X6, Record No: 2714,Age 33 Two transitions alone are seen in this record, near t=150 and t=350. At 40,150, 300 and 350 we get transitions actually, as noted in the nonlinear wavelets (Fig.10- bior.4.4.,Mexican hat).
The oscillations are still linear only, except for the transition at t=150. In a continuous seizure –like oscillatory record of this type, though it is inter-ictal, the record is linear in its oscillatory groups. Therefore, seizure is not likely to occur in the immediate time.
Fig. 10
New wavelets as applicable for specific patient diagnosis:
Using the Van Der Pol Wavelet¹
In what follows, the newly designed Van der Pol wavelet is applied.
The following is the trial on a typical spike
pattern tested on the newly designed wavelet – the Van der Pol wavelet (fig.5b).
The signal is analysed using CWT scalogram plot for this new wavelet as shown below in fig.11 (b).
It is noted that the Van der pol wavelet localizes better the components of the signal spike and wave-bits. The two components are noted individually at scale 8 and 32 whereas the dB4 was noted to merge the two somewhat (not shown in fig.11).
This indicates the presence of such a nonlinear oscillatory pattern in the EEG signals in general, during such spike, spike and wave activity at t=70,140,275,350.
As an example of its feature in showing spike-waves alone better, the following figure shows the wavelet scalogram and the time plot at scale 3.
This brings out the peaks clearly amidst alpha wave pattern.
Fig. 11: (a) spiky EEG record segment (b) Showing the Vander Pol wavelet CWT (c) wavelet coefficients plot at scale 3
Designing a Wavelet from the signal itself Using the pattern taken from the EEG itself, the Fig.12 shows how a new wavelet can be designed. Here a typical spike wave pattern is assumed to be a wavelet and then it is decomposed into a conventional wavelet (fig.12 top right) as well with the designed wavelet. The scalogram segments indicate the usefulness of the method in dispreading the scalogram in such cases. The spike wave signal is first approximated to a 6th order polynomial for
best fit and giving the integral a near zero value.
Then, it is fitted as a wavelet, though a crude one, for CWT. The CWT analysis can then be performed with it for the entire record. That will certainly highlight wherever such a typical pattern occurs in the time record (Fig.13).
Fig. 13: The typical EEG recorded segment of signal for a patient (a) Signal (b) Morelet scalogram (c) Picked wavelet scalogram Fig. 12: Showing how the chosen wavelet self designed for a signal can
points of high intensity. The significant pattern is noted in scale 7.5, at timing near 240, 670. From the comparison of the two, it will be possible to identify the regions where such activity is present more uniformly by the tiny bright spots in fig.(c) Such specific locations are not noticeable in the Morlet wavelet spectrogram plot (b).
In fig.14, the same is performed on a spike onset record. Here, the middle Db6 wavelet scalogram has too many patches, while the regions which need to be observed for scale plots are better localized and they are for instance, noteworthy at the 130,. 260 and 170 sample points.
Appendix: Pick a wavelet from the eeg record -a new technique
The picked signal segment of the EEG record, which is noticed to be repeated often for this patient during recording, is taken and saved a file.
Load the file of EEG record extracted from EEG machine Software. Load a:sy3.mat % load this file previous saved from Matlab after recording.
s1= sy3(104:124);
% these 20 data points belong to the picked EEG wave pattern for wavelet formation!
S1=s1-mean(s1);
% this eliminates DC component. Save this s1 as a picked data file in a: as
Save a:\s1 s1
Then the wavemenu is opened by typing Wavemenu from Matlab 7.0 [1]
The GUI menu appears. In that, from FILE the s1 file is loaded from a:
The the pattern appears on screen. Then the approximation of “ortho.const” is chosen and the same figure is approximated as a wavelet. Then we have to perform analysis with this new wavelet.
First we save this adapted wavelet in a .mat file by going to the FILE menu, picking the SAVE option and saving it in a:\pickeegwv.mat
load a:pickeegwv
[psi,xval,nc]=pat2cwav(Y,’orthconst’,6,’continuous’) plot(xval,nc*psi)
0 100 200 300 400 500 600
0 500 1000
Absolute Values of Ca,b Coefficients for a = 1 2 3 4 5 ...
time (or space) b
s
c
al
es
a
50 100 150 200 250 300 350 400 450 500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Absolute Values of Ca,b Coefficients for a = 1 2 3 4 5 ...
time (or space) b
s
c
al
es
a
50 100 150 200 250 300 350 400 450 500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Fig. 14: The typical EEG recorded segment of signal for a patient (a) Signal (b) DB6 Scalogram (c) Picked wavelet scalogram
The second line above evaluates the psi function for this picked wavelet.
The psi function and the numerical constant nc are used to plot the wavelet as above in the next line.
Then we have to perform the wavelet analysis as usual with this new wavelet.
We first plot the signal itself in top box. Subplot(311);plot(sy3);
Then we perform usual wavelet analysis for a chosen scale.
Sca =1:.1:10 Fr =1./sca
sy3 = sy3-mean(sy3)
c = cwt(sy3,sca,’morl’,’plot’);
Then we perform the wavelet analysis using our picked new adapted wavelet.
Subplot (313)
c=cwt(sy3,sca,nc*psi,’plot’);
CONCLUSION
Inter-ictal spikes and sharp waves in human EEG are characteristic signatures of epilepsy. When we use wavelet transform to analyze the properties of EEG manifestations of epilepsy, it has been demonstrated that the behavior of wavelet transform of epileptic spikes across scales can constitute the foundation of a relatively simple yet effective comparison algorithm. However, the interpretation of wavelet transform data needs an equally intensive training. Present day EEG machines are almost computerized with appropriate softwar2,3 incorporated with provision for additional
processing and hence these wavelet techniques might well prove to aid EEG signal analysis better.
REFERENCES
1. B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1: 701-710, 754-762 (1920). 2. Matlab tool box reference 7.0, Mathworks
