NDT.net • Apr 2006 • Vol. 11 No.4

Resolution of Stress Wave Based Acoustic Tomography

F. Divos
University of West Hungary, Sopron, E-mail: divos@fmk.nyme.hu
P.Divos
Technical University Budapest, E-Mail: pdivos@gmail.com

14th International Symposium on Nondestructive Testing of Wood
May 2005, University of Applied Sciences, Germany, Eberswalde.
Published by Shaker Verlag (ISBN 3-8322-3949-9).


Abstract

The 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.

Introduction

Why 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 [1]. 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 frequency

From physics we know that the resolution of optical systems is in the range of the wavelength l of the light [2]. 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).

l = V/f         (1)

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 [3]. 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.


Figure 1. Oscilloscope image of the beginning of a stress wave signal in larch. The distance between the source and the receiver is 7 cm. Vertical and horizontal axes are amplitude and time respectively, and the grid size is 50µs. A 40 kHz signal is superimposed on the 2 kHz signal.

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 [4], 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 theoretical calculation of the resolution is difficult. Resolution computations can actually be significantly more difficult than computing the image itself [5]. Instead of a computation, we selected an experimental method for the resolution determination. For the resolution experiment 50µs we used a 34 cm dia. larch disk. A 10 mm circular artificial cavity was created around the pith, and then progressively enlarged to 25, 47, 76 and 100 mm diameters. The detection of each cavity was attempted using 6, 8, 10,12,16, 24 and 30 sensors.

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.

Figure 2a: Acoustic tomographic image of a 34 cm dia. spruce disk with a 2.5 cm cavity (0.5% of the cross-section area.)

Figure 2b: Acoustic tomographic image of a 34 cm dia. spruce disk with a 4.7 cm cavity (2% of the cross-section area.)

Figure 2c: Acoustic tomographic image of a 34 cm dia. spruce disk with a 7.6 cm cavity (5% of the cross-section area.)

Figure 2d: Acoustic tomographic image of a 34 cm dia. spruce disk with a 10.7 cm cavity (10% of the cross-section area.)


Figure 3. Acoustic tomographic images of the spruce disk with a 4.7 cm cavity using 30, 24, 16, 12, 8 and 6 sensors. The "low" and "high" velocity limits (m/s) are shown below the image.


Figure 4. The difference between the low and high velocities as a function of the sensor number and the resolution (expressed as a percentage of the total cross-section)

The influence of the inversion method
Relative line velocity decrease method.


Figure 5. Example image.

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.

Cell-based backprojection
The theoretical background of this algorithm comes from Berryman [6]. Let ti be the time measured between the ith sensor pairs. Let Ca and Cb be the positions of the ath and bth sensors. Let s(x,y) be the slowness (the reciprocal value of velocity) of wood at the (x,y) point. So we can write:

The goal of tomography is to determine the s(x,y) function and line integral along the wave path. Fermat’s law states that the real wave path between two sensors is the one of least overall travel time. Linear tomography – used in our evaluation – neglects this and approximates the wave path with a straight line. If we split the polygon of sensors into cells, and we assume that the slowness is constant in each cell, we can write:
where sj means the slowness of the jth cell and lij means the length of the ith wave path through the jth cell. Wave path lengths lij are zero for most cells because a given wave path will typically intersect only a few of the cells.

We can rewrite (3) in matrix notation by defining the vectors s and t and the matrix M as follows:

So for given cells and slowness values we can calculate the travel times in our model. The goal is to determine vector s so that vector t gets as close as possible to the measured travel time data. The method we used is called elementary backprojection approximation. We determine sj by the following formula:
where sgn(x) = 1 if x>0 and sgn(x) = 0 if x = 0 and Si is the ith line slowness. This means that the slowness of the jth cell equals to the average of the line slowness values of lines intersecting the cell. We split the polygon of sensors into square cells. The resolution is determined by the number of cells. We can raise the number of cells as long as every cell has at least one line intersecting it. This is the theoretical limit. See an example image in figure 6.


Figure 6. Cell-based backprojection

Filtered backprojection
The theoretical background comes from Deans [7]. This method tries to restore the s(x,y) function defined in (3). This method has been invented by J. Radon in 1917. It has been used in the first computer tomographs. The Radon transform (R) of f(x,y) is defined by the following formula:

where the line integral is along a line L whose distance from the origin is p and f is the angle between the line and the y axis. So this method is also linear. Note the similarity between (2) and (6). The Radon transform of the slowness function provides the travel-times. (p,f) denotes the measured time on the (p,f) line and f(x,y) is the slowness function s(x,y). The goal is to perform a so called Inverse Radon transform to obtain the slowness function from the travel-time data. There is an important mathematical theorem in this field:
This means that the radon transform can be substituted by a 2-dimensional Fourier transform (I2) and a 1-dimensional Inverse Fourier transform ( I1-1). The filtered backprojection algorithm has been developed based on this theorem. First we have to calculate the Fourier transform of (p,f) at a given f:
After this we have to calculate the Inverse Fourier transform of abs(q)Ff(q)
These two steps are called filtering. We have to do this step for several f values. The last step is called backprojection:
This method delivers an arbitrary image resolution because the final step (10) can be calculated at any (x,y) point, but it doesn’t mean that the resolution is unlimited. The limit is because of the finite number of the lines along which (p,f) i.e. the travel-time data is known. So below a certain resolution, the image will not contain any new information.

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.


Figure 7. The reconstruction failure at a deep crack using linear filtered backprojection.

Conclusions

The 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.

References

  1. Wang X., Divos F., Pilon C, Brashaw B.K., Ross R.J., Pellerin R.F.: Assessment of Decay in Standing Timber Using Stress Wave Timing Nondestructive Evaluation Tool, Forest Products Laboratory, General Technical Reports FPL-GTR-147, 2004
  2. Feynman R. P., Leighton R. B., Sands M.: Feynman Lectures on Physics, Section 27.7, ISBN: 0-201-50064-7, Publisher: Benjamin Cummings, 1991,
  3. Pellerin R.F., Ross R.J.: Nondestructive Evaluation of Wood, Section 3. ISBN: 1-892529-26- 2, Forest Products Society, 2002
  4. Bucur V.: Ultrasonic techniques for nondestructive testing of standing trees. Ultrasonics.;43(4):237-9, Feb 2005.
  5. Berryman J. G.: Computing resolution in acoustic tomography, The Journal of the Acoustical Society of America, Volume 107, Issue 5, p. 2805-- May 2000.
  6. Berryman, J. G.: lecture notes on Nonlinear Inversion and Tomography, Massachusetts Institute of Technology 1990.
  7. Deans, S. R.: The Radon Transform and Some of Its Applications, Wiley 1983.

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