·Home ·Table of Contents ·Methods and Instrumentation  Ultrasonic Tomography of closely spaced defects
S. K. Rathore, N. N. Kishore and P. Munshi,
Department of Mechanical Engineering
S. K. Singhal,
Department of Aerospace Engineering
Indian Institute of Technology, Kanpur  208 016 (India)
Email: skrathor@iitk.ac.in
Contact

Abstract
The ultrasonic wave undergoes refraction when it encounters an inhomogeneity in its path of propagation. This phenomenon becomes more complex if there are more than one inhomogeneity present in a solid object. The use of digital ray tracing method has facilitated the determination of the actual path of wave propagation between the source and receiver under such situation. By combining raytracing method with algebraic reconstruction techniques, it is possible to develop tomographic imaging procedures for refracting objects. In present work, a simulated study of ultrasonic tomography is carried out to realize the reconstruction if two closely spaced defect geometries exist in an object. Also, the effect of contrast variation on the accuracy of reconstruction is studied. The relative performance of two approaches i.e. straightray and curvedray approaches is presented. It is also attempted to establish the resolution or closeness of two defects so that these can be separated.
Introduction
In contrast with Xrays, tomographic imaging with ultrasound is made difficult by the fact that rays of sound energy do not necessarily travel along the straight line when an interface is encountered. This phenomenon of refraction while using ultrasound as a source has posed major challenges in the field of tomographic reconstruction. In acoustic tomography, the problem of refraction has necessitated some means to deal it. This has led to development of ray tracing method to find actual ray path of ultrasonic wave. However, the potential of these methods is yet to be exploited.
Kak [1979] reviewed developments in ultrasound CT and has presented the reconstruction of velocity and attenuation coefficient images in medical application. The effects of refraction were ignored as only soft tissue structures studied. Digital ray tracing algorithms have been developed by many workers to account for refraction in ultrasonic tomography. They have used Snell's law, Fermat principle or the eikonal equation in geometrical optics for it. A review of digital ray tracing in a general twodimensional refractive index field, is presented by Anderson and Kak (1982). The various aspects of numerical implementation like discretisation, interpolation schemes, step size, use of smoothing windows etc have been dealt within the above work. Bold and Birdsall (1986) have compared the different integration methods with regards to the three algorithms reported by Anderson and Kak. They have found that the fourth order methods give good combination of accuracy, simplicity and speed. Accuracy increases as order of integration increases and as step size decreases.
Denis et al (1995) have shown that the reconstruction quality can be improved significantly for refractive index variations of up to 10% using curvedray methods. Both simulation and experimental results are presented to verify the reduction of the refraction artifacts by this method.
In the present work, the tomographic reconstruction of the object containing two closely spaced defects is carried out. The distance between these two defects is varied so as to establish the minimum gap at which the two defects can be resolved. The raypaths are determined by the digital ray tracing. The contrast i.e. the relative difference in material velocities or refractive indices between the defect and base material, is varied within a range to study the reconstruction of two defects.
Methodology
Tomography is a reconstruction of a crosssection from the projection data. There are algorithms to accomplish the reconstruction from the limited projection data viz. ART (Algebraic reconstruction techniques) and MART (Multiplicative ART). The MART being the most efficient (Verhoeven, 1993 and Mishra et al, 1999), is chosen in which the image is updated at each iteration by multiplying the correction term as below.
where
and are updated and previous image vectors respectively,
l
is relaxation parameter,
is an intercept length of i^{th} ray with j^{th} pixel or an element of weight matrix,
and are measured and calculated projection data respectively.
The time of flight (TOF) i.e. the transit time from the source to receiver is taken as the projection data and the receiver is exactly opposite to the transmitter. The reconstruction gives slowness image (the inverse of velocity). Hence, any defect is characterized by the difference in velocity from the base material.
To account for raybending, it is necessary to find out the actual path of travel between the source and receiver, which is curved in the presence of refraction. Thus, the digital ray tracing is made use of in determining the exact weight matrix as required in the algorithm for curvedray approach. In straightray approach, the ray paths are assumed as straight line. By combining digital ray tracing algorithms with ART and MART, it will be possible to realise the shape and size of defect geometry to a greater accuracy. A code has been developed for ray tracing based on Fermat principle and validated by satisfying the characteristics of Maxwell's "fish eye" function.
Algorithm
The complete process of reconstruction involves many steps, which are presented below.
Step I: Find TOF data numerically using the weight matrix, which is obtained by performing ray tracing on simulated image.
Step II: For initial guess to next iteration, obtain tomographic image using MART by considering weight matrix obtained by straight ray approach.
Step III: Perform ray tracing on the image obtained from previous iteration and find weight matrix thereof.
Step IV: Obtain tomographic image using weight matrix as found in step III.
Step V: Repeat steps III & IV till errors are minimized.
The step II and V give the solution by straightray and curvedray approaches respectively. The performance of the reconstruction technique is evaluated both quantitatively and qualitatively. The error or performance index used in this work for comparison, is defined as:
Normalised RMS Error,
Problem Definition
The object to be reconstructed is considered to have two defect geometries of circular shape. Both the defects are of same size and the size of both the defects is kept fixed for all the cases. The three cases of different gap between the edges of defects 'b' i.e. 4, 8 and 12mm respectively are taken for the present study. The complete geometry of the object is shown in the figure 1.
Fig 1: Geometry of the object 
The base material of the object has longitudinal velocity of 2700 m/s and the velocity in the defect material is varied in the range from 0.5% to 50% of the base material.
Results and Discussion
There are two objectives of the present study. First is to establish minimum gap (resolution) between the two defects, which can be detected by ultrasonic tomography. Another being the determination of capability of curvedray method to detect the two defects for very low and very high velocity contrasts. The lowest and highest contrast taken are 0.5% and 50% respectively as it is assumed that real life situations are well within this limit.
In the tomographic reconstruction method, the relaxation parameter (l
) equal to 0.1 is found to be the best compromise between the convergence rate and reconstruction accuracy and is adopted for all the cases. The performance of straightray and curvedray approaches is evaluated qualitatively and quantitatively.
Straightray Approach
The straightray approach has given poor reconstruction for low velocity contrast (0.5%) and very high contrast (50%) even when the defects are situated 12mm apart. It is also found that the degradation of reconstruction quality starts above 15% contrast. Figure 2 shows the reconstruction for velocity contrast of 2% and 20%. A thin black circle depicts actual hole size and location in the images shown. It is evident from these images that the image quality at 20% contrast is inferior. As the contrast level is decreased below 2% or increased above 15%, the reconstruction quality diminishes and after a certain value, no solution is obtained. Also, the size of the reconstructed defect is more than actual one and the errors are higher in the straightray reconstruction. Hence, it can be said that when the two defects have the gap of the order of defect size, they can be located and separated just barely by straightray based ultrasonic tomography except for very low and high velocity contrast.
(a) Contrast 2%
 (b) Contrast 20%

Fig 2: Reconstructed Images from straightray method, for 12 mm gap 
When the gap between the defects is decreased to 8mm, it shows similar trends as that for 12mm gap but the reconstruction quality is poorer as can be seen from Fig. 3. Also, valid solution is not obtained for the contrast of 20%. Thus, it can be concluded that when the defects are closer, the range of contrast for which the reconstruction is feasible, becomes smaller. With further reduction of the gap to 4mm, it is not possible to make out anything from the reconstruction solution for the total contrast range considered. So, there is a limit of gap below which the straightray approach can not give any reconstruction solution. The solutions obtained by straightray approach are quantitatively inferior and sizing remains unresolved.
(a) For 8 mm gap
 (b) For 12 mm gap

Fig 3: Reconstructed Images from straightray method (Velocity Contrast 10%) 
Curvedray Approach
The curvedray approach is necessary for a more realistic case of ultrasonic tomography. This is adopted in the present work to achieve the reconstruction of higher accuracy. The significant improvement in the reconstruction quality is observed with this approach in comparison to straightray approach (Table 1).
 Velocity Contrast %age

0.5
 2
 5
 10
 15
 20
 50

Straightray method
 25.30
 13.30
 15.15
 15.17
 15.61
 15.59
 17.89

Curvedray method
 12.17
 9.85
 9.98
 9.19
 9.97
 10.49
 15.00

Table 1: Normalised RMS Error (EL2) for 12mm gap

It is found that at a given contrast, the shape and size of the defects in the reconstructed image improve as the gap between the two defects is increased from 4mm to 12mm. Figure 4 shows a good quality image at 10% contrast. From the comparison between the images in fig. 3(b) and 4, the improvement with curvedray method over straightray method is quite obvious as the RMS error (EL2) drops by ~6%. This clearly shows advantage of curvedray method in ultrasonic tomography. Besides, the reconstruction for the velocity contrast as low as 0.5% and as high as 50% have become possible. However, at 50% velocity contrast, the errors are high which are indicative of distortion in shape and size in the reconstructed image. Thus, for the contrast range of 0.5% to 20%, good quality reconstruction with regard to shape, size and location, are obtained if the gap is of the order of the size of defects.
Fig 4: Reconstruction with curved ray method, for 12mm gap (contrast 10%)

Fig 5: Variation of RMS error with velocity contrast, for different defect gaps

Figure 5 shows the rise in the error as the two defects come closer. It can also be seen that at high and low contrast, the error is higher for all the three cases which indicates poor reconstruction or no solution. When the gap is reduced to 8mm, the defects are not accurately located for high and low velocity contrast. If the two defects are brought further closer i.e. 4mm, then the contrast range giving good reconstruction further shrinks. The defects are separable for the contrast range from 2% to 15% (Fig. 6(a)) and whereas, figure 6(b) shows poor reconstruction quality with regards to size and location of the defects at 20% contrast. It is found that below this gap it is not possible to separate two defects whatever may be the velocity contrast. Thus, this study forms a basis for the detection limits for ultrasonic tomography.
(a) Contrast 2%
 (b) Contrast 20%

Fig 6: Reconstruction with curvedray method, for 4mm gap 
Conclusions
 Improvement in quality of reconstruction is achieved for all contrast levels with the curvedray approach.
 Curvedray Approach has made it possible the reconstruction at very low contrast and very high contrast when the gap is of order of the defect size or more.
 Two defects are separable for as low as 4mm gap for the velocity contrast in the range 2% to 15%. Below 4mm, the separation is not possible in the present case for any contrast level.
 Velocity contrast range giving good reconstruction shrinks as the gap between the defects reduces.
References
 Anderson A.H. and Kak A.C., Digital ray tracing in twodimensional refractive index fields, Jr. of the Acoustical Society of America, 72(5), 1982, 15931606
 Bold G.E.J. and Birdsall T.G., A topdown philosophy for accurate numerical ray tracing, Jr. of the Acoustical Society of America, 80(2), 1986, 656660
 Denis F., Basset O. and Gimenez G., Ultrasonic transmission tomography in refracting media: Reduction of refraction artifacts by curvedray techniques, IEEE transactions on Medical Imaging, 14(1), 1995, 173188
 Herman G.T., Image reconstruction from projections, Academic Press, New York, 1980
 Kak A.C., Computerized tomography with Xray, Emission and Ultrasound sources, Proceedings of the IEEE, 67(9), 1979, 12451275
 Mishra D., Muralidhar K., and Munshi P., A robust MART algorithm for tomographic applications, Numerical Heat Transfer, Part B35(4), 1999, 485506
 Verhoeven D., Multiplicative algebraic computed tomographic algorithms for the reconstruction of multidimensional interferometric data, Optical Engineering, 32(2), 1993, 410419