**4. Experimental results**

This section shows the wavelet analysis of each A-scan signals detected by the sensor at positions indicated in **Table 1**. Three different types of mother wavelets are used to analyze the signal, namely the Mexican hat, Morlet, and db4. The three mother wavelets are very popular for ultrasound wave analysis due to their high correlation with the ultrasound wave form.

**Figure 4** shows the cross section front view at XY plane of the cylindrical defect embedded at depth of the object. The position of the three scan points at the surface of the object are superimposed on same Figure for clarification the horizontal and

**103**

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound*

**Scan point** *X* **position** *Y* **position True time of flight** R1 11 51 19.65 2 31 51 18.30 3 81 51 17.10

vertical spacing from cylindrical defect. The exact data of the position of each scan

**Figure 5** shows the absolute position of the defect cylinder represented by top view for the three scanning points. In addition, the locus ellipsoid estimation based on distance corresponding to TOF from exciter to defect point scatter and back to the receiving sensor for the three scanning points. It can be seen that the intersection between the ellipsoid and the cylinder happens at the point of back scattering

It is shown in **Figure 6** the wavelet contour map generated using Morlet WT. It is clear that the WT analysis resulted in clustering the signal into groups of segregated echoes. Each echo is governed by its intensity level, time duration, and scale levels. Scale levels are inversely proportional to the frequency spectrum. Hence, we can see at the top of the WT spectrum lies the echoes with low frequencies, while the echoes at the bottom correspond to high frequency components. Each of these echoes starts at a certain time shift, and it is clear that the start of the echo is occurring at lower frequencies with less intensity, and later the higher frequency components start to appear with their intensity level increasing. TOF of the first echo is corresponding to the direct surface propagation of the signal from the excitation point to the receiving sensor position, while TOF of the second large echo signal is corresponding to the reflected signal from the defect. It is possible to estimate the corresponding TOF based on that conclusion to be 20, 19, and 18.5 μs

*DOI: http://dx.doi.org/10.5772/intechopen.84964*

*Scan points considered for samples inspection.*

**Table 1.**

**Figure 4.**

point is represented in **Table 1**.

*Scan points considered for sample inspection and synthetic defect location.*

from the defect to the sensor.


*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound DOI: http://dx.doi.org/10.5772/intechopen.84964*

### **Table 1.**

*Wavelet Transform and Complexity*

The TOF corresponding to the presence of the defect will be equal to the sum of TOF from source of excitation to the defect scatterer and the TOF from the defect scatterer to the receiving sensor. IF this TOF is converted to distance by multiplying by the longitudinal velocity of sound in the material, we can see that the position of the defect scatter would be any point at the surface of a locus ellipsoid whose two

*Aluminum specimen, internal damage, and laser scan area. All dimensions are in millimeters. (a) Isometric view.* 

drical defect is considered to investigate the detection capabilities of the wavelet and synchrosqueezed transforms. The sample's structure and the position of the defect are shown in the next figure. The hole under investigation is the one on the

This section shows the wavelet analysis of each A-scan signals detected by the sensor at positions indicated in **Table 1**. Three different types of mother wavelets are used to analyze the signal, namely the Mexican hat, Morlet, and db4. The three mother wavelets are very popular for ultrasound wave analysis due to their high

**Figure 4** shows the cross section front view at XY plane of the cylindrical defect embedded at depth of the object. The position of the three scan points at the surface of the object are superimposed on same Figure for clarification the horizontal and

top around the scan area of the laser-generated ultrasound (**Figure 3**).

, and with an embedded cylin-

foci are the exciter and sensor positions [24].

correlation with the ultrasound wave form.

**4. Experimental results**

An aluminum cube, with dimensions of 200 mm3

**102**

**Figure 3.**

*(b) Front view. (c) Top view.*

*Scan points considered for samples inspection.*

### **Figure 4.**

*Scan points considered for sample inspection and synthetic defect location.*

vertical spacing from cylindrical defect. The exact data of the position of each scan point is represented in **Table 1**.

**Figure 5** shows the absolute position of the defect cylinder represented by top view for the three scanning points. In addition, the locus ellipsoid estimation based on distance corresponding to TOF from exciter to defect point scatter and back to the receiving sensor for the three scanning points. It can be seen that the intersection between the ellipsoid and the cylinder happens at the point of back scattering from the defect to the sensor.

It is shown in **Figure 6** the wavelet contour map generated using Morlet WT. It is clear that the WT analysis resulted in clustering the signal into groups of segregated echoes. Each echo is governed by its intensity level, time duration, and scale levels. Scale levels are inversely proportional to the frequency spectrum. Hence, we can see at the top of the WT spectrum lies the echoes with low frequencies, while the echoes at the bottom correspond to high frequency components. Each of these echoes starts at a certain time shift, and it is clear that the start of the echo is occurring at lower frequencies with less intensity, and later the higher frequency components start to appear with their intensity level increasing. TOF of the first echo is corresponding to the direct surface propagation of the signal from the excitation point to the receiving sensor position, while TOF of the second large echo signal is corresponding to the reflected signal from the defect. It is possible to estimate the corresponding TOF based on that conclusion to be 20, 19, and 18.5 μs

### **Figure 5.**

*Ellipsoid locus of defect position based on true time of Flight estimation resulting from the scan point and acoustic transducer positions. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

**Figure 6.**

*Resulting wavelet contour maps with Morlet wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

**105**

**Figure 8.**

**Figure 7.**

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound*

*Resulting wavelet contour maps with Mexican hat wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

*Resulting wavelet contour maps with db24 wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

*DOI: http://dx.doi.org/10.5772/intechopen.84964*

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound DOI: http://dx.doi.org/10.5772/intechopen.84964*

**Figure 7.**

*Wavelet Transform and Complexity*

**104**

**Figure 6.**

**Figure 5.**

*Ellipsoid locus of defect position based on true time of Flight estimation resulting from the scan point and* 

*Resulting wavelet contour maps with Morlet wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

*acoustic transducer positions. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

*Resulting wavelet contour maps with Mexican hat wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

**Figure 8.**

*Resulting wavelet contour maps with db24 wavelet. (a) Scan point 1. (b) Scan point 2. (c) Scan point 3.*

### **Figure 9.**

*Synchrosqueezing wavelet transform contour map of the scan point 1. (a) Time-based signal. (b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing* 

*transform contour map with an initial presence of acoustic activity at 21.08 μs, 2.87 MHz @−80.31 dB.*

### **Figure 10.**

*Synchrosqueezing wavelet transform contour map of the scan point 2. (a) Time-based signal. (b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing transform contour map with an initial presence of acoustic activity at 23.56 μs, 2.15 MHz @−84.04 dB.*

**107**

**Table 4.**

**Figure 11.**

**Table 2.**

**Table 3.**

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound*

*Synchrosqueezing wavelet transform contour map of the scan point 3. (a) Time-based signal. (b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing transform contour* 

**Wavelet Scan point 1 Scan point 2 Scan point 3** Morlet 20.22 18.58 18.28 Mexican hat 23.05 19.98 20.98 Db4 21.65 18.95 18.50

**Wavelet Scan point 1 Scan point 2 Scan point 3** Morlet 0.57 0.28 1.18 Mexican hat 3.40 1.68 3.88 Db4 2.00 0.65 1.40

Synchrosqueezing transform 21.08 23.56 20.24

**Scan point 1 Scan point 2 Scan point 3**

*map with an initial presence of acoustic activity at 20.244 μs, 2.57 MHz @−89.64 dB.*

*Resulting time of flight error compared with the true time of flight, in microseconds.*

*Resulting time of flight from the internal defect in microseconds.*

*Resulting time of flight from the internal defect in microseconds.*

*DOI: http://dx.doi.org/10.5772/intechopen.84964*

*Wavelet Transform Applied to Internal Defect Detection by Means of Laser Ultrasound DOI: http://dx.doi.org/10.5772/intechopen.84964*

### **Figure 11.**

*Wavelet Transform and Complexity*

**106**

**Figure 10.**

**Figure 9.**

*Synchrosqueezing wavelet transform contour map of the scan point 1. (a) Time-based signal.* 

*transform contour map with an initial presence of acoustic activity at 21.08 μs, 2.87 MHz @−80.31 dB.*

*(b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing* 

*Synchrosqueezing wavelet transform contour map of the scan point 2. (a) Time-based signal. (b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing transform contour* 

*map with an initial presence of acoustic activity at 23.56 μs, 2.15 MHz @−84.04 dB.*

*Synchrosqueezing wavelet transform contour map of the scan point 3. (a) Time-based signal. (b) Synchrosqueezing transform contour map. (c) Detail of the time-based signal. (d) Detail of the synchrosqueezing transform contour map with an initial presence of acoustic activity at 20.244 μs, 2.57 MHz @−89.64 dB.*


### **Table 2.**

*Resulting time of flight from the internal defect in microseconds.*


### **Table 3.**

*Resulting time of flight error compared with the true time of flight, in microseconds.*


### **Table 4.**

*Resulting time of flight from the internal defect in microseconds.*


**Table 5.**

*Resulting time of flight error compared with the true time of flight, in microseconds.*

for **Figure 6a**–**c**, respectively. **Figure 7** and **Figure 8** represent the same wavelet echo analysis for Mexican hat and db4 mother wavelets respectively. **Table 2** show the resulting TOF from the internal defects at three scan points with reference to the different mother wavelets while **Table 3** shows the error of the resulting TOF compared to the calculated true TOF for each of the cases in (**Table 2**). It is found that the use of the Morlet mother wavelet gives the least estimation error and it is apparently the most accurate mother wavelet to use for this kind of analysis.

For the case of the TFR that results from the application of the SSWT, it can be observed that by the accurate redistribution of the energy that compose to the signal, the obtained images achieve an improved depiction of their modal frequencies in comparison with the conventional CWT, aiding to superiorly identify the behavior of the phenomenon. Moreover, for the scope of application of this study, by identifying the first instant of time when the bi-dimensional manifold created by means of the contour mapping of the SSWT apparently becomes closed by connecting all the modal frequencies of the signal of interest, it is possible to determine the onset of said signal.

As is well known, the accurate determination of this instant of time is critical for the TOF-related methods; hence, by means of this methodology, the required precision for the onset pick is achieved when only the signal waveform is used for this purpose. Synchrosqueezing wavelet transform contour map for the scan points of interest and TOF estimation are calculated **Figures 9**–**11**. **Tables 4**–**5**. show the resulting TOF for the three scan points and Table 5 shows the corresponding error with comparison to the true TOF.

Nevertheless, considerations must be taken in order to not analyze a very small signal, this with the aim to avoid the negative effects of the Cone of Influence (COI) of the CWT, since the SSWT still leads to inaccuracies for these areas.
