chapter 7 features extraction using discrete wavelet...

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CHAPTER 7

FEATURES EXTRACTION USING DISCRETE

WAVELET TRANSFORM (DWT) AND FAST

FOURIER TRANSFORM (FFT)

7.1 FEATURE EXTRACTION

Once the ultrasonic test signals acquired in a form of digitized data

are preprocessed, we need to determine features from the raw signal by the

use of digital processing techniques. This process is named ‘feature

extraction’. Feature extraction is a special form of dimensionality reduction.

Feature extraction involves simplifying the amount of resources required to

describe a large set of data accurately.

Since not all features that can be extracted from ultrasonic signals

for a given classification problem need to be used, due to their redundancy, a

further process is needed for redundancy reduction by retaining only an

informative subset of them. This stage of processing is called ‘feature

selection’.

7.1.1 Need for Feature Extraction

When the input data to an algorithm is too large to be processed

and it is suspected to be notoriously redundant (much data, but not much

information) then the input data will be transformed into a reduced

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representation set of features (also named features vector). Transforming the

input data into the set of features is called features extraction. Thus the

extraction of discriminatory features in the signal enhances the reduction of

the length of the data vector by eliminating redundancy in the signal and

compressing the relevant information into a feature vector of significantly

lower dimension.

The ultrasonic oscillograms are a graphical representation in which

it depicts some information regarding the variations in the pattern according

to the type of the flaws. So these ultrasonic oscillograms can be described by

pertinent features allowing the defect classification. After pertinent features

extraction, it is normally useful to elaborate a recognition procedure

(identification) of the detected defect type.

7.2 FEATURES EXTRACTION USING DWT

Discrete wavelet transform is used to extract characteristics from a

signal on various scales proceeding by successive high pass and low pass

filtering. The wavelet coefficients are the successive continuation of theapproximation and detail coefficients

The basic feature extraction procedure consists of

1. Decomposing the signal using DWT into N levels using

filtering and decimation to obtain the approximation and

detailed coefficients

2. Extracting the features from the DWT coefficients

The features extracted from the Discrete wavelet transform (DWT)

coefficients of ultrasonic test signals are considered useful features for input

into classifiers due to their effective time–frequency representation of non-

stationary signals.

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7.2.1 Feature Extraction Algorithm

Initially, it is verified that the digitized flaw data are available in

the powers of 2 for making the effective decomposition.

The various steps involved in the feature extraction algorithm are as

follows:

Step 1: The ultrasonic flaw data are decomposed into four detail subbands

using Discrete Wavelet Transform (DWT). The subbands are high frequency

detail band coefficients and low frequency approximation band coefficients.

Step 2: The approximation co-efficients are further decomposed using DWT

to extract localized information from the subband of detail coefficients. In this

work, four levels of decomposition have been done using biorthogonal

wavelet (bior 4.4).

Four level approximation and detail coefficients of six classes of

defect are graphically represented in Appendix 1 as Figures A1.1 to A1.6.

Step 3: For further analyzing and processing, all the four level detail band

coefficients have been taken.

Step 4: The frequency vector (in radians/sample) is extracted for four detail

subbands using periodogram function in Matlab.

Step 5: The features are computed either by using syntax or by implementing

the formulae. They are mean, variance, mean of energy, maximum amplitude,

minimum amplitude, maximum energy, minimum energy, average frequency,

mid frequency, maximum frequency, minimum frequency, half point of the

function.

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The M-file program for four level signal decomposition and

features extraction using DWT are provided in Appendix 2.

Step 6: Finally, the extracted features for the six classes of defects are

tabulated and analyzed for classification.

7.2.2 Extracted Features

In this work, twelve features are extracted from the Discrete

wavelet transform (DWT) coefficients of ultrasonic test signals obtained from

the six classes of defect. The extracted features from the signal are as below:

1. Mean: It is nothing but an average value.

n

ii 1

1m xn

2. Variance: The variance is defined as the sum of square

distances of each term in the distribution from the mean,

divided by the number of terms in the distribution.

n2

ii 1

1 x mn 1

3. Mean of the energy: It is the average value of the energy.

n2

e ii 1

1m xn

where x Sequence, m Mean, n Number of Samples

4. Maximum Amplitude: It is the peak value of amplitude of the

signal.

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5. Minimum Amplitude: It is the lowest value of amplitude of

the signal.

6. Maximum Energy: It is the highest energy value obtained

from the signal.

7. Minimum Energy: It is the lowest energy value obtained from

the signal.

8. Average Frequency:

n

i ii 1

avg n

ii 1

f xpf

p

where p Power spectral density, f Frequency vector

9. Mid Frequency: It is the frequency value which is obtained

when the power spectral density is at the maximum value.

10. Maximum frequency: It is the maximum frequency value of

the energy in the spectrum.

11. Minimum frequency: It is the minimum frequency value of the

energy in the spectrum.

12. Half Point of the energy (HaPo): It is a very valuable variable

as it represents the frequency that divides up the spectrum into

two parts of same area.

7.2.3 Feature Extraction Results

The extracted features from the ultrasonic flaw signal for crack in

each of the four level sub bands are shown in the table 7.1. D1 represents 1st

level detail band. D2 represents 2nd level detail band. D3 represents 3rd level

detail band. D4 represents 4th level detail band. The extracted features for

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each of the other 5 classes of defect in each of the four level sub bands are

shown in Appendix 3 as tables A 3.1 to A 3.5.

Table 7.1 Extracted Features for crack in four detail sub bands

Four level detail sub bands of waveletcoefficientsSl.

NoFeatures D1

/2 - D2

/4 - /2D3

/8 - /4D4

/16 - /81. Mean 0.2331 0.0191 0.5567 -0.46172. Variance 6.6855 68.5707 1045.37 389.10793. Mean of energy 6.7366 68.5041 1043.6377 387.80114. Max. amp 25.7044 63.4938 205.2167 82.11985. Min. amp -23.7403 -86.6635 -316.108 -193.9046. Max. energy 660.717 7510.559 99924.09 37598.67. Min.energy 0.0000 0.0000 0.0003 0.00008. Avg. frequency 2.6154 1.7991 1.8754 1.70949. Mid frequency 2.8777 2.0433 2.1967 1.448110. Max. frequency 0.0031 0.0123 0.0245 0.073611. Min.frequency 0.3283 2.3746 1.4849 2.454412. Half pt. 1.4205 2.0064 2.5525 1.988

7.3 FEATURE ANALYSIS OF DWT FEATURES

7.3.1 Feature values for Six Classes of Defect

Among the extracted twelve features, the average values of each

feature for each classes of defect are determined and tabulated. The average

values of the each of the twelve features obtained from DWT coefficients of

the crack signal in 1st level detail sub band are shown in the Table 7.2. This

average value is calculated for the 30 signals obtained from crack. The

average values of all features for other classes of defect are provided in

Appendix 3 as tables A 3.6 to A 3.10.

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7.3.2 Selection of Features

As the relationship between ultrasonic signal characteristics and

flaw classes is not straightforward, the extraction of features plays a critical

role in classification accuracy and this becomes the important basis of

decision-making for classification.

The variation of the twelve features with respect to each classes of

defect is analysed and for each defect, the average values for all the features

are determined and are plotted in graphs. The average feature values for the

six classes of defect in the first level sub band are marked in the graph

representing defects in the x axis and average values in the y axis. The

variation in the feature values for the six classes of defect are shown in the

following graphs as Figures 7.1(a) to 7.1(k).

Figure 7.1(a) Variation of mean for six classes of defect

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Figure 7.1(b) Variation of variance for six classes of defect

Figure 7.1(c) Variation of mean of energy for six classes of defect

Figure 7.1(d) Variation of maximum amplitude for six classes of defect

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Figure 7.1(e) Variation of minimum amplitude for six classes of defect

Figure 7.1(f) Variation of maximum energy for six classes of defect

Figure 7.1(g) Variation of average frequency for six classes of defect

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Figure 7.1(h) Variation of mid frequency for six classes of defect

Figure 7.1(i) Variation of maximum frequency for six classes of defect

Figure 7.1(j) Variation of minimum frequency for six classes of defect

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Figure 7.1(k) Variation of half point of energy for six classes of defect

In the feature analysis, the variation of the twelve features with

respect to each classes of defect is analysed and for each defect, the average

values for all the features is determined. By analyzing and comparing the

graphical results, it is inferred that among the extracted twelve features; only

eight features have given faithful information and also good discrimination

between the flaws.

They are

1. Mean

2. Variance

3. Maximum amplitude

4. Minimum amplitude

5. Maximum energy

6. Average frequency

7. Minimum frequency

8. Half point of the function

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7.3.3 Inputs to ANN and SVM

The selected eight features which are giving good discrimination

between material defects are considered as the main parameters and hence

these eight features are taken combinely as the input to the ANN and SVM for

the classification of defects. Based on feature analysis, other four features

such as mean of energy, minimum energy, mid frequency and maximum

frequency are neglected because of the following reasons:

Mean of energy : The feature values are same as the variance.

Minimum energy : The feature values are zero for all six classes of defect

Mid frequency : It gives closer values for all six classes of defect

Max. frequency : It gives similar values for all six classes of defect

The selected features extracted from each ultrasonic signal are used

as the input to the ANN and SVM. The input must be representative of its

respective ultrasonic oscillogram. Here, the input of the ANN and SVM is

eight component vector. The 4th level detail DWT coefficients representation

of a defect signal (left) and its respective input of the ANN (right) are shown

in Figures 7.2 (a) to 7.2 (f).

mean of the samplesvariancemaximum amplitudeminimum amplitudemaximum energyaverage frequencyminimum frequencyhalf point

0.0103262.7837124.6363 -127.516256.28 1.8955 3.1416 1.8408

Figure 7.2(a) DWT coefficients representation of a crack signal (left) and

its respective input of the ANN (right)

=

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-1.254798.6577147.7117-204.86341968.951.58253.09252.1844

Figure 7.2(b) DWT coefficients representation of a porosity signal (left)

and its respective input of the ANN (right)

0.2553481.575194.0681-169.08337662.451.74322.57711.4972


Figure 7.2(c) DWT coefficients representation of a lack of fusion signal

(left) and its respective input of the ANN (right)


-1.4214274.805494.3939-191.45436654.631.63563.14161.7426

Figure 7.2(d) DWT coefficients representation of a lack of penetration

signal (left) and its respective input of the ANN (right)

=

=

=

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-0.4579862.8073207.6305-162.66643110.441.28532.30111.5953

Figure 7.2(e) DWT coefficients representation of a tungsten inclusion

signal (left) and its respective input of the ANN (right)

-0.0574592.2208173.8241-188.83335657.821.83671.20261.939


Figure 7.2(f) DWT coefficients representation of a non defect signal (left)and its respective input of the ANN (right)

7.4 FEATURES EXTRACTION USING FFT

7.4.1 Feature Extraction Algorithm

The various steps involved in the feature extraction algorithm are asfollows:

Step 1: The ultrasonic flaw data are down sampled in four stages. The downsampled data are in different sample length. Four stages down sampledsignals for six classes of flaws are graphically represented in Appendix 4 asFigures A 4.1 to A 4.6.

Step 2: Fast Fourier Transform is applied to the four stages of down sampledsignals.

=

=

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Step 3: The frequency vector (in radians/sample) is extracted for four stagedown sampled signals using periodogram function in Matlab.

Step 4: The features are computed using syntax and implementing theformulae. The extracted features are mean, variance, mean of energy,maximum amplitude, minimum amplitude, maximum energy, minimumenergy, average frequency, mid frequency, maximum frequency, minimumfrequency and half point of the function.

The M-file program for four stage down sampling and featuresextraction after applying FFT are shown in Appendix 5.

Step 5: Finally, the extracted features for the six classes of defects aretabulated and analyzed for classification.

7.4.2 Extracted Features

Twelve features are extracted from the each signal of the six classesof defects. The extracted features from the signal are as below:

1 Mean: It is nothing but an average value.

n

ii 1

1m xn

2 Variance: The variance is defined as the sum of squaredistances of each term in the distribution from the mean,divided by the number of terms in the distribution.

n2

ii 1

1 x mn 1

3 Mean of the energy: It is the average value of the energy.

n2

e ii 1

1m xn

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where x Sequence, M mean, n Number of Samples

4 Maximum Amplitude: It is the peak value of amplitude of the

signal.

5 Minimum Amplitude: It is the lowest value of amplitude of

the signal.

6 Maximum Energy: It is the highest energy value obtained

from the signal.

7 Minimum Energy: It is the lowest energy value obtained from

the signal.

8 Average Frequency:

n

i ii 1

avg n

ii 1

f xpf

p

where p Power spectral density, f Frequency vector

9 Mid Frequency: It is the frequency value which is obtained

when the power spectral density is at the maximum value.

10 Maximum frequency: It is the maximum frequency value of

the energy in the spectrum.

11 Minimum frequency: It is the minimum frequency value of the

energy in the spectrum.

12 Half Point of the energy (HaPo): It is a very valuable variable

as it represents the frequency that divides up the spectrum into

two parts of same area.

7.4.3 Feature Extraction Results

The extracted features from the ultrasonic flaw signal for crack ineach of the four stage down samplings are shown in the Table 7.3. S1represents 1st stage down sampling. S2 represents 2nd stage down sampling.

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S3 represents 3rd stage down sampling. S4 represents 4th stage downsampling. The extracted features for each of the other five classes of defect ineach of the four stage down samplings are shown in Appendix 6 asTables A 6.1 to A 6.5.

Table 7.3 Extracted Features for crack in four stage down sampling

Four stage down samplingSl.No Features S1 S2 S3 S41. Mean 127 127 127 127

2. Variance 66579581 33376311 16710952 8381750

3. Mean of energy 65780248 32970438 16504812 8282812

4. Max. amplitude 519237 259935 130044 65142

5. Min. amplitude 2.1927 -5.9101 -22.7786 10

6. Max. energy 269607062 67566204 16911441 4243480

7. Min. energy 4.6023 -335.1817 502.0688 100

8. Avg. frequency 3.1725 3.1721 3.1732 3.1799

9. Mid frequency 6.1666 6.1666 6.1666 6.0746

10. Max. frequency 5.8429 5.8414 5.8414 5.7678

11. Min. frequency 5.889 5.8905 5.8905 5.7923

12. Half point 4.4409 4.4393 4.4363 4.4301

7.5 FEATURE ANALYSIS OF FFT FEATURES

7.5.1 Feature Values for Six Classes of Defect

Among the extracted twelve features, the average values of each

feature for each classes of defect are determined and tabulated. The average

values of the each of the twelve features obtained from FFT coefficients of

the crack signal in 1st stage down sampling are shown in the Table 7.4. This

average value is calculated for thirty number of signals obtained from crack.

The average values of all features for other classes of defect are provided in

Appendix 6 as Tables A 6.6 to A 6.10.

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7.5.2 Selection of features

The average feature values for the six classes of defect in the first

stage down sampled signal are marked in the graph representing defects in the

x axis and average values in the y axis. The variation in the feature values for

the six classes of defects is shown graphically in Figures 7.3 (a) to 7.3 (l).

Figure 7.3(a) Variation of mean for six classes of defect

Figure 7.3(b) Variation of variance for six classes of defect

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Figure 7.3(c) Variation of mean of energy for six classes of defect

Figure 7.3(d) Variation of maximum amplitude for six classes of defect

Figure 7.3(e) Variation of minimum amplitude for six classes of defect

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Figure 7.3(f) Variation of maximum energy for six classes of defect

Figure 7.3(g) Variation of minimum energy for six classes of defect

Figure 7.3(h) Variation of average frequency for six classes of defect

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Figure 7.3(i) Variation of mid frequency for six classes of defect

Figure 7.3(j) Variation of maximum frequency for six classes of defect

Figure 7.3(k) Variation of minimum frequency for six classes of defect

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Figure 7.3(l) Variation of half point for six classes of defect

In the feature analysis, the variation of the twelve features with

respect to each classes of defect is analysed and for each defect, the average

values for all the features is determined. By analyzing and comparing the

graphical results, it is inferred that among the extracted twelve features; only

eight features have given faithful information and also good discrimination

between the flaws. They are

1. Variance

2. Mean of Energy

3. Maximum Amplitude

4. Minimum Amplitude

5. Minimum Energy

6. Mid Frequency

7. Maximum Frequency

8. Minimum Frequency

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7.5.3 Inputs to ANN and SVM

The selected eight features are giving good discrimination between

material defects and are considered as the main parameters which influence

the classification of defects and hence these eight features are taken

combinely as the input to the ANN and SVM. Based on feature analysis, other

four features such as Mean, Maximum Energy, Average Frequency, Half

Point are neglected as it gives similar values for all six classes of defect.

The selected features extracted from each ultrasonic signal are used

as the input to the ANN and SVM. Here, the input of the ANN and SVM is

eight component vector.

7.6 SUMMARY

Feature extraction procedure and the various features extracted

from the ultrasonic flaw signals using Discrete Wavelet Transform (DWT)

and Fast Fourier Transform (FFT) are described in this section. The extracted

features for each of the six classes of defect in each of the four level sub

bands are tabulated. Selection of features based on feature analysis is also

clearly described. Lastly, the critical features which give the best

classification results are presented.

chapter 7 features extraction using discrete wavelet...

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