·Table of Contents ·Materials Characterization and testing | Defect Identification by Ultrasonic Tomography in PolymersS.K.SinghalH.B.Technological Institute, Kanpur S.K.Rathore, N.N.Kishore, N.G.R.Iyengar and P.Munshi Indian Institute of Technology, Kanpur-208016 India e-mail:sks@iitk.ac.in Contact |
In present work, an automated ultrasonic experimental set-up has been developed to record time of flight data in through transmission mode. The Time-of-flight data has been obtained for perspex samples containing artificially implanted defects of different shapes, namely, circular and elliptical. Reconstruction for both experimental and simulated data is presented with and without ray bending consideration. A significant quantitative and qualitative improvement was observed when ray bending was taken into account. Fairly good reconstruction was obtained for circular and elliptical defect in respect of size, shape and location. The study is also carried out for a symmetrically placed defect and a defect with eccentricity.
Key-words: Ultrasonic Computed Tomography,Digital Ray Tracing, Fermat Principle, MART
In acoustic tomography, one can use Rayliegh, Lamb, shear or longitudinal waves depending upon the application. But to reconstruct internal details of thick materials, only longitudinal and shear wave can be used. In the present investigation, longitudinal wave has been used. In this case, the most convenient acoustical parameter is the transit time from source to receiver. The reconstructed image is a distribution of acoustic slowness (the inverse of velocity) over an object cross section.
However, due to ray bending, ultrasonic tomography poses challenges. Kak [2] presented a detailed review on major development in imaging with Computed Tomography (CT). Censor [3] documented an article in the field of series expansion reconstruction method. CT has revolutionized X-ray imaging for medical purpose because it offers rapid, multiple plane views of area of interest. However, due to significant differences between X-ray propagation and acoustic propagation, (curved paths followed by acoustic wave in nonhomogeneous media) errors are introduced in the reconstruction. To take into account the effect of ray bending in ultrasonic, Anderson and Kak [4] have presented a review of digital ray tracing using Fermat principle in a two dimensional refractive index field. Denis et al [5] have shown that the reconstruction quality can be improved significantly for soft tissue using curved ray method. Industrial application of ultrasonic tomography is relatively sparse, with some success observed in defect imaging in variety of media including wood [6], SiO_{2} ingots for semiconductor applications [7] and geological media [8].
The objective of present work is to apply such techniques for NDE of polymers. In this work, the use of ultrasonic CT to characterize polymer properties is explored. Digital ray tracing has been used to incorporate the effect of curved ray path in ultrasonic CT. This has been applied on simulated and experimental data.
The performance of the tomographic reconstruction algorithms, ART (Algebraic Reconstruction Technique) and MART (multiplicative-ART) have been evaluated by Subbarao et al [9], in the context of interferometric tomography. MART was found to be superior to ART. Hence in this study, MART has been used .The difference between ART and MART algorithms lies in the manner in which the corrections are applied to the current field values. Corrections are additive in former and multiplicative in later.
where,
f_{i} is projection data (TOF)
f_{j} is image vector (slowness)
Two specimens were made of perspex of size 60mm x 60mm having a circular hole of 20mm and an eccentric elliptical hole. The holes were filled with polystyrene. Longitudinal velocity was found to be 2700m/s and 2400m/s respectively in perspex and polystyrene. To achieve normal incidence for 45^{0} angle of projections, a square block of 90 mm x 90 mm and 12.5 mm thick with a square hole of 60 mm x 60 mm oriented at 45^{0} was made and the square specimen was encapsulated in it. Necessary corrections were applied for water and outer block material.
where,
fi is the measured projection (TOF)
is the calculated projection (TOF)
l is relaxation parameter,
The performance of reconstruction technique is evaluated both quantitatively and qualitatively. The average EL1 and RMS EL2 error are used to measure performance and defined as:
The digital ray tracing employing Fermat principle is made use of in determining the exact weight matrix W_{ij} as required in MART algorithm. To accomplish the reconstruction, following steps are involved.
Step 1: Find TOF data numerically using the weight matrix, which is obtained by performing ray tracing on simulated image.
Step 2: For initial guess to next iteration, obtain tomographic image using MART by considering weight matrix obtained by straight ray approach.
Step 3: Perform ray tracing on the image obtained from pervious iteration and find weight matrix thereof.
Step 4: Obtain tomgraphic image using weight matrix as found in step 3.
Step 5: Repeat step 3 and 4 till errors are minimized
Reconstruction of circular flaw
In case of simulated study, the time of flight (TOF) is numerically calculated on the basis of actual path of the wave propagation using digital ray tracing for each ray of all projection angles. First, the reconstruction is carried out by considering weight matrix W_{ij}, obtained from straight-line assumption and the same is shown in Fig. 1(a). The size of the reconstructed flaw is more than actual one, which is represented by black circle. The average error EL1 and RMS EL2 error are also high i.e. 0.61 ´10^{-3} and 18.52% respectively. It is due to bending of rays near the edge of flaw and the same phenomenon is not accounted in straight-line assumption. This incompatibility between the straight-line ray path and actual ray path based TOF becomes the source of noise in TOF data and hence, gives rise to higher errors.
(a) With straight ray | (b) With ray tracing |
Fig 1: Reconstruction of circular flaw from simulation study |
When the reconstruction is performed with curved ray-path based weight matrix W_{ij}, then the average error EL1 drops by 0.1 ´10^{-3}and RMS error drops by ~6%. This is reflected in realization of reconstructed flaw size and slowness value inside flaw is close to actual one (Fig.1(b)). The artifacts are also reduced when ray tracing is considered.
Fig 2 (a) and (b) depicts the reconstructed images for the experimental TOF data. It can be seen that the reconstructed flaw is smaller than actual one. It is observed in TOF data that there is some zone in the receiving side in which no ray arrives through the flaw because of ray bending and this zone is around the flaw edge. However, a ray going around the insert may reach receiving end [6]. This causes spurious projection data in the neighborhood of the flaw edge, hence the noise in projection, which is responsible for smaller reconstructed flaw size. The improvement in the reconstructed image with ray-tracing in terms of artifacts is evident from the figure.
(a) With straight ray | (b) With ray-tracing | |
Fig 2: Reconstruction of circular flaw from experimental data. | Fig 3: Reconstruction of eccentric elliptical flaw from experimental data |
Reconstruction of Elliptical Flaw
The flaw of elliptical shape located with some eccentricity is also investigated with experimental TOF. In contrast to circular flaw, the size realization is better in elliptical flaw. It can be attributed to the fact that the flaw boundary is less oblique to incident wave whereas in case of circular flaw, it is highly oblique to the incident wave near edge for all the angles of projection. This results into less bending near edges in elliptic flaw. Also, the reconstruction with eccentricity is possible (Fig.3).
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