GaBi

The most important part of DWT is that it uses the discrete data as a scale parameter. In the DWT, the scale and the time as described in the last equation above are discretised as follow [181]:

Where and are integers, thus the CWT function , in the equation above

converted to the DWT by the following formula:

The discretisation of the scale parameter and time parameter leads to the discrete wavelet transform, which defined as:

The DWT has two important approaches to discrete the signal at different scale and position (resolution levels and different frequency), which are decomposing the signal into approximations (A) and details (D). The approximation information could be obtained from the low pass filter, while the detail information could be obtained from the high pass filter as explained in figure 5-4.

, (5-18)

_, (5-19)

Figure 5-4: DWT decomposition signal to approximation and detail using filters [181].

Figure 5-5 shows how to analyse and synthesis the signal, the input signal goes through two one-dimensional filters, one for high pass filter (H0), and one for low pass filter (H1). These filters have filtering operation and followed by subsampling by factor of 2. Then, the signal will be reconstructed by first up sampling, after that, filtering and summing the sub bands will be followed.

Figure 5-5: DWT two channel filters [185].

The synthesis filters F0 and F1 have to be adapted for analysing the H0 and H1 filters in order to achieve perfect reconstruction. It is very easy to obtain satisfying relationship between the 2-channal filters by considering Z-transform function. After analysis, the two sub bands will be as follows [186]:

The combination of these filters are:

In order to overcome the problem of aliasing and distortion, the following conditions should be considered [186]:

&

The multiscale pyramid decomposition and reconstruction of an image or signal with high and low pass filters has been illustrated in figure 5-6 and figure 5-7 below.

Figure 5-6: Filter bank structure of the DWT analysis [185].

Figure 5-7: Filter bank structure of the reverse DWT synthesis [185].

After one level of decomposition, there will be four frequency bands, which are Low- Low (LL), Low-High (LH), High-Low (HL), and High-High (HH). The next decomposition level will be applied to the LL band, which forms a recursive

decomposition procedure. Consequently, an N-level of decomposition will have 3N+1 different frequency bands, which includes 3N high frequency and one LL bands. Table 5-11 illustrates a brief comparison between the performance of CWT and DWT.

Table 5-11: Comparison of the performance of CWT and DWT [187].

CWT DWT a) It uses exponential scales with a base

smaller than 2.

a) It uses exponential scales with the base equal to 2.

b) Large computational resources required to compute the CWT.

b) Less computational resources required to compute the DWT. c) It is a shift-invariant. c) It not shift-invariant.

d) It is highly redundant transform. d) Is also redundant but less than the CWT.

e) It is orthonormal transform. e) It is orthonormal vector. f) The outputs are not down sample but

not better than DWT.

f) The outputs are down sampled, but better than CWT.

g) The inverse of CWT could be implemented but usually the signal reconstruction is not perfect.

g) It provides perfect signal reconstruction upon inversion, which means that the DWT of signal coefficients could be used to synthesise and exact reproduction of the signal with numerical precision.

The DWT has been widely used for analysing the induction motor signal (thermal image, current and vibration signals) due to its excellent decorrelation property, it has been used as a transform stage in many modern image and video compression systems [188]. Image compression is one of the most important visible applications of wavelets.

Traditionally, in the field of image processing a Discrete Wavelet Transform (DWT) has been adopted for image compression due to its simplicity and practicality. It has been applied for many different types of images such as JPEG, MPEGZ, PNG, etc.

In this work, the DWT technique has been adopted for extracting the best features from the IM thermal image. Among the various DWT techniques, Daubechies wavelet is considered for analysing the thermal image, as it is multilevel decomposition wavelet. The names of the Daubechies family wavelets are written as dbN, where N is the order, and db is the “surname” of the wavelet. The db1 wavelet is the same as the Haar wavelet, which is one of the wavelet functions as mentioned above. Figure 5-8 illustrates the nine members of the db family [189].

The wavelet toolbox in MATLAB software (version R2015a) has been used for analysing the thermal image, current signature and vibration signal.

Two-dimensional discrete wavelet analysis tool based on the Daubechies wavelet (db7) with 7 vanishing moments and 3 levels, db73 has been used for analysing the thermal images.

Procedure for thermal image analysis using DWT are below and explained in figure 5-9 :

a) Read the thermal image and convert it to the HSV colour in order to obtain the discrete pixel values.

b) Transformation: apply two-dimensional DWT using db7 with level 3. c) Save the extracted features from the image to the MATLAB workspace for

further processing.

d) Calculate the histogram for the approximation and the details coefficients. e) Save the data of the histogram.

f) Repeat the same process for other images.

Import thermal image

Convert it to HSV colour

Apply two-dimensional DWT (db73)

Export the image coefficient to the MATLAB workspace

Calculate the histogram for the image details and approximation

Save the histogram data

Repeat the process for other images

Figure 5-9: Procedure for thermal image analysis in MATLAB using wavelet toolbox.

One-dimensional discreet wavelet analysis tool has been used for analysing the current and vibration signals, the procedure of analysing these signals are as follows:

b) Apply the DWT to extract the signal features.

c) Save the extracted features for further processing.

d) Repeat the process for all faulty signals.

In terms of current/vibration signal, figure 5-10 illustrates the procedure of the signal analysis for the current and vibration in more details.

Import current/ vibration signal

Apply one-dimensional DWT (db75)

Export the new signal coefficient to the MATLAB workspace

Calculate the histogram for the signal details and approximation

Save the histogram data

Repeat the process for other signals

Figure 5-10: Current and vibration signals processing procedure.

Moving on now to consider feature selection method, because it plays a vital role in the field of classification. It has the ability to choose the best features among all dataset. Two important reasons for using the feature selection method: the first one is to reduce the data dimensionality, which is also reduce the processing time. The

second reason is to choose the features that increase the classification accuracy by removing the unwanted ones.