research article wavelets and morphological operators based...

Research ArticleWavelets and Morphological Operators Based Classification ofEpilepsy Risk Levels

Harikumar Rajaguru1 and Vijayakumar Thangavel2

1 ECE Bannari Amman Institute of Technology Sathyamangalam 638401 India2 IT Bannari Amman Institute of Technology Sathyamangalam 638401 India

Correspondence should be addressed to VijayakumarThangavel vishal 16278yahoocoin

Received 4 February 2014 Accepted 3 March 2014 Published 7 April 2014

Academic Editor P Karthigaikumar

Copyright copy 2014 H Rajaguru and V Thangavel This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The objective of this paper is to compare the performance of Singular Value Decomposition (SVD) Expectation Maximization(EM) and Modified Expectation Maximization (MEM) as the postclassifiers for classifications of the epilepsy risk levels obtainedfrom extracted features through wavelet transforms and morphological filters from EEG signals The code converter acts as a levelone classifier The seven features such as energy variance positive and negative peaks spike and sharp waves events averageduration and covariance are extracted from EEG signals out of which four parameters like positive and negative peaks spikeand sharp waves events and average duration are extracted using Haar dB2 dB4 and Sym8 wavelet transforms with hard andsoft thresholding methods The above said four features are also extracted through morphological filters The performance of thecode converter and classifiers are compared based on the parameters such as Performance Index (PI) and Quality Value (QV)ThePerformance Index andQuality Value of code converters are at low value of 3326 and 1274 respectivelyThe highest PI of 9803and QV of 2382 are attained at dB2 wavelet with hard thresholding method for SVD classifier All the postclassifiers are settled atPI value of more than 90 at QV of 20

1 Introduction

The Electroencephalogram (EEG) is a measure of cumu-lative firing of neurons in various parts of the brain [1]It contains information regarding changes in the electri-cal potential of the brain obtained from a given set ofrecording electrodes These data include the characteristicwaveforms with accompanying variations in amplitude fre-quency phase and so forth as well as brief occurrenceof electrical patterns such as spindles sharps and spikewaveforms [2] EEG patterns have shown to be modified bya wide range of variables including biochemical metaboliccirculatory hormonal neuroelectric and behavioral factors[3] In the past the encephalographer by visual inspectionwas able to qualitatively distinguish normal EEG activityfrom localized or generalized abnormalities contained withinrelatively long EEG records [4] The most important activitypossibly detected from the EEG is the epilepsy [5] Epilepsy ischaracterized by uncontrolled excessive activity or potential

discharge by either a part or all of the central nervoussystem [5] The different types of epileptic seizures arecharacterized by different EEG waveform patterns [6] Withreal-time monitoring to detect epileptic seizures gainingwidespread recognition the advent of computers has madeit possible to effectively apply a host of methods to quantifythe changes occurring based on the EEG signals [4]The EEGis an important clinical tool for diagnosing monitoring andmanaging neurological disorders related to epilepsy [7] Thisdisorder is characterized by sudden recurrent and transientdisturbances of mental function andor movements of bodythat results in excessive discharge group of brain cells [8]The presence of epileptiform activity in the EEG confirmsthe diagnosis of epilepsy which sometimes is confused withother disorders producing similar seizure-like activity [9]Between seizures the EEG of a patient with epilepsy maybe characterized by occasional epileptic form transients-spikes and sharp waves [10] Seizures are featured by shortepisodic neural synchronous discharges with considerably

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 813197 13 pageshttpdxdoiorg1011552014813197

2 Mathematical Problems in Engineering

enlarged amplitude This uneven synchrony may happenin the brain accordingly that is partial seizures visibleonly in few channels of the EEG signal or generalizedseizures which are seen in every channel of the EEGsignal involving the whole brain [11] Epileptic seizure isan abnormality in EEG gathering and is featured by shortand episodic neuronal synchronous discharges with severelyhigh amplitude This anomalous synchrony may happen inthe brain locally (partial seizures) and is visible only infewer channels of the EEG signal or including the entirebrain that is visible in all the channels of the EEG signal[12]

11 Related Works In the last three decades the analysisand classification of epilepsy from EEG signal have becomea fascinating research A huge volume of research wasperformed which includes spike detection classificationepilepsy seizures ictal and interictal analysis nonlinear andlinear analysis and soft computing methods Gotman [9]discussed the improvement of epileptic seizure detectionand evaluation Pang et al [10] summarized the history andevaluation of various spike detecting algorithms Reference[13] discussed the different neural network as functionapproximation and universal approximation for epilepsydiagnosis Sarang [14] encapsulates the performance of spikedetecting algorithms in terms of sensitivity specificity andaverage detection Reference [14] ordered the performanceof spike detecting algorithms in terms of good detectionratio (GDR) McSharry et al [8] discussed and enumeratedthe nonlinear methods and its relevance to predict epilepsyby considering EEG samples as time series Majumdar [15]reviews various soft computing approaches of EEG signalswhich emphasize more on pattern recognition techniquesReference [15] mainly focuses on dimensionality reductionSNR problems and linear and soft computing techniques forEEG signal processing Kaushik concludes that the neuralnetwork and Bayesian approaches are two popular choiceseven though linear statistical discriminants are easier toimplement A large number of Support Vector Machines(SVM) are also discussed in this paper for their classificationaccuracy Hence the EEG signal occupies a great deal of dataregarding the working of the brain However classificationand estimation of the signals are inadequate As there is noexplicit category suggested by the experts visual examinationof EEG signals in time domain may be deficient Routineclinical diagnosis necessitates the analysis of EEG signals[13] Hence automation and computer methods have beenutilized for this reason Current multicenter clinical analysisindicates confirmation of premonitory symptoms in 62of 500 patients with epilepsy [16] Another interview basedstudy found that 50 of 562 patients felt ldquoaurasrdquo beforeseizures Those clinical data provide a motivation to searchfor premonitoring alterations on EEG recordings from thebrain and to employ a device that can act without humanintervention to forewarn the patient [17] On the other handdespite decades of research existing techniques do not yieldto better performance This paper addresses the applicationand comparison of SVD EM and MEM classifiers towards

optimization of code converter outputs in the classificationof epilepsy risk levels

Webber et al [18] have proposed the three-stage designof an EEG seizure detection system The first stage ofthe seizure detector compresses the raw data stream andtransforms the data into variables which represent the stateof the subjectrsquos EEG These state measures are referred toas context parameters The second stage of the system isa neural network that transforms the state measures intosmaller number of parameters that are intended to representmeasures of recognized phenomena such as small seizurein the EEG [9 10] The third stage consists of a few simplerules that confirm the existence of the phenomena underconsideration Similarly this paper also presents a three-stagedesign for epilepsy risk level classification The first stageextracts the required seven distinct features from raw EEGdata stream of the patient in time domain The next stagetransforms these features into a code word through a codeconverter with seven alphabets which represents the patientrsquosstate in five distinct risk levels for a two-second epoch of EEGsignal per channel The last stage is a SVD EM or MEMwhich optimizes the epilepsy risk level of the patient Theorganization of the paper is as follows Section 1 introducesthe paper and materials and its methods are discussed inSection 2 Section 3 describes the SVD EM and MEM aspostclassifiers for epilepsy risk level classification Resultsare discussed in Section 4 and the paper is concluded inSection 5

2 Materials and Methods

21 Data Acquisition of EEG Signals For the comparativestudy and to analyze the performance of the pre- andpostclassifiers we have obtained the raw EEG data of 20epileptic patients in European Data Format (EDF) whounderwent treatment in the Neurology Department of SriRamakrishna Hospital Coimbatore An issue that has beengiven great attention is the preprocessing stage of the EEGsignals because it is important to use the best techniqueto extract the useful information embedded in the nonsta-tionary biomedical signals The obtained EEG records werecontinuous for about 30 seconds each of them was dividedinto epochs of two-second duration A two-second epochis long enough to detect any significant changes in activityand presence of artifacts and also short enough to avoidany redundancy in the signal [19] For a patient we have 16channels over three epochs Having a frequency of 50Hzeach epoch was sampled at a frequency of 200Hz Eachsample corresponds to the instantaneous amplitude values ofthe signal totaling to 400 values for an epoch Figure 1 showsthe model of flow diagram of epilepsy risk level classificationsystem Four types of artifacts were present in our dataThey included eye blink electromyography (EMG) artifactchewing and motion artifacts [20] Approximately 1 ofthe data was artifacts We did not make any attempt toselect certain number of artifacts and of a specific natureThe objective of including artifacts was to have spikes ver-sus nonspike categories of waveforms The latter could be

Mathematical Problems in Engineering 3

Features

Code converters Code converters

Samples

Wavelet transform Morphological filtering

So threshold

Code converters

SVD EM and MEM used as classifiers

Epilepsy risk levels

(3) sensitivity (4) specificity

PatternsPatterns

Hard threshold

Features

Raw EEG signal

(1) Performance Index (2) Quality Value

Figure 1 Flow diagram for epilepsy risk level classification system

a normal background EEG andor artifacts [21] In orderto train and test the feature extractor and classifiers weneed to select a suitable segment of EEG data In ourexperiment the training and testing were selected througha short sampling window and all EEG signals were visuallyexamined by a qualified EEG technologist A neurologistrsquosdecision regarding EEG features (or normal EEG segment)was used as the gold standardWe choose a sample window of400 points corresponding to 2 seconds of the EEG data Thiswidth can cover almost all types of transient epileptic patternsin the EEG signal even though seizure often lasts longer [22]

In order to classify the risk level of the patients certainparameters were chosen which are detailed below

(1) For every epoch the energy is calculated as [4]

119864 =

119899

sum

119894=1

1199092

119894 (1)

where 119909119894mdashsample value of signal and 119899mdashnumber of

such samples(2) One of the simplest linear statistics that may be used

for investigating the dynamics underlying the EEG isthe variance of the signal calculated in consecutivenonoverlapping windows The variance (1205902) is givenby

1205902

=sum119899

119894=1(119909119894minus 120583)2

119899 (2)

where 120583mdashaverage amplitude of the epoch

(3) For the average variance the covariance of duration isdetermined by using the equation below

CD =sum119901

119894=1(119863 minus 119905

119894)2

1199011198632 (3)

The following are the four parameters which are extractedusing morphological filters and wavelet transforms

(1) The total number of positive and negative peaks isfound above the threshold

(2) For a zero crossing function if it lies between 20milliseconds and 70 milliseconds then the spikes canbe detected If the zero crossing function lies between70 milliseconds and 200 milliseconds then the sharpwaves are detected when the zero crossing functionlies between 70 milliseconds and 200 milliseconds

(3) The total number of spikes and sharp waves aredetermined as the events

(4) The duration for these waves is determined by therelation

119863 =sum119901

119894=1119905119894

119901 (4)

where 119905119894mdashpeak to peak duration and 119901mdashnumber of

such durations

22 Wavelet Transforms for Feature Extraction The brainsignals are nonstationary in nature In order to capture thetransients and events of the waveforms we are in dire stateto visualize the time and frequency simultaneously Hencethe wavelet transforms are the better choice to extract thetransient features and events from the EEG signals Thewavelet transform based feature extraction is discussed asfollows

Let us consider a function 119891(119905) The wavelet transform ofthis function is defined as [23]

119908119891 (119886 119887) = int

infin

minusinfin

119891 (119905) 120595lowast

119886119887(119905) 119889119905 (5)

where 120595lowast

(119905)mdashcomplex conjugate of the wavelet function120595(119905)

With the set of the analyzing function the wavelet familyis deduced from the mother wavelet 120595(119905) by [24]

120595lowast

119886119887(119905) =

1

radic2

120595(119905 minus 119887

119886) (6)

where 119886mdashdilation parameter and 119887mdashtranslation parameterThe feature extraction process is initialized by studying

the effect of simpleHaar thresholdTheHaarwavelet functioncan be represented as [25]

120595 (119905) =

1 0 le 119905 lt1

2

minus11

2le 119905 lt 1

0 otherwise

(7)


Table 1 Parameter ranges for various risk levels

Normalized parameters Risk levelsNormal Low Medium High Very high

Energy 0-1 07ndash36 29ndash82 76ndash11 92ndash30Variance 0ndash03 015ndash045 04ndash22 16ndash43 38ndash10Peaks 0ndash2 1ndash4 3ndash8 6ndash16 12ndash20Events 0ndash2 1ndash5 4ndash10 7ndash16 15ndash28Sharp waves 0ndash2 1ndash5 4ndash8 7ndash11 10ndash12Average duration 0ndash03 015ndash045 04ndash24 18ndash46 36ndash10Covariance 0ndash005 0025ndash01 009ndash04 028ndash064 054ndash1

Wavelet thresholding is a signal estimation techniquethat exploits the capabilities of wavelet transform for signaldenoising or smoothing It depends on the choice of athreshold parameter which determines to great extent theefficacy of denoising

120588119879(119909) =

119909 if |119909| gt 1198790 if |119909| le 119879

(8)

where 119879 is the threshold levelTypical threshold operators for denoising include hard

threshold soft threshold and affine (firm) threshold Hardthreshold is defined as [24] Soft thresholding (waveletshrinkage) is given by

120588119879(119909) =

119909 minus 119879 if (119909 ge 119879) 119909 + 119879 if (119909 le 119879) 0 if |119909| lt minus119879

(9)

Haar Db2 Db4 and Sym8 wavelets with hard thresh-olding and four types of soft thresholding methods suchas Heursure Minimaxi Rigrsure and Sqtwolog are used toextract the parameters from EEG signals With the help ofexpertrsquos knowledge and our experiences with [5 20 26]we have identified the following parametric ranges for fivelinguistic risk levels (very low low medium high and veryhigh) in the clinical description for the patients which isshown in Table 1

The output of code converter is encoded into the stringsof seven codes corresponding to each EEG signal parameterbased on the epilepsy risk levels threshold values as set inTable 1 The expert defined threshold values as containingnoise in the form of overlapping ranges Therefore we haveencoded the patient risk level into the next level of risk insteadof a lower level Likewise if the input energy is at 34 then thecode converter output will be at medium risk level instead oflow level [26]

23 Code Converter as a Preclassifier The encoding methodprocesses the sampled output values as individual codeSince working on definite alphabets is easier than processingnumbers with large decimal accuracy we encode the outputsas a string of alphabetsThe alphabetical representation of thefive classifications of the outputs is shown in Table 2

The ease of operation in using characteristic representa-tion is obviously evident than in performing Cumber some

Table 2 Representation of risk level classifications

Risk level Coded RepresentationNormal ULow WMedium XHigh YVery high Z

Table 3 Output of code converter for patient 2

Epoch 1 Epoch 2 Epoch 3YYYYXXX ZYYWYYY YYYXYZZYYYXYYY ZZYZZZZ YYYXYZZYYYYYYY ZZYZZZZ ZYYYZZZZYYYZZZ ZZYZYYY YYYXXZZYYYYYYY YYYXYYY YYYYYZZYYYYYYY YYYXYYY YYYXYYXYYYYYYY YYYYYYY YYYYYYZZZYZYZZ ZZYZZZZ ZZYZZZZ

operations of numbers By encoding each risk level from oneof the five states a string of seven characters is obtained foreach of the sixteen channels of each epoch A sample outputwith actual patient readings is shown in Table 3 for eightchannels over three epochs

It can be seen that channel 1 shows low risk levels whilechannel 7 shows high risk levels Also the risk level clas-sification varies between adjacent epochs There are sixteendifferent channels for input to the system at three epochsThisgives a total of forty-eight input and output pairs Since wedeal with known cases of epileptic patients it is necessaryto find the exact level of epilepsy risk in the patient Thiswill also aid towards the development of automated systemsthat can precisely classify the risk level of the epileptic patientunder observation Hence an optimization is necessary Thiswill improve the classification of the patient and can providethe EEG with a clear picture [20] The outputs from eachepoch are not identical and are varying in condition such as[YYZXXXX] to [WYZYYYY] to [YYZZYYY] In this caseenergy factor is predominant and thus results in the highrisk level for two epochs and low risk level for middle epochChannels five and six settle at high risk level Due to this type


of mixed state output we cannot come to proper conclusiontherefore we group four adjacent channels and optimize therisk level The frequently repeated patterns show the averagerisk level of the group channels Same individual patternsdepict the constant risk level associated in a particular epochWhether a group of channel is at the high risk level or notis identified by the occurrences of at least one Z pattern inan epoch It is also true that the variation of the risk levelis abrupt across epochs and eventually in channels Hencewe are in a dilemma and cannot come up with the finalverdict The five risk levels are encoded as Z gt Y gt X gt

W gt U in binary strings of length five bits using weightedpositional representation as shown in Table 4 Encoding eachoutput risk level gives us a string of seven alphabets thefitness of which is calculated as the sum of probabilities of theindividual alphabets For example if the output of an epochis encoded as ZZYXWZZ its fitness would be 0419352

The Sensitivity Se and Specificity Sp are represented as[19]

PI = PC minusMC minus FAPC

times 100 (10)

where PImdashPerformance Index PCmdashperfect classificationMCmdashmissed classification and FAmdashfalse alarm

The performance of code converter is 4481The perfectclassification represents when both the physician and codeconverter agrees with the same epilepsy risk level Missedclassification represents a high level as low level False alarmrepresents a low level as high level with respect to physicianrsquosdiagnosis The other performance measures are also definedas below

The sensitivity Se and specificity Sp are represented as [19]

Se = [ PC(PC + FA)

] lowast 100

(05

06) lowast 100 = 8333

Sp = [ PC(PC +MC)

] lowast 100

(05

07) lowast 100 = 7142

AverageDetection (AD) =(Sensitivity + Specificity)

2

AD = 78875

Relative Risk =SensitivitySpecificity

Relative Risk = 1166(11)

The relative risk factor indicates the stability and sen-sitivity of the classifier For an ideal classifier the relativerisk will be unity More sensitive classifier will have thisfactor slightly above unity whereas slow response classifier

Table 4 Binary representation of risk levels

Risk level Code Binary string Weight ProbabilityVery high Z 10000 1631 = 051612 0086021High Y 01000 831 = 025806 0043011Medium X 00100 431 = 012903 0021505Low W 00010 231 = 006451 0010752Normal U 00001 131 = 003225 0005376

11111 = 31 sum = 1

Table 5 Performance of code converter output based on wavelettransform along hard thresholding

Wavelets Perfectclassification

Missedclassification

Falsealarm

PerformanceIndex

Haar 6145 15625 2291 3758Db2 6118 1614 2265 3644Db4 6457 1249 2291 4472Sym8 6352 1144 2395 4481

makes this factor lower than unity We have obtained alow value of just 40 for Performance Index and 83337142 7887 and 1166 for sensitivity specificity averagedetection and relative risk for the code converter Due tothe low performance measures it is essential to optimize theoutput of the code converter Performance Index of codeconverters output using different wavelet transforms for hardthresholding methods is tabulated in Table 5

24 Rhythmicity of Code Converter Now we are about toidentify the rhythmicity of code converter techniques whichis associated with nonlinearities of the epilepsy risk levels Letthe rhythmicity be defined as [10]

119877 =119862

119863 (12)

where 119862 = number of categories of patterns and 119863 =

total number of patterns which is 960 in our case For anideal classifier 119862 is to be one and 119877 = 0001042 Table 6shows the rhythmicity of the code converter classifier forhard thresholding of each wavelet Table 6 shows that thevalue of 119877 is highly deviated from its ideal value Hence it isnecessary to optimize the code converters outputs to endurea singleton risk level In the following section we discuss themorphological filtering of EEG signals

25 Morphological Filtering for Feature Extraction of EEG Sig-nals Morphological filtering was chosen over othermethodssuch as the temporal approach of the EEG signal and waveletbased approach due to the fact that morphological filteringcan precisely determine the spikes with a very high accuracyrate [14] Let us call it as a function 119891(119905) Let us also take intoaccount a structuring element 119892(119905) which together with 119891(119905)is the subsets of Euclidean space 119864


Table 6 Rhythmicity of code converter for wavelets with hardthresholding

Wavelets Number of categoriesof patterns

Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875

Accordingly theMinkowski addition and subtraction [6]for the function 119891(119905) is given by the relation

Addition (119891 oplus 119892) (119905) = max⏟⏟⏟⏟⏟⏟⏟

119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)

Subtraction (119891Θ119892) (119905) = min⏟⏟⏟⏟⏟⏟⏟

119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)

The opening and closing functions of the morphologicalfilter are given as

Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)

Closing (119891 sdot 119892) (119905) = [(119891 oplus 119892119878

)Θ119892] (119905)

(14)

The abovementioned equations help us in determiningthe peaks and valleys in the original recording [7] Theopening function (erosion-dilation) is used in smoothing ofthe convex peak of the original signal and the closing function(dilation-erosion) is used in smoothing the concave peak ofthe signal Combinations of opening and closing functionlead to the formation of a new filter which when fed withthe original signal can divide it into two the first signal beingdefined by a structuring element and the second signal beingthe residue of 119891(119905) This type of filtering is done in orderto detect the spikes with high accuracy For two structuringelements say 119892

1(119905) and 119892

2(119905) the open-close (OC) and close-

open (CO) functions are defined as

OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)

When considered separately the OC and CO functionsresult in a variation in amplitude that is while OC resultsin lower amplitude the CO function yields higher amplitudeFor easier interpretation and calculation we go for the aver-age of the two defined as opening-closing-closing-opening(OCCO) function The same is depicted below as

OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)

where 119891(119905) is the original signal represented as

119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)

where 119909(119905)mdashspiky part of the signal

Performance Index sensitivity and specificity of codeconverter outputs through morphological filter based featureextraction arrived at the low value of 3346 7623 and7742 respectively This scenario impacts the optimizationof code converter outputs using postclassifier to accomplish asingleton resultThe following section describes the outcomeof SVD EM and MEM techniques as postclassifier

3 Singular Value DecompositionExpectation Maximization and ModifiedEM as Postclassifier for Classification ofEpilepsy Risk Levels

In this section we discuss the possible usage of SVD EMand MEM as a postclassifier for classification of epilepsyrisk levels The Singular Value Decomposition (SVD) wasestablished in the 1870s by Beltrami and Jordan for real squarematrices [27] It is used mainly for dimensionality reductionand determining the modes of a complex linear dynamicalsystem [27] Since then SVD is regarded as one of the mostimportant tools of modern numerical analysis and numericallinear algebra

31 SVD Theorem Let us have an 119898 times 119899 matrix 119860 =

[1198861 1198862 1198863 119886

119899] The SVD theorem states that [28]

119860 = USV119879 (18)

where 119860 isin 119877119898 times 119899 (with 119898 ge 119899) 119880 isin 119877119898 times 119899 119881 isin 119877119899 times 119899and 119878 is a diagonal matrix of size 119877119899 times 119899

Equation (18) can be further realized as

119860 = sum119875120590119896119906119896V119879119896 (19)

The columns of 119880 are called the left singular vectors ofmatrix 119860 and the columns of 119881 are called the right singularvectors of 119860 119875 = min(119898 119899) sum is called the singular valuematrix along with the diagonal

We have taken the EEG records of twenty patients for ourstudy Each patientrsquos sample is composed of a 16 times 3 matrixas code converter outputs depicted in Table 3 Consideringthis to be as matrix 119860 SVD is computed The so-obtainedEigen value is eventually regarded as the patientrsquos epilepsy risklevel The similar procedure is carried out in finding out theremaining Eigen values of other patients as well

32 Expectation Maximization as a Postclassifier The Expec-tation Maximization (EM) is often defined as a statisticaltechnique for maximizing complex likelihoods and handlingincomplete data problem EMalgorithm consists of two stepsnamely the following

Expectation Step (119864 Step) say for data 119909 having anestimate of the parameter and the observed data the expectedvalue is initially computed [29] For a given measurement1199101and based on the current estimate of the parameter the

expected value of 1199091is computed as given below

119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)

Maximization Step (119872 Step) from the Expectation Stepwe use the data which was actually measured to determinethe ML estimate of the parameter

Considering the code converter outputs let us take a set ofunit vectors to be as119883Wewill have to findout the parameters120583 and 120581 of the distribution Md(120583 119896) Accordingly we canform the equation as [30]

119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)

Considering 119909119894isin 119883 the likelihood of119883 is

119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)

The log likelihood of (19) can be written as

119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)

In order to obtain the likelihood parameters 120583 and 120581 wewill have tomaximize (22) with the help of Lagrange operator120582 The equation can be written as

119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)

Derivating (23) with respect to 120583 120582 and 120581 and equatingthese to zero will yield the parameter constraints as

120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)

In the Expectation Step the threshold data are estimatedgiven the observed data and current estimate of the modelparameters [31]This is achieved using the conditional expec-tation explaining the choice of terminology In the 119872-Stepthe likelihood function is maximized under the assumptionthat the threshold data are knownThe estimate of themissingdata from the 119864-Step is used in lieu of the actual thresholddata

33 Modified Expectation Maximization Algorithm AModi-fied Expectation Maximization (EM) algorithm which usesmaximum likelihood (ML) approach is discussed in thispaper for pattern optimization Similar to the conventional

EM algorithm this algorithm alternated between the estima-tion of the complete log-likelihood function (119864-Step) and themaximization of this estimate over values of the unknownparameters (119872-Step) [32] Because of the difficulties in theevaluation of the ML function [33] modifications are madeto the EM algorithm as follows

The method of maximum likelihood corresponds tomanywell-known estimationmethods in statistics For exam-ple one may be interested in the heights of adult femalegiraffes but be unable due to cost or time constraints tomeasure the height of every single giraffe in a populationAssuming that the heights are normally (Gaussian) dis-tributed with some unknown mean and variance the meanand variance can be estimated withMLE while only knowingthe heights of some samples of the overall population

Given a set of samples 119883 = 1199091 1199092 119909

119899 the complete

data set 119878 = (119883 119884) consists of the sample set 119883 and aset 119884 of variable indicating from which component of themixtures the samples came The description is given belowof how to estimate the parameters of the Gaussian mixtureswith the maximization algorithm After optimization ofthe patterns maximum likelihood is adopted to redesignthe intracranial area into two clusters Basically maximumlikelihood algorithm is a statistical estimation algorithmused for finding log-likelihood estimates of parameters inprobabilistic models [30]

(1) Find the initial values of the maximum likelihoodparameters which are means covariance and mixingweights

(2) Assign each 119909119894to its nearest cluster centre 119888

119896by

Euclidean Distance (119889)(3) Inmaximization step useMaximization119876(120579 1205791015840)The

likelihood function is written as

119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)

(4) Repeat iterations until 120579119894+1 minus 120579119894

becomes smallenough

The algorithm terminates when the difference betweenthe log likelihood for the previous iteration and currentiteration fulfills the tolerance For 120583 = 0 and 120590 = 1 thelikelihood function was applied to the 16 times 3 matrix of thecode converter output by having truncated to the knownendpoints

4 Results and Discussion

To study the relative performance of these code convertersand SVD EM and MEM we measure two parameters thePerformance Index and the Quality Value These parametersare calculated for each set of twenty patients and are com-pared


Table 7 Performance Index for morphological based feature extraction

Classifiers Morphological operators based feature extractionPerfect classification Missed classification False alarm Performance Index

Code converter 626 1825 1913 3326With SVD optimization 9122 731 142 8948With EM optimization 8268 1293 438 801With MEM optimization 8532 1095 372 8335

Table 8 Performance analysis of wavelet transforms with hard thresholding

Classifiers Perfect classification Missed classification False alarm Performance IndexHaar wavelet


DB2 waveletCode converter 6118 1614 2265 3644With SVD optimization 9813 09465 0946 9803With EM optimization 8729 81 459 8542With MEM optimization 8958 581 461 8781


Sym8 waveletCode converter 6352 1144 2395 4481With SVD optimization 9735 1512 1135 9723With EM optimization 8727 778 548 8503With MEM optimization 8871 69 567 8695

41 Performance Index A sample of Performance Indexof morphological filter based feature extraction with codeconverters Singular Value Decomposition EM and MEMfor an average of twenty known epilepsy data sets is shown inTable 7 As shown in Table 7 the morphological filter basedfeature extraction along with SVD optimization is ranked atfirst with high PI of 8948 against the 801 and 8335 ofEM andMEMmethods But themorphological filter pluggedinto more missed classification rather than less false alarmwhich is a dangerous trend Therefore this method will beconsidered as a lazy and high threshold classifier

Table 8 depicts the performance analysis of wavelet trans-form with hard thresholding method In case of hard thresh-olding while code converter has got an average classificationrate and false alarm of 6268 and 18105 EM optimizerhas 8739 of perfect classification with a false alarm rate of443Notmuch of deviationsMEMhave 8936 and 446of average perfect classification and false alarm respectivelySVD optimization has got the highest value of perfectclassification rate of 9658 with zero false alarms HenceSVD optimizer can be regarded as the best postclassifier Inall the four wavelet transforms SVD postclassifier is the bestsuited one to achieve the high classification rate EM and

MEMtechniques failmiserably to achieve better classificationaccuracy when compared with SVD classifier

Table 9 represents the performance analysis of wavelettransforms with soft thresholding with code converter SVDEM and MEM respectively It can be found that in softthresholding the code converter has got an average perfectclassification of 656 and false alarm of 1194 SVD has gota classification rate of over 85 with comparatively highervalues of false alarms MEM optimizer claims to be the bestoptimizer as it has a classification rate of 9397 with a falsealarm rate of 35 only This is obtained when Haar wavelet isused with mini max soft thresholding

42 Quality Value This parameter determines the overallquality of the classifiers used The relation for Quality Valueis given by [19]

QV =119862

(119877fa + 02) lowast (119879dly lowast 119875dct + 6 lowast 119875msd) (29)

where 119862mdashscaling constant 119877famdashfalse alarm per set 119879dlymdashaverage delay of on-set classification 119875dctmdashpercentage of


Table 9 Performance analysis of Haar wavelet transforms with soft thresholding

Classifiers Perfect classification Missed classification False alarm Performance IndexHeursure soft thresholding


Mini max soft thresholdingCode converter 6463 2059 151 4452With SVD optimization 8522 0 1477 7943With EM optimization 8948 592 46 8815With MEM optimization 9397 274 35 934

Rig sure soft thresholdingCode converter 6634 1988 1378 4911With SVD optimization 8849 0 115 8418With EM optimization 909 384 548 8993With MEM optimization 9222 362 416 9125

Sqtwolog soft thresholdingCode converter 6534 2769 696 4689With SVD optimization 7769 2088 142 6658With EM optimization 8465 1085 449 8212With MEM optimization 8838 888 274 8687

Table 10 Quality Value of wavelet transforms with hard thresholding

Wavelets Quality ValueWithout optimization With SVD optimization With EM optimization With MEM optimization

Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203

Table 11 Performance analysis of twenty patients using dB2 wavelet hard thresholding with SVD EM and MEM postclassifiers

ParametersCode convertermethod beforeoptimization

SVD optimization With EMoptimization

With MEMoptimization

Risk level classification rate () 6145 9813 8729 8958Weighted delay (s) 2189 2017 2233 214False alarm rateset 2265 09463 459 46Performance Index 3645 9803 8542 8781Sensitivity 7543 9905 954 954Specificity 8194 991 9189 9419Average detection 78875 99075 93645 94795Relative risk 1166 09999 1038 10128Quality Value 1257 2382 1972 2049

Table 12 Quality Value of Haar wavelet transforms with soft thresholding

Haar wavelet with soft thresholding Quality ValueCode converter With SVD optimization With EM optimization With MEM optimization

Heursure 1354 2016 2012 2085Mini max 1211 1938 2009 2254Rig sure 1291 2044 2032 2142Sqtwolog 1322 1782 1877 2022


Table 13 Performance analysis of twenty patients using Haar wavelet in soft thresholding with SVD EM and MEM postclassifiers




Heursure soft thresholdingRisk level classification rate () 661 8721 8903 9046Weighted delay (s) 247 191 219 208False alarm rateset 1193 994 416 493Performance Index 5282 8264 8788 8982Sensitivity 8539 9005 9627 9507Specificity 7828 9716 9276 954Average detection 78875 9361 9451 9523Relative risk 1166 0926 1037 0996Quality Value 1354 2016 2012 2085

Mini max soft thresholdingRisk level classification rate () 6463 8522 8948 9397Weighted delay (s) 253 17 215 206False alarm rateset 151 1477 46 35Performance Index 4452 7943 8815 934Sensitivity 8231 8525 954 9671Specificity 7681 100 9408 9726Average detection 78875 92625 9474 9698Relative risk 1166 085 1014 0994Quality Value 1211 1936 2009 2254

Rig sure soft thresholdingRisk level classification rate () 6634 8849 909 9222Weighted delay (s) 252 177 201 21False alarm rateset 1378 115 548 416Performance Index 4911 8418 8993 9125Sensitivity 8435 8849 9474 9583Specificity 7823 100 9616 9683Average detection 78875 9445 9545 9633Relative risk 1166 088 0992 0989Quality Value 1291 2044 2032 2142

Sqtwolog soft thresholdingRisk level classification rate () 6534 7769 8465 8838Weighted delay (s) 296 2806 234 23False alarm rateset 696 142 449 274Performance Index 4689 6658 8212 8687Sensitivity 9134 9857 955 9726Specificity 7082 7911 8915 9112Average detection 78875 8884 92325 9419Relative risk 1166 1245 107 106Quality Value 1322 1782 1877 2022

perfect classification and 119875msdmdashpercentage of perfect risklevel missed

By setting the value of ldquoCrdquo to a constant value consideras 10 The classifier with the highest Quality Value is thebetter one Table 10 depicts the Quality Value of wavelettransforms with hard thresholding and SVD EM and MEM

optimization methods It was observed that SVD with dB2wavelet in hard thresholding attained the maximum value ofQV at 2382 and EM with Haar wavelet has the low value ofQV at 1832

Table 11 shows the performance analysis of twentypatients using dB2 wavelet hard thresholding with SVD


Table 14 Performance analysis of twenty patients using morphological filters with SVD EM and MEM postclassifiers

Parameters Code converter method SVD optimization With EM optimization With MEM optimizationRisk level classification rate () 626 9122 8727 8871Weighted delay (s) 234 226 22 218False alarm rateset 1913 142 547 567Performance Index 3326 8948 8503 8695Sensitivity 7784 9857 9559 9897Specificity 7891 9265 9811 9767Average detection 78875 9561 9685 9832Relative risk 1166 1063 0974 1013Quality Value 1274 2062 1952 203

Table 15 Performance analysis of postclassifiers in terms of weighted delay and quality value

MethodswaveletsSVD optimization EM optimization MEM optimization

Weighteddelay (sec) Quality Value Weighted

delay (sec) Quality Value Weighteddelay (sec) Quality Value

Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203

Soft thresholdheursure

Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22

Soft thresholdmini max

Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073

Soft thresholdrig sure

Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106

Soft thresholdsqtwolog

Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997

Morphological filters 226 2062 22 1952 218 203

EM and MEM as postclassifiers The evaluation parametersachieved an appreciable value in the case of SVDpostclassifierwhen compared to the other two classifiers Hence we canchoose SVD as a good postclassifier for epilepsy risk levelclassification All the three postclassifiers are bestowed withthe best sensitivity and specificity measures EM and MEMclassifiers are plugged into the higher false alarm rate and thisleads to the lower QV and PI for the system

Since the Haar wavelet is a predominant wavelet we hadchosen this wavelet for the four types of soft thresholdingmethods and the same is depicted in Table 12 As seen inTable 12 the highest QV of 2254 is attained in the mini maxsoft thresholding with MEM as a postclassifier

Table 13 exhibits the performance analysis of twentypatients using Haar wavelet in soft thresholding with SVDEM and MEM postclassifiers MEM postclassifier with minimax soft thresholding reached the better QV and PI whencompared to SVD and EM classifiers A slight incrementaltradeoff in the weighted delay forMEM is responsible for thisperformance when compared with SVD and EM classifiersSVD fails to achieve a good performance in thismethodologydue to more false alarm rate EM is struck in the middle pathas far as Performance Index is concerned

Table 14 shows the performance analysis of twentypatients using morphological filters with SVD EM andMEM postclassifiers In this method SVD outperforms other


classifiers in terms of QV and PIThismorphological filteringis inherited with slow response and is considered to be ahigh threshold classifier SVD classifier is summed with lowfalse alarm andweighted delays All thesemethods in averagepositioned at more than 90 of Performance Index andaround Quality Value of 18 Since for all these classifiers thefact that the obtained weighted delay is more than 2 secondsleads to larger threshold and slow response system

We wish to analyse the time complexity of the postclas-sifiers in terms of weighted delay and quality value Table 15shows the performance analysis of postclassifiers in termsof weighted delay and Quality Value It is observed that thefour types of wavelet transforms in hard thresholdingmethodalong with SVD postclassifier attained low weighted delayand high value of QV

In the case of Table 15 the EM and MEM classifiers areeither plugged into more missed classification or false alarmsand subsequently lead to lower value of QV less than 20 inmost of the wavelet transforms In case of soft thresholdingdB2 wavelet in rig sure thresholding for MEM postclassifieroutperforms other fifteen methods Morphological filters arestacked at higher delay with QV set at near 20

5 Conclusion

The objective of this paper is to classify the risk level ofthe epileptic patients from the EEG signals The aim is toobtain high classification rate Performance Index QualityValue with low false alarm and missed classification Dueto the nonlinearity obtained and also a poor performancefound in the code converters an optimization was vital forthe effective classification of the signals We went for SVDEM and MEM as postclassifiers Morphological filters werealso used for the feature extraction of the EEG signals Afterhaving computed the values of PI and QV discussed underthe results column we found that SVDwas working perfectlywith a high classification rate of 9122 and a false alarmas low as 142 Therefore SVD was chosen to be the bestpostclassifier The accuracy of the results obtained can bemade even better by using extreme learning machine asa postclassifier and further research will be in this direc-tion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors express their sincere thanks to the Managementand the Principal of Bannari Amman Institute of TechnologySathyamangalam for providing the necessary facilities forthe completion of this paper This research is also fundedby AICTE RPS F no 8023BORRIDRPS-412009-10 datedDecember 10 2010

References

[1] Y Yuan ldquoDetection of epileptic seizure based on EEG signalsrdquoin Proceedings of the 3rd International Congress on Image andSignal Processing (CISP rsquo10) pp 4209ndash4211 Yantai ChinaOctober 2010

[2] S Herculano-Houzel ldquoThe human brain in numbers a linerarlyscaledmdashup primate brainrdquo Frontiers in Human Neurosciencevol 3 pp 1ndash11 2009

[3] Z Zainuddin L Kee Huong and O Pauline ldquoOn the use ofwavelet neural networks in the task of epileptic seizure detectionfrom electroencephalography signalsrdquo in Proceedings of the 3rdInternational Conference on Computational SystemsmdashBiologyand Bioinformatics vol 11 pp 149ndash159 2012

[4] A A Dingle R D Jones G J Carroll and W R Fright ldquoAmultistage system to detect epileptiform activity in the EEGrdquoIEEE Transactions on Biomedical Engineering vol 40 no 12 pp1260ndash1268 1993

[5] R Harikumar R Sukanesh and P A Bharathi ldquoGenetic algo-rithmoptimization of fuzzy outputs for classification of epilepsyrisk levels from EEG signalsrdquo Journal of Interdisciplinary PanelsIE vol 86 no 1 pp 1ndash10 2005

[6] P Xanthopoulos S Rebennack C-C Liu et al ldquoA novelwavelet based algorithm for spike and wave detection inabsence epilepsyrdquo in Proceedings of the 10th IEEE InternationalConference on Bioinformatics and Bioengineering (BIBE rsquo10) pp14ndash19 Philadelphia Pa USA June 2010

[7] A Mirzaei A Ayatollahi P Gifani and L Salehi ldquoEEGanalysis based on wavelet-spectral entropy for epileptic seizuresdetectionrdquo in Proceedings of the 3rd International Conference onBioMedical Engineering and Informatics (BMEI rsquo10) pp 878ndash882 Yantai China October 2010

[8] P E McSharry L A Smith L Tarassenko et al ldquoPredictionof epileptic seizures are nonlinear methods relevantrdquo NatureMedicine vol 9 no 3 pp 241ndash242 2003

[9] J Gotman ldquoAutomatic seizure detection improvements andevaluationrdquo Electroencephalography and Clinical Neurophysiol-ogy vol 76 no 4 pp 317ndash324 1990

[10] C C C Pang A R M Upton G Shine and M V KamathldquoA comparison of algorithms for detection of spikes in theelectroencephalogramrdquo IEEE Transactions on Biomedical Engi-neering vol 50 no 4 pp 521ndash526 2003

[11] L Tarassenko Y U Khan and M R G Holt ldquoIdentificationof inter-ictal spikes in the EEG using neural network analysisrdquoin Proceedings of the IEE Proceedings Science MeasurementTechnology vol 145 pp 270ndash278 November 1998

[12] M van Gils ldquoSignal processing in prolonged EEG recordingsduring intensive carerdquo IEEE EMB Magazine vol 16 no 6 pp56ndash63 1997

[13] E Sezer H Isik and E Saracoglu ldquoEmployment and compari-son of different artificial neural networks for epilepsy diagnosisfrom EEG signalsrdquo Journal of Medical Systems vol 36 no 1 pp347ndash362 2012

[14] R Sarang ldquoA strong adaptive and comprehensive evaluationof wavelet based epileptic EEG spike detection methodsrdquo inProceedings of the International Conference on Biomedical andPharmaceutical Engineering (ICBPE rsquo06) pp 432ndash437 Singa-pore December 2006

[15] K Majumdar ldquoHuman scalp EEG processing various softcomputing approachesrdquoApplied Soft Computing Journal vol 11no 8 pp 4433ndash4447 2011


[16] R P Costa P Oliveria G Rodrigues B Leitao and D AntonioldquoEpileptic seizure classfication using neural networks with 14featuresrdquo in Proceedings of the 12th International Conferenceon Knowledge Based Intelligent Information and EngineeringSystem Part II Lecture Notes of Computer Science Series pp281ndash288 Springer Zageed Croatia September 2008

[17] H Adeli ldquoChaos-wavelet- neural network models for auto-mated EEG based diagnosis of the neurological disordersrdquo inProceedings of the 17th International Conference on SystemsSignals and Image Processing (IWSSIP rsquo10) pp 45ndash48 RiodeJanerio Brazil June 2010

[18] W R S Webber R P Lesser R T Richardson and KWilson ldquoAn approach to seizure detection using an artificialneural network (ANN)rdquo Electroencephalography and ClinicalNeurophysiology vol 98 no 4 pp 250ndash272 1996

[19] H Qu and J Gotham ldquoA patient-specific algorithm for thedetection of seizure onset in long- term EEG monitoring pos-sible use as a warning devicerdquo IEEE Transactions on BiomedicalEngineering vol 44 no 2 pp 115ndash122 1997

[20] R Harikumar T Vijayakumar and M G Sreejith ldquoPerfor-mance analysis of SVD and support vector machines foroptimization of fuzzy outputs in classification of epilepsy risklevel from EEG signalsrdquo in Proceedings of the IEEE RecentAdvances in Intelligent Computational Systems (RAICS rsquo11) pp718ndash723 Thiruvananthapuram India September 2011

[21] R M Rangayyan Bio-Medical Signal Analysis a Case StudyApproach IEEE Press-JohnWiley amp Sons New York NY USA2002

[22] L M Patnaik and O K Manyam ldquoEpileptic EEG detectionusing neural networks and post-classificationrdquoComputerMeth-ods and Programs in Biomedicine vol 91 no 2 pp 100ndash1092008

[23] A T Tzallas M G Tsipouras and D I Fotiadis ldquoA time-frequency based method for the detection of epileptic seizuresin EEG recordingsrdquo in Proceedings of the 20th IEEE Interna-tional Symposium on Computer-Based Medical Systems (CBMSrsquo07) pp 135ndash140 Maribor Slovenia June 2007

[24] V V K DV Prasad P Siddiah and B Prabhakara Rao ldquoAnew wavelet based method for denoising biological signalsrdquoInternational Journal of Computer Science andNetwork Securityvol 8 no 1 pp 238ndash244 2008

[25] P Kumar and D Agnihotri ldquoBiosignal Denoising via waveletthresholdsrdquo IETE Journal of Research vol 56 no 3 pp 132ndash1382010

[26] RHarikumar andB SNarayanan ldquoFuzzy techniques for classi-fication of epilepsy risk level from EEG signalsrdquo in Proceedingsof the IEEE Confernce on Covergent Technologies for the Asia-Pacific Region (TENCON rsquo03) pp 209ndash213 Bangalore IndiaOctober 2003

[27] V C Klema and A J Laub ldquoSingular value decompositionits computation and some applicationsrdquo IEEE Transactions onAutomatic Control vol AC-25 no 2 pp 164ndash176 1980

[28] P K Sadasivan and D Narayana Dutt ldquoSVD based techniquefor noise reduction in electroencephalographic signalsrdquo SignalProcessing vol 55 no 2 pp 179ndash189 1996

[29] T K Moon ldquoThe expectation-maximization algorithmrdquo IEEESignal Processing Magazine vol 13 no 6 pp 47ndash60 1996

[30] C E McCulloch ldquoMaximum likelihood algorithms for gener-alized linear mixed modelsrdquo Journal of the American StatisticalAssociation vol 92 no 437 pp 162ndash170 1997

[31] G Tian Y Xia Y Zhang and D Feng ldquoHybrid genetic andvariational expectation-maximization algorithm for Gaussian-mixture-model-based brain MR image segmentationrdquo IEEETransactions on Information Technology in Biomedicine vol 15no 3 pp 373ndash380 2011

[32] Y Zhou and J P Y Lee ldquoA modified expectation maximizationalgorithm for maximum likelihood direction-of-arrival estima-tionrdquo in Proceedings of the 34th Asilomar Conference pp 613ndash617 November 2000

[33] J Xian J Li and Y Yang ldquoA new EM acceleration algorithmfor multi-user detectionrdquo in Proceedings of the 3rd InternationalConference onMeasuring Technology andMechatronics Automa-tion (ICMTMA rsquo11) pp 150ndash153 Shanghai China January 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


enlarged amplitude This uneven synchrony may happenin the brain accordingly that is partial seizures visibleonly in few channels of the EEG signal or generalizedseizures which are seen in every channel of the EEGsignal involving the whole brain [11] Epileptic seizure isan abnormality in EEG gathering and is featured by shortand episodic neuronal synchronous discharges with severelyhigh amplitude This anomalous synchrony may happen inthe brain locally (partial seizures) and is visible only infewer channels of the EEG signal or including the entirebrain that is visible in all the channels of the EEG signal[12]

11 Related Works In the last three decades the analysisand classification of epilepsy from EEG signal have becomea fascinating research A huge volume of research wasperformed which includes spike detection classificationepilepsy seizures ictal and interictal analysis nonlinear andlinear analysis and soft computing methods Gotman [9]discussed the improvement of epileptic seizure detectionand evaluation Pang et al [10] summarized the history andevaluation of various spike detecting algorithms Reference[13] discussed the different neural network as functionapproximation and universal approximation for epilepsydiagnosis Sarang [14] encapsulates the performance of spikedetecting algorithms in terms of sensitivity specificity andaverage detection Reference [14] ordered the performanceof spike detecting algorithms in terms of good detectionratio (GDR) McSharry et al [8] discussed and enumeratedthe nonlinear methods and its relevance to predict epilepsyby considering EEG samples as time series Majumdar [15]reviews various soft computing approaches of EEG signalswhich emphasize more on pattern recognition techniquesReference [15] mainly focuses on dimensionality reductionSNR problems and linear and soft computing techniques forEEG signal processing Kaushik concludes that the neuralnetwork and Bayesian approaches are two popular choiceseven though linear statistical discriminants are easier toimplement A large number of Support Vector Machines(SVM) are also discussed in this paper for their classificationaccuracy Hence the EEG signal occupies a great deal of dataregarding the working of the brain However classificationand estimation of the signals are inadequate As there is noexplicit category suggested by the experts visual examinationof EEG signals in time domain may be deficient Routineclinical diagnosis necessitates the analysis of EEG signals[13] Hence automation and computer methods have beenutilized for this reason Current multicenter clinical analysisindicates confirmation of premonitory symptoms in 62of 500 patients with epilepsy [16] Another interview basedstudy found that 50 of 562 patients felt ldquoaurasrdquo beforeseizures Those clinical data provide a motivation to searchfor premonitoring alterations on EEG recordings from thebrain and to employ a device that can act without humanintervention to forewarn the patient [17] On the other handdespite decades of research existing techniques do not yieldto better performance This paper addresses the applicationand comparison of SVD EM and MEM classifiers towards

optimization of code converter outputs in the classificationof epilepsy risk levels

Webber et al [18] have proposed the three-stage designof an EEG seizure detection system The first stage ofthe seizure detector compresses the raw data stream andtransforms the data into variables which represent the stateof the subjectrsquos EEG These state measures are referred toas context parameters The second stage of the system isa neural network that transforms the state measures intosmaller number of parameters that are intended to representmeasures of recognized phenomena such as small seizurein the EEG [9 10] The third stage consists of a few simplerules that confirm the existence of the phenomena underconsideration Similarly this paper also presents a three-stagedesign for epilepsy risk level classification The first stageextracts the required seven distinct features from raw EEGdata stream of the patient in time domain The next stagetransforms these features into a code word through a codeconverter with seven alphabets which represents the patientrsquosstate in five distinct risk levels for a two-second epoch of EEGsignal per channel The last stage is a SVD EM or MEMwhich optimizes the epilepsy risk level of the patient Theorganization of the paper is as follows Section 1 introducesthe paper and materials and its methods are discussed inSection 2 Section 3 describes the SVD EM and MEM aspostclassifiers for epilepsy risk level classification Resultsare discussed in Section 4 and the paper is concluded inSection 5

2 Materials and Methods

21 Data Acquisition of EEG Signals For the comparativestudy and to analyze the performance of the pre- andpostclassifiers we have obtained the raw EEG data of 20epileptic patients in European Data Format (EDF) whounderwent treatment in the Neurology Department of SriRamakrishna Hospital Coimbatore An issue that has beengiven great attention is the preprocessing stage of the EEGsignals because it is important to use the best techniqueto extract the useful information embedded in the nonsta-tionary biomedical signals The obtained EEG records werecontinuous for about 30 seconds each of them was dividedinto epochs of two-second duration A two-second epochis long enough to detect any significant changes in activityand presence of artifacts and also short enough to avoidany redundancy in the signal [19] For a patient we have 16channels over three epochs Having a frequency of 50Hzeach epoch was sampled at a frequency of 200Hz Eachsample corresponds to the instantaneous amplitude values ofthe signal totaling to 400 values for an epoch Figure 1 showsthe model of flow diagram of epilepsy risk level classificationsystem Four types of artifacts were present in our dataThey included eye blink electromyography (EMG) artifactchewing and motion artifacts [20] Approximately 1 ofthe data was artifacts We did not make any attempt toselect certain number of artifacts and of a specific natureThe objective of including artifacts was to have spikes ver-sus nonspike categories of waveforms The latter could be


Features


Samples


So threshold

Code converters




PatternsPatterns

Hard threshold

Features

Raw EEG signal






119864 =

119899

sum

119894=1

1199092

119894 (1)




1205902

=sum119899

119894=1(119909119894minus 120583)2

119899 (2)



CD =sum119901

119894=1(119863 minus 119905

119894)2

1199011198632 (3)






119863 =sum119901

119894=1119905119894

119901 (4)


such durations



119908119891 (119886 119887) = int

infin

minusinfin

119891 (119905) 120595lowast

119886119887(119905) 119889119905 (5)

where 120595lowast



120595lowast

119886119887(119905) =

1

radic2

120595(119905 minus 119887

119886) (6)



120595 (119905) =

1 0 le 119905 lt1

2

minus11

2le 119905 lt 1

0 otherwise

(7)






120588119879(119909) =

119909 if |119909| gt 1198790 if |119909| le 119879

(8)



120588119879(119909) =


(9)
















times 100 (10)




Se = [ PC(PC + FA)

] lowast 100

(05

06) lowast 100 = 8333

Sp = [ PC(PC +MC)

] lowast 100

(05

07) lowast 100 = 7142


2

AD = 78875






11111 = 31 sum = 1




Falsealarm

PerformanceIndex

Haar 6145 15625 2291 3758Db2 6118 1614 2265 3644Db4 6457 1249 2291 4472Sym8 6352 1144 2395 4481



119877 =119862

119863 (12)







Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875



119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)


119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)


Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)


)Θ119892] (119905)

(14)


1(119905) and 119892



OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)


OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)


119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)






[1198861 1198862 1198863 119886


119860 = USV119879 (18)



119860 = sum119875120590119896119906119896V119879119896 (19)






119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Features


Samples


So threshold

Code converters




PatternsPatterns

Hard threshold

Features

Raw EEG signal






119864 =

119899

sum

119894=1

1199092

119894 (1)




1205902

=sum119899

119894=1(119909119894minus 120583)2

119899 (2)



CD =sum119901

119894=1(119863 minus 119905

119894)2

1199011198632 (3)






119863 =sum119901

119894=1119905119894

119901 (4)


such durations



119908119891 (119886 119887) = int

infin

minusinfin

119891 (119905) 120595lowast

119886119887(119905) 119889119905 (5)

where 120595lowast



120595lowast

119886119887(119905) =

1

radic2

120595(119905 minus 119887

119886) (6)



120595 (119905) =

1 0 le 119905 lt1

2

minus11

2le 119905 lt 1

0 otherwise

(7)






120588119879(119909) =

119909 if |119909| gt 1198790 if |119909| le 119879

(8)



120588119879(119909) =


(9)
















times 100 (10)




Se = [ PC(PC + FA)

] lowast 100

(05

06) lowast 100 = 8333

Sp = [ PC(PC +MC)

] lowast 100

(05

07) lowast 100 = 7142


2

AD = 78875






11111 = 31 sum = 1




Falsealarm

PerformanceIndex

Haar 6145 15625 2291 3758Db2 6118 1614 2265 3644Db4 6457 1249 2291 4472Sym8 6352 1144 2395 4481



119877 =119862

119863 (12)







Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875



119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)


119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)


Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)


)Θ119892] (119905)

(14)


1(119905) and 119892



OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)


OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)


119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)






[1198861 1198862 1198863 119886


119860 = USV119879 (18)



119860 = sum119875120590119896119906119896V119879119896 (19)






119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120588119879(119909) =

119909 if |119909| gt 1198790 if |119909| le 119879

(8)



120588119879(119909) =


(9)
















times 100 (10)




Se = [ PC(PC + FA)

] lowast 100

(05

06) lowast 100 = 8333

Sp = [ PC(PC +MC)

] lowast 100

(05

07) lowast 100 = 7142


2

AD = 78875






11111 = 31 sum = 1




Falsealarm

PerformanceIndex

Haar 6145 15625 2291 3758Db2 6118 1614 2265 3644Db4 6457 1249 2291 4472Sym8 6352 1144 2395 4481



119877 =119862

119863 (12)







Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875



119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)


119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)


Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)


)Θ119892] (119905)

(14)


1(119905) and 119892



OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)


OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)


119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)






[1198861 1198862 1198863 119886


119860 = USV119879 (18)



119860 = sum119875120590119896119906119896V119879119896 (19)






119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














times 100 (10)




Se = [ PC(PC + FA)

] lowast 100

(05

06) lowast 100 = 8333

Sp = [ PC(PC +MC)

] lowast 100

(05

07) lowast 100 = 7142


2

AD = 78875






11111 = 31 sum = 1




Falsealarm

PerformanceIndex

Haar 6145 15625 2291 3758Db2 6118 1614 2265 3644Db4 6457 1249 2291 4472Sym8 6352 1144 2395 4481



119877 =119862

119863 (12)







Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875



119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)


119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)


Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)


)Θ119892] (119905)

(14)


1(119905) and 119892



OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)


OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)


119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)






[1198861 1198862 1198863 119886


119860 = USV119879 (18)



119860 = sum119875120590119896119906119896V119879119896 (19)






119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Rhythmicity119877 = 119862119863

Haar 31 0032292Db2 41 0042708Db4 30 003125Sym8 45 0046875



119905minus119906isin119865

119906isin119866

119891 (119905 minus 119906) + 119892 (119906)


119905=119906isin119865

119906isin119866

119891 (119905 minus 119906) minus 119892 (119906)

(13)


Opening (119891 ∘ 119892) (119905) = [(119891Θ119892119878

) oplus 119892] (119905)


)Θ119892] (119905)

(14)


1(119905) and 119892



OC (119891 (119905)) = 119891 (119905) ∘ 1198921(119905) sdot 119892

2(119905)

CO (119891 (119905)) = 119891 (119905) sdot 1198921(119905) ∘ 119892

2(119905)

(15)


OCCO (119891 (119905)) =[OC (119891 (119905)) + CO (119891 (119905))]

2 (16)


119891 (119905) = 119909 (119905) +OCCO (119891 (119905)) (17)






[1198861 1198862 1198863 119886


119860 = USV119879 (18)



119860 = sum119875120590119896119906119896V119879119896 (19)






119909[119896+1]

1= 119864 [119909

1| 1199101 119901119896

] (20)


This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










This implies

119909[119896+1]

1= 1199101

14

(14) + (119901[119896]2)

(21)



119883 = 119883119894| 119883119894sim Md (120583 119896) for 1 le 119894 le 119899 (22)


119875 (119883 | 120583 119896) = 119875 (119909119894 119909

119899| 120583 119896)

=

119899

prod

119894=1

119891 (119909119894| 120583 119896) =

119899

prod

119894=1

119888119889(119896) 119890119896120583119879119909119894

(23)


119871 (119883 | 120583 119896) = ln119875 (119883 | 120583 119896) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 (24)

where

119903 = sum 119894119909119894 (25)


119871 (120583 120582 120581 119883) = 119899 ln 119888119889(119896) + 119896120583

119879

119903 + 120582 (1 minus 120583119879

120583) (26)


120583 =

2

119903

120583119879

120583 = 1

1198991198881015840

()

119888119889()

= minus120583119879

119903

(27)






119899 the complete




119896by



119876(120579119894+1

120579119894

) = max119876(120579119894

120579) 120579119894+1

= arg max 119876(120579 120579119894

)

119889 (119901 119902) = 119889 (119901 119902) = radic

119899

sum

119894+1

(119902119894minus 119901119894)2

(28)


becomes smallenough



















QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























QV =119862












Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Haar 1156 235 1832 1924Db2 1257 2382 1972 2049Db4 1249 2315 2132 2211Sym8 1284 2337 1952 203





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Hard threshold

Haar 214 235 2431 1832 236 1924dB2 2017 2382 223 1972 214 2049dB4 1974 2315 211 2132 201 2211Sym8 2038 2337 22 1952 218 203


Haar 194 2016 219 2012 208 2085dB2 282 1601 237 2024 227 2054dB4 241 1999 231 1979 227 2057Sym8 216 1895 226 2044 213 22


Haar 17 1936 215 2009 206 2254dB2 223 1939 23 1887 216 2041dB4 197 2013 217 1997 214 2066Sym8 251 1951 227 202 222 2073


Haar 177 2044 201 2032 21 2142dB2 162 1877 207 194 204 1995dB4 153 1674 208 2042 208 2204Sym8 165 1951 218 2002 209 2106


Haar 208 1782 234 1877 23 2022dB2 325 1673 242 1917 236 201dB4 276 191 237 1962 236 1935Sym8 297 1752 241 1874 239 1997










5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusion




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article wavelets and morphological operators based...

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