|NDT.net Apr 2006 Vol. 11 No.4|
Resolution of Stress Wave Based Acoustic TomographyF. Divos
University of West Hungary, Sopron, E-mail: firstname.lastname@example.org
Technical University Budapest, E-Mail: email@example.com
14th International Symposium on Nondestructive Testing of Wood
AbstractThe resolution of images obtained by stress wave based acoustic tomography depends on the acoustic wave frequency, number of sensors used and the applied evaluation algorithm. The resolution of the image is better if the applied frequency is higher, the number of sensor is higher and the applied algorithm is more advanced. The paper deals with this issue.
IntroductionWhy is the resolution of acoustic tomography images important? This information is necessary when evaluating the usefulness of the image, and the user of the tomography needs to decide upon and choose the required resolution that fit the application. The definition of resolution used in this article is the smallest size of a round shape object that appears in the image. In optics, resolution is defined as the smallest possible distance between two objects, that appear separately on the image. The two definitions are not the same, but there are fairly close. Stress wave based acoustic tomography is a rather new technique, applied in tree evaluation both in research and practical applications like the assessment of the stability of trees in urban areas. This technique provides a two-dimensional image of the cross-section of a tree trunk . 6 to 32 stress wave sensors are used. Sensors are coupled to the wood material of the tree by a spike or screw. Stress wave is generated by a hammer impact on a sensor. All the other sensors are receiving the signal and en electronic clock measures the transit times. The input time data is created by triggering all of the sensors. Using N sensors, the velocity is calculated in N(N-1)/2 different directions. For this calculation, the geometric arrangement of the sensors is used as well. The final step is the inversion, so called Radon transformation. The resolution of stress wave based acoustic tomography depends on the applied frequency, the number of sensors and the applied inversion technique.
The influence of the frequencyFrom physics we know that the resolution of optical systems is in the range of the wavelength l of the light . In electron microscopy the theoretical resolution limit is the wavelength of the electron. Each time we use waves for imaging, the resolution is limited by the applied wavelength. The wavelength can be determined from the velocity (V) and frequency (f).
The velocity of stress waves perpendicular to the grain in green condition is between 1000 and 2000 m/s, depending on the tree species. The frequency of the stress wave is not a well-defined value. A sharp hammer impact on steel generates a wide frequency range. What we measure depends on the material, the applied sensor and the geometry, e.g. the distance between the source and the receiver. Figure 1 shows an oscilloscope image of the beginning of a stress wave signal. The material is larch, and the distance between the source and the receiver is 7 cm. The horizontal axis is time, and the grid size is 50µs. A high frequency (40 kHz) signal is superimposed on the 2 kHz signal. In the vicinity of the start sensor, the presence of the high frequency component of the wave is clear. With increasing distance, the high frequency component becomes smaller and smaller, because the attenuation is stronger at higher frequencies.
The attenuation is 20 dB/m and 200 dB/m at 2 kHz and 40 kHz, respectively. The attenuation increases exponentially with frequency. The highest frequency level usable for wood is about 100-200 kHz . At 35 cm from the start sensor, the low frequency component of the wave becomes dominant, but the high frequency component is still there, detectable only with strong signal amplification. Stress wave timers use high amplification. For this reason, in wavelength calculation 40 kHz frequency value is used. Selecting 1000 m/s for stress wave velocity, the calculated wavelength is 25mm. This is the theoretical resolution limit of the stress wave based tomography.
Increasing the resolution would be possible by using higher frequencies. This is limited by the high material attenuation. An interesting experiment is reported by Bucur , where 1MHz ultrasonic probes were used in 16-channel acoustic tomography, on a 60 cm dia. tree. The resolution turned out to be 50 mm, even though the theoretical resolution limit is around 1mm.
The influence of the sensor number
The images on figures 2 and 3 represent the radial stress-wave velocity map. Changing the “low” and “high” velocity value, the images change, so using suitable radial velocity limits is important. Spots where the reconstructed radial velocity falls below the “low” value and those where it increases the “high” value are marked yellow and red, respectively. The filtered backprojection technique (explained later) was used as an evaluation tool, and the number of sensors was 30. The smallest – 10 mm dia. – cavity is not detectable; the image shows intact material. This defect size is below the resolution limit. 25 mm cavities and larger appear on the image as red spot (dark grey in figure 2). The size and location of the spot is in good coincidence with the actual location of the defect. The gap between “low” and “high” velocity limits increases with cavity size.
The same cavities were also evaluated using reduced sensor numbers. The results for 47 mm cavity size are given in figure 3. The “low” and “high” velocity settings are given below the images. Higher sensor number allows a higher gap between low” and “high” velocity limits. This means that the contrast of the image is better when more sensors are used. Determination of the proper velocity limits is easy if we know the cut surface. In practice, we do not have this advantage. High contrast is the only guarantee of correct evaluation.
We did the evaluation under 7 different sensor number and 5 different cavity size conditions and the velocity difference between the velocity at the weakest intact spot and the velocity at the cavity area was measured. The results are shown in figure 4, as a surface graph. The graph provides the velocity difference as a function of the sensor number and the resolution (expressed as a percentage of the total cross-section.) 300 m/s velocity difference was selected arbitrarily as a reasonable difference from a practical point of view, and marked by a thick line. A relatively quick improvement is observed in the resolution to up to 12 sensors. Above 12, the improvement slows down.
The influence of the inversion method
In the first step, the reference velocity Vref is determined, by calculating the average of the line velocities of the neighbouring sensors. This velocity can be used as reference, because the outer part of the tree usually is intact. In the second step, every line velocity is divided by the reference velocity Vref. If this ratio is lower than 0.8, the program marks this line as a “defect line”. A spot is drawn at the intersecting points of every defect line, to produce the final image (Figure 5). A spot indicates defect in the tree at that place. The resolution of the image is directly determined by the number of sensors because a spot can only be drawn at the intersection of two lines.
We can rewrite (3) in matrix notation by defining the vectors s and t and the matrix M as follows:
Neglecting the Fermat principle, i.e. using a straight wave path, we got unusual results on a dry and cracked poplar disk. Reconstruction of the deep crack was not possible. The development of a bent wave path algorithm for detecting such defects is currently underway.
ConclusionsThe resolution of stress wave based acoustic tomography is influenced by the applied frequency, the number of sensors and the applied inversion technique. The theoretical limit of the resolution is the wave length of the stress wave. The calculated limit is 25 mm. Artificial cavities of various sizes were evaluated in a 34 cm diameter larch disk. Using 30 sensors, a 25 mm diameter circular cavity was successfully detected. When increasing the number of sensors in acoustic tomography, the resolution and the contrast of the image improves as well. The linear filtered backprojection technique provides a good result: the size and location of the circular defect is correct. Reconstruction of deep cracks is difficult by linear backprojection; a bent wave path approach is necessary.