electroencephalogram ( eeg ) is the recording of the brain s spontaneous electrical activity captured non - invasively by placing electrodes on the scalp .
eeg has been utilized in many medical diagnosis , prognosis and therapies including epilepsy , sleep disorder , coma , encephalopathy and brain deaths .
eeg signals are often corrupted by two sources of artifacts : physiologic such as eye , muscle , and cardiac activities , and extraphysiologic such as line interference and electrode noise .
extraphysiologic artifact can often be removed using appropriate filtering techniques as there is spectral separation .
however physiologic artifact removal requires careful attention as they can be within the same frequency range of the eeg signal and are aperiodic .
ocular artifacts ( oa ) due to eye movement and eye blinks are dominant over other contaminating physiologic artifacts . as eeg signal can be used for analyzing different diseases  , monitoring brain engagement , , different techniques have been proposed for the removal of oa from eeg to make it more reliable for different purposes .
the widely used methods for removing oas are based on regression in time domain   and frequency domain .
but these methods need the recording of electrooculogram ( eog ) , and can also result in the elimination of neural activities .
statistical techniques like principal component analysis ( pca ) , kurtosis , independent component analysis ( ica )   and multiscale sample entropy   are also shown to be effective to remove oa , but they rely on multiple channel data .
one of the robust and promising ocular artifact removal techniques for single channel eeg data is wavelet transform ( wt ) , .
discussed applying stationary wavelet transform with coif3 wavelet filters to denoise the eeg signal . in their proposed study ,
use of haar wavelet is explored in detecting changes in the state of the eye ( eye - blinks and eyeball movements ) .
stationary wavelet transform with coif3 as a basis function with various non - adaptive thresholding methods have also been demonstrated .
another wavelet - based approach of removing ocular artifact was to use stationary wavelet transform with sym3 as a basis function and to use the coefficient of variation to detect and denoise the artifact . with the advent of ambulatory and miniaturist body - worn eeg systems with few channels for routine monitoring 
, there is a growing need to develop effective oa removal technique that can operate on few channel eeg data . for real - time applications like mental state classification , comfort sensing , emotion sensing , movement prediction etc .
many brain - computer interfacing ( bci ) systems are utilized for routine and continuous monitoring of brain activities for epilepsy , autistic spectrum disorder ( asd ) , and alzheimer s patients .
hence , removal of artifacts in real - time with the access of few channel ( especially single channel ) is of research interest .
though there is a vast amount of literature available on using wavelet transform for ocular artifact removal from eeg data  , a little have investigated the effects of using various possible forms of wt for single channel oa removal .
the evaluation of existing wt techniques is critical in order to find the efficient , reliable and unsupervised way of denoising oa for real time bci applications .
specifically , we present a comparative study of discrete and stationary wavelet transform using four basis functions : haar , coif3 , sym3 , and bior4.4 with universal and statistical thresholding for oa removal from single channel eeg data .
these combinations are carefully selected as they are commonly used for oa removal  , .
furthermore , we present objective performance metrics using multiple statistical measures in time domain and frequency domain .
wt can be applied to any single channel eeg data to remove oas without information from any other eeg or eog channels .
wt decomposes a time - varying signal into its set of basis functions known as wavelets .
these basis functions known as wavelets are obtained by performing dilations and shifting of the mother wavelet:\documentclass[12pt]{minimal } 
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 } { } \begin{equation * } \psi _ { \mathrm { a , b}}\mathrm { ( t)}=\psi \left({\frac { t - b}{\mathrm { a}}}\right ) \end{equation*}\end{document } where a is the scaling parameter and b is the shifting parameter . in this study , we have implemented multi - level wavelet decomposition in order to get precise information about the wavelet coefficients .
in addition to ocular artifact removal , wt is proven to be a robust tool in several applications like machine condition monitoring , hologram analysis , pitch detection of speech signals , multi - modality medical image fusion , fault detection in a spur gear , power quality analysis , signal processing in white - light interferometry .
dwt is considered non - redundant and highly efficient wavelet transform to obtain discrete wavelet representation of signals , . in dwt
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this process is repeated until the desired frequency range is obtained . at each stage ,
the filter output is down - sampled by 2 , later up - sampled to reconstruct the signal . in this study , we have used in - built wavedec function in matlab ( mathworks , natick , ma ) to implement dwt in the denoising algorithm . 
 figure 1.graphical representation of dwt decomposition .
( hpf : high pass filter , lpf : low pass filter . ) graphical representation of dwt decomposition .
the major drawback of dwt is its time - variance , which is particularly important in statistical signal processing applications such as eeg .
swt overcomes this translation - invariance drawback of dwt , but has redundant information and is relatively slow .
the approximate and detail sequences at each level of decomposition are of the same length as the original sequence . after obtaining the coefficients at \documentclass[12pt]{minimal } 
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wt of the eeg signals yields the wavelet coefficients which represents the correlation between eeg signal and the wavelet basis functions .
figure 2 represents some commonly used wt basis functions utilized in the literature for oa removal . for eye - blink removal ,
these wavelets perform well as they resemble the characteristics of these eye blinks , , . in this paper , we have compared the performance of widely used symlet ( sym3 ) , haar ( haar ) , coiflet ( coif3 ) , and biorthogonal ( bior4.4 ) wavelets . 

figure 2.examples of common wavelet basis functions that can be applied for artifact removal from eeg data .
examples of common wavelet basis functions that can be applied for artifact removal from eeg data .
the approximate and detail coefficients of the decomposed eeg needs to be denoised to separate the artifactual coefficients from neural signal coefficients . for thresholding the wavelet coefficients , two commonly used metrics : universal threshold ( ut ) and statistical threshold ( st )
ut is implemented as:\documentclass[12pt]{minimal } 
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 } { } \begin{align } k=&\sqrt { \mathrm { 2logn } } \sigma \\
\sigma ^{2}=&\mathrm { median}\left({\frac { \left | { c_{a } } \right |}{0.6745}}\right ) \end{align } 
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 } { } $ a^{th}$ 
 \end{document } level of decomposition that undergoes thresholding , and 0.6745 is the constant value for gaussian noise .
we also implemented st ( proposed by krishnaveni et al . in ) in our study which is based on the statistics of the signal .
as discussed in , the proposed threshold produces better de - noised results than the other conventional thresholds .
mathematically , the proposed st is formulated as:\documentclass[12pt]{minimal } 
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 } { } \begin{equation * } t = 1.5^\ast std(h_{k } ) \end{equation*}\end{document } where t is the estimation of neuronal wide band signal magnitude using st , std(\documentclass[12pt]{minimal } 
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 } { } $ k^{th } $ 
 \end{document } level . in both cases ,
hard thresholding is applied , where wavelet coefficient is removed if the absolute value of wavelet coefficient is greater than the threshold .
different statistical performance metrics have been used to objectively compare various combinations of oa removal in time and frequency domain . for time domain comparison ,
correlation coefficient , mutual information , signal to artifact ratio and normalized mean square error have been evaluated . for frequency domain comparison
correlation coefficient ( cc ) is a statistical method that shows the degree of association between two variables .
suppose c(t1 , t2 ) is the auto - covariance of a process x(t ) , or in another way , c ( t1 , t2 ) is the covariance of the random variables x ( t1 ) and x ( t2 ) , then correlation coefficient of the process \documentclass[12pt]{minimal } 
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 } { } \begin{equation * } \mathrm { r}\left ( { \mathrm { t1 , t2 } } \right ) = \frac { \mathrm { c}\left ( { \mathrm { t1 , t2 } } \right ) } { \sqrt { \mathrm { c}\left ( { \mathrm { t1 , t1 } } \right ) \mathrm { c}\left ( { \mathrm { t2 , t2 } } \right ) } } \end{equation*}\end{document } mutual information ( mi ) is used statistically to measure how much information one random variable contains about the other random variable .
if u and v are two partitions of sample space s , then information about u contained in v or information about v contained in u is:\documentclass[12pt]{minimal } 
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 } { } \begin{equation * } \text { i ( u , v ) } = \mathrm { h}\left ( { \mathrm { u } } \right ) + \mathrm { h}\left ( { \mathrm { v } } \right ) -\mathrm { h ( u.v ) } \end{equation*}\end{document } i ( u , v ) represents mutual information .
signal to artifact ratio ( sar ) is a quantification method to measure the amount of artifact removal in a specific signal after processing with an algorithm .
if \documentclass[12pt]{minimal } 
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 } { } \begin{equation * } sar=10\log \left({\frac { std\left ( { z } \right ) } { \mathrm { std}\left ( { \mathrm { z}-\hat { \textrm { z } } } \right ) } } \right ) \end{equation*}\end{document } normalized mean square error ( nmse ) approximates the difference between the ideal and actual data .
nmse is computed in db using the equation:\documentclass[12pt]{minimal } 
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 } { } \begin{equation * } nmse=20loge\left\{{\frac { \sum \left [ { x1\left ( { i } \right ) -x2\left ( { i } \right ) } \right ] ^{2}}{\sum \left [ { x1\left ( { i } \right ) } \right ] ^{2}}}\right\ } \end{equation*}\end{document } time and frequency components can be analyzed simultaneously using the wavelet decomposition tool of eeglab toolbox ( matlab , ca , us ) .
for eeg acquisition , a 14-channel referential montage epoc headset ( emotiv , eveleigh , nsw , australia ) at a sampling rate of 128 sps was used in the lab setting . before data acquisition
, the skin of the subject was cleaned using nuprep skin preparing gel ( weaver and company , aurora , co ) and mild abrasive strips to remove the dead skin and thereby moisten the skin .
data was collected from four subjects ( two males , two females ) at af3 , af4 , f3 , f4 , f7 , f8 , fc5 , fc6 , p7 , p8 , t7 , t8 , o1 , and o2 channel locations in a closed room for 1 min 45 s. during the recording , subjects were instructed to blink 9 times with a 5 s hiatus . as oas
are prominent in the frontal lobe , only af3 channel data was used for analysis in this study . out of two sessions
/ subject data recording , 7 datasets were used for this study using an approved institutional review board protocol ( university of memphis irb # 2289 ) . as oas are prominent in the frontal lobe , most of the comparison plots in this study are from af3 channel .
however , as discussed in the literature , wt can be applied to denoise any channel location , as depicted in figure 3 .
oas occur due to eye movement and eye - blinks and have frequency ranges of 0 - 7 hz and 8 - 13 hz , respectively .
to accurately identify artifact related wavelet coefficients , we have implemented multi - level wavelet decomposition using swt or dwt
. the decomposition was done up to level 8 ( level 8 : 0.25 - 0.5 hz , level 7 : 0.5 - 1 hz , level 6 : 1 - 2 hz , level 5 : 2 - 4 hz , level 4 : 4 - 8 hz , level 3 : 8 - 16 hz , level 2 : 16 - 32 hz and level 1 : 32 - 64 hz ) to obtain the frequency range of interest . for denoising the wavelet coefficients ,
thresholding has been done over the detail coefficients from level 8 up to level 3 . as the sampling rate of the dataset is 128 sps ,
decomposing up to level 3 gives us the required ocular related wavelet coefficients for denoising .
either ut or st is implemented for thresholding and sym3 , haar , coif3 or bior4.4 have been used as mother wavelet . with two wavelet decomposition techniques ( dwt / swt ) , two thresholds ( ut / st ) and four mother wavelets ( sym3/haar / coif3/bior4.4 ) , 16 combinations or methods are possible to remove oa from eeg .
figure 3.12-channel eeg data from the dataset 4 ( top ) raw ( below ) oa artifact - free .
x - axis is time in seconds and y - axis is amplitude in microvolts .
12-channel eeg data from the dataset 4 ( top ) raw ( below ) oa artifact - free .
x - axis is time in seconds and y - axis is amplitude in microvolts .
figure 4 compares raw and oa - artifact free eeg data using coif3 wavelet basis function with statistical threshold to denoise single channel eeg data using swt and dwt techniques .
careful observation indicates that swt produced cleaner processed signals , however , dwt is a faster method , which is an important aspect of real time data processing ( e.g. streaming data ) .
most oa removal algorithms affected neuronal signals where there is no blink artifact ( non - blink regions ) . to the distinguish performance of an algorithm to remove oa , while preserving neuronal signals , we have segregated the raw eeg data into  blink regions  and  non - blink regions .  

figure 4.comparison of a section of eeg data from af3 channel ( dataset 1 ) before and after denoising using swt and dwt decomposition techniques with coif3 wavelet .
comparison of a section of eeg data from af3 channel ( dataset 1 ) before and after denoising using swt and dwt decomposition techniques with coif3 wavelet .
cc and mi have been calculated between raw and re - constructed signals for blink and non - blink regions of the entire eeg data .
the ideal eye blink removal algorithm would produce high cc and mi values in the non - blink region , while low cc and mi values in the blink region , .
figure 5 and figure 6 show the statistical analysis of the cc and mi metrics ( averaged over 7 datasets ) for both blink and non - blink regions of af3 channel eeg data . 

figure 5.correlation coefficient comparison ( \documentclass[12pt]{minimal } 
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figure 6.mutual information comparison ( \documentclass[12pt]{minimal } 
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 } { } $ \text{n}=7 $ 
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correlation coefficient comparison ( \documentclass[12pt]{minimal } 
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 } { } $ \text{n}=7 $ 
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mutual information comparison ( \documentclass[12pt]{minimal } 
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 } { } $ \text{n}=7 $ 
 \end{document } ) for blink and non - blink eeg data using ut and st thresholds with swt and dwt methods . in both cases ,
dwt with ut performs better in an eye - blink region than all other wt threshold combinations . among them
, dwt+ut+sym3 gives the lowest value of cc , while dwt+ut+haar gives the lowest value of mi .
however , the efficacy of dwt with ut to preserve neuronal information in a non - blink region is poor .
similarly , among all cases , the swt with st generates higher values of cc and mi than all other wt threshold combinations for non - blink regions . among these
, swt+st+haar provides the highest values of cc and mi in non - blink regions .
based on cc and mi metric , swt with ut reveals that neuronal signals retention is poor , but removes or changes the blink zone in a greater amount .
swt with st retains neuronal signals based on both cc and mi metrics , even though oa removal performance based on cc and mi is not as good as other methods . applying dwt with ut
does nt show the power of preserving neural information based on cc and mi metric .
dwt with st is excellent to retain neuronal information based on cc metric for all wavelets , but performance is dependent on wavelet types based on mi metric , where haar wavelet outperforms other basis functions .
according to the definitions , the technique that produces the higher value in the sar and the lower value in the nmse is considered to be more effective .
it is noted from table 1 that , in general , sar values are higher when st is used with any types of wavelet , indicating st is aggressive in eliminating probable artifacts , while ut is conservative .
table 1sar on eeg datasets using ut and st thresholds with swt and dwt methodswt+thresh.sym3haarcoif3bior4.4swt + ut1.420.611.480.581.400.611.420.61swt + st2.330.862.180.792.320.852.280.85dwt + ut1.280.631.130.491.310.631.30.62dwt + st1.930.821.680.651.890.801.890.78 table 2 depicts the values of nmse for different methods .
based on table 2 , swt+st again outperforms other methods , while dwt+st is the second best , compared to other wt threshold combinations .
as the lower nmse indicates better technique , both swt and dwt show superior performance with st for any type of wavelet basis functions .
nmse values are lower when st is applied with both swt and dwt using any type of wavelet basis functions .
table 2nmse on eeg datasets using ut and st thresholds with swt and dwt methodswt+thresh.sym3haarcoif3bior4.4swt + ut5.882.376.112.235.792.365.852.37swt + st8.923.258.472.968.843.28.773.21dwt + ut5.312.54.71.945.432.495.392.45dwt + st7.973.186.92.497.83.17.823.02 time - frequency analysis results are shown along with the raw signal ( dataset 1 , af3 channel location ) in figure 7 .
few dwt methods are observed to have introduced new artifactual noise in the processed data during the oa removal throughout the spectrum ( e.g. dwt+ut+haar , dwt+st+haar ) .
dwt with ut is also noted to decrease the overall magnitudes of neuronal signals . within dwt results ,
further , in order to analyze over frequency domain , magnitude squared coherence measure is calculated between the raw and oa - artifact free eeg data and is plotted for these two combinations in figure 8 .
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\end{document}(f ) is power spectral density of raw eeg signal and \documentclass[12pt]{minimal } 
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it has been observed that all of the swt combinations show efficacy in preserving neuronal signals and do not introduce new artifacts like some of the dwt combinations . 

figure 7.time-frequency comparison plots of various wavelets and thresholds for oa artifact denoising technique using ( a ) swt ( b ) dwt .
figure 8.magnitude squared coherence measure plot for one subject from af3 location after denoising with ( a ) dwt+st+coif3 ( b ) dwt+st+bior4.4 combination .
time - frequency comparison plots of various wavelets and thresholds for oa artifact denoising technique using ( a ) swt ( b ) dwt .
magnitude squared coherence measure plot for one subject from af3 location after denoising with ( a ) dwt+st+coif3 ( b ) dwt+st+bior4.4 combination .
brain monitoring using a few eeg sensors in non - clinical settings has drawn a lot of attention lately .
these real time bci applications like cognitive load assessment , diagnosis of brain disorders , fatigue prediction , and cognitive biometrics , need fast and efficient pre - processing algorithms in order to process and analyze raw brain signals reliably in real time .
this study , therefore , focusses on comparing the effectiveness of commonly used wavelet based techniques for ocular artifact removal in a single channel eeg system . this will allow us to determine the optimal wavelet decomposition technique and corresponding threshold to denoise eeg signal effectively . in this paper ,
data from af3 channel has been presented as a representative of eeg signal contaminated with artifacts to compare several wt based methods .
however , the algorithm is not specific to the channel and is applicable to eeg recordings from any channel . based on cc and mi metrics , dwt with st using haar wavelet
is found to be more effective than dwt+st+coif3 or dwt+st+bior4.4 , but time - frequency analysis shows higher distortion in the processed data using this combination . as these metrics analyze different performances of the algorithm , so none of the combinations was superior based on all metrics .
based on these results , dwt with st using coif3 and bior4.4 wavelet basis functions have performed well for oa removal while preserving neuronal signals in the non - blink regions based on cc , mi , sar , nmse and time - frequency analysis . as fast algorithms are required for real - time systems , a trade - off must be considered for an efficacious method with low distortion . it is known that dwt is a faster technique requires less computational resources than swt for real - time analysis . according to the results presented in this paper , dwt+st+coif3 or dwt+st+bior4.4 could be an optimum choice .
for the applications , where computation time is not critical , swt with haar wavelet can be used with the statistical threshold . instead of applying oa removal algorithm over the whole dataset as outlined in this paper ,
another approach is to identify blink regions and apply oa removal algorithm only to these oa regions to develop a faster oa removal technique , which we have reported previously .
our future research directions include hardware implementation and optimization of efficacious oa removal technique for a single channel eeg system , real - time oa removal , feature extraction , and cognitive load classification to monitor brain engagement in natural environment within a wearable embedded system .
 electroencephalogram ( eeg ) is a technique for recording the asynchronous activation of neuronal firing inside the brain with non - invasive scalp electrodes . 
 artifacts , such as eye blink activities , can corrupt these neuronal signals . 
 while ocular artifact ( oa ) removal is well investigated for multiple channel eeg systems , in alignment with the recent momentum toward minimalistic eeg systems for use in natural environments , we investigate unsupervised and effective removal of oa from single - channel streaming raw eeg data . in this paper , 
 the unsupervised wavelet transform ( wt ) decomposition technique was systematically evaluated for the effectiveness of oa removal for a single - channel eeg system . 
 a set of seven raw eeg data set was analyzed . 
 two commonly used wt methods , discrete wavelet transform ( dwt ) and stationary wavelet transform ( swt ) , were applied . 
 four wt basis functions , namely , haar , coif3 , sym3 , and bior4.4 , were considered for oa removal with universal threshold and statistical threshold ( st ) . to quantify oa removal efficacy from single - channel eeg , 
 five performance metrics were utilized : correlation coefficients , mutual information , signal - to - artifact ratio , normalized mean square error , and time - frequency analysis . 
 the temporal and spectral analysis shows that the optimal combination could be dwt with st with coif3 or bior4.4 to remove oa among 16 combinations . 
 this paper demonstrates that the wt can be an effective tool for unsupervised oa removal from single - channel eeg data for real - time applications . 
Introduction
Wavelet Transform and Performance Metrics
Experimental Method
Results and Analysis
Discussion and Conclusions