|NDT.net - December 2002, Vol. 7 No.12|
The problem of image reconstruction from incomplete X-ray data, applied to in-situ pipes wall thickness assessment, is considered, its main critical points being the following:
There are many practical cases, in which obtained projected x-ray image is blurry, the x-ray cross section is fuzzy and the fuzziness is non uniform along the coordinate. The classical example is x-ray inspection of pipes, which are used in many industries like chemical, power engineering, etc.. It appears a practical requirement for accurate wall thickness measurements of corroded and depleted pipes in service. But usually for those pipes, commonly covered by isolation and filled with a process liquid, the access to the object and the applicability of complex equipment, used for regular Computer Tomography (CT), are strongly limited as to accessibility, cost and labor inputs. A reasonable formulation of the problem could guess the development of the technique for pipes inner surface image restoration in the following conditions:
Mathematically that causes an induced uncertainty in the original image under restoration given particular data. This situation requires application of some non trivial approach to the reconstruction procedure like, e.g., recently developed Multi Step Image Restoration Technique using Bayesian image reconstruction at the final step [1-4].
The present article exhibits the developed technique being evaluated using simulated and experimental data. In all cases at the end from only two to five projections, exposed within an observation angle 60° - 90°, were made for data acquisition. 2D and 3D cases for image reconstruction were considered. Both regular x-ray films and imaging plates combined with corresponding digitization procedures were explored in the experiments. All computations are supported by PC platforms. Finally the images of restored pipes were compared with the originally exposed to estimate the reconstruction errors.
The Bayesian image restoration technique with prior knowledge is applied to the inversion of projected data, the corresponding procedure being described below. Two approaches are applied to the sampling of the object space. In the first one, dealing with the 2D cross section, the object space is discretized on the uniform grid in polar coordinates yielding quasi-rectangular cells in this coordinate system (voxel model). In the second one the object is separated in space only by its surface coordinates in the Cartesian or polar coordinate systems.
As the problem is binary, i.e. there are only two types of substance to be considered which are air and metal, in the first approach each cell i holds only one of two possible values of the attenuation coefficient m i. The process model, given by the absorption law, can be represented in discrete form as follows
where ln,ij is the path length of the ray in the cell j, which is projected onto pixel i in projection n.
The prior functional applied in this work is taken as the square of the first derivative of the attenuation coefficient in voxel j along the basic coordinate directions with respect to the center of each voxel. This functional can be written in the form
where the index a designates cells adjacent to the voxel j so that m j,a gives the attenuation coefficient in that voxel. Additional constraints are introduced into the a priori restrictions in order to create the integral structure of the inner surface and to allow internal cavities in a pipe wall. The resulting solution for the stated reconstruction problem, i.e. the determination of the unknown distribution of the attenuation coefficient, is given by the following optimization problem:
The minimization is carried out by applying the original algorithm based on the gradient descent. The input data for the reconstruction procedure (3) are the ray sums pmn,i which are obtained from the scanned data Dmn,i. From physical point of view the relation between D and P is similar to the relation between D and I, where I gives the intensity of the incident radiation, which is linear
with I = Io exp(-ml) . Generally, the coefficients A and B can not be calculated theoretically because of the complexity of the physical processes taking place during x-ray propagation in matter and x-ray image detection, which includes the film development and the digitization procedure. It is found that in case of pipes the process model, which is necessary to calculate the ray sums pmn,i, from the data, and finally the relation between D and pmn,i, can be approximated with sufficient accuracy by
with xn,i giving the x-coordinate of pixel i in projection n, xc being the x-coordinate of theprojection of the pipe center on the detector plane, and a, b, c representing unknown parameters, which have to be determined by means of available optimization procedure. The third summand in Equation (5) is introduced to compensate the observed asymmetry of the X-ray image. While calculating the ray sums pmn,i, the attenuation coefficient m is calculated rigorously taking into account the phenomenon of beam hardening.
The parameters a, b, and c are calculated from the mean square optimization of the experimental data Dn,i, provided by exposing the corresponding perfect pipe: the discrepancy between the measured D and the one obtained from Equation (5) is minimized.
The next step is to take into account the scattered radiation, which contribution to the projected image can be substantial . The following considerations have been used for the present elaboration:
where a and b unknown constants, calculated using data from the perfect pipe. Background radiation part is approximated by linear function:
where g - unknown constant, calculated using data from the perfect pipe.
The next paragraphs illustrate the application of the technique for 2D and 3D pipes image reconstruction from limited data, and consequent pipes wall thickness assessment.
We shall separately consider two most critical reasons for the data uncertainty in pipes:
1. Non uniformity of x-ray track length
In case of non uniformity of x-ray track length the apparent attenuation coefficient can change dramatically along the coordinate and cause pipes projected image blurring. Given obtained fuzzy data, simulated with attenuation law in the form (1), the quality of restored image, using usually applied least square minimization technique (first term in eq.(3)), is corrupted due to the enormous mean square discrepancy between measured and simulated data. Although integrating a priori knowledge in the reconstruction algorithm in the form (3) positively influences the quality of the restored image , it still appears to be useless for wall thickness assessment. The above mentioned conditions are realized in the data acquisition system, which geometrical setup is shown in Figure 1. The pipe phantom with two defects was used for the validation of the image restoration technique. Depending upon the pipe diameter and thickness the ratio of the maximum/minimum track length can reach the value of one order and more, causing enormous blurring. S0, S4 - are successive positions of the x-ray source.
|Fig 1: Geometrical setup for data acquisition.|
|Fig 2: Exploited 2D cross section extracted from projection (a) together with optimized data set region (b) for reconstruction purposes.|
In our experiments the exposed pipe had a radius of 57 mm and a nominal wall thickness of 6 mm. The machined notches 1 and 2 had the width of 6 mm and a depth of 3 mm and 1 mm, respectively. For the exposure a 0.8 mm focal spot X-ray tube with a voltage of 200 kV and a current of 2 mA was used for the 5 projections within the observation angle between extreme sources positions of 90°. The exploited 2D cross section, extracted from the image, is shown in Figure 2a. Large difference of track lengths in the metal (the longest path length belongs to the ray tangential to inner pipe surface and is close to 51 mm, the smallest is close to zero near the outer tangent and 9mm in the center), and small tube voltage used for the exposure resulted in image blurring effect in the pipe region bounded by the outer and inner tangents.
One of the key point of the proposed technique is the procedure of excluding from the iteration step of those experimental data, which are located in the area of large blurring effect. In our example only the data set similar to that shown in Figure 2b were used for the reconstruction. The optimal cut-off was estimated by minimizing the reconstruction error at the level of about 20 % of the total pipe projection. It reduces the amount of input data and essentially improves the quality of restoration. The result of image restoration is shown in the Figure 3b which can be compared with the actual pipe image given in Figure 3a. The marginal reconstruction error was less than 0.16 mm.
|Fig 3: Comparison of reconstruction result (b) with original image (a).|
2. Compensation of scattered radiation
The idea of the compensation of a scattered and background radiation from an x-ray image, described above, assumes the following steps:
where superscripts mean calculated and measured values respectively.
Figure 4 illustrates all five profiles which are logically follow the procedure described above. The final profile used for reconstruction is shown by curve 5.
|Fig 4: Original x-ray profiles for the pipe (100 mm diameter) defected by notch (1), simulated absorbed radiation (2), extracted scattered radiation and background (3), simulated scattered radiation and background (4), profile used for restoration (5).|
The development profile of the reconstructed pipes inner surface is shown in Figure 5.
|Fig 5: The development profile of the reconstructed pipes inner surface. The reconstruction was done after extraction of the scattered and background radiation. The mean square error of reconstruction is 291mk.|
The technique for wall thickness measurements using 3D tomographic pipes image,
reconstructed from limited projections and observation angle, was first described in . The
advantage has been taken from introduced elaboration, which assumed the presentation of the
defected pipes image with a hull model. Due to the model the reconstructed object is
represented by two hulls: outer and inner. The outer hull is assumed to be a cylinder with
known radius. The inner surface is considered as a set of elementary surface elements
Su : u =1,L, U, S =YSu, Su1 I Su2 = 0, u1šu2 given by triangle with vertexes, defining the nodes of a discrete grid. The partition of the surface is implemented in such a way, that the osculating sides of adjacent triangles and their common vertexes belong only to one of them.
The forward operator sequentially searches through all inner hull elements for:
represents the discrepancy between the current inner surface S , which is given by a set of
discrete grid knots
Mi, i = 1,L, l and the inner surface, being the best fit to measured data. The reconstruction results are shown in Figure 6.
|Fig 6: Reconstruction results: (a) gray value representation of inner pipe surface together with circumferencial wall thickness profile 1 and profile 2 along pipe axis.|
Investigating the results, shown in the wall thickness profiles 1 and 2, it turns out that the average pipe wall thickness is accurately restored with an error, which does not exceed 200mm. This value holds for smooth wall thickness variations as known from corrosion and erosion. The accuracy of the reconstruction of the notches is smaller and the error is estimated at the level of about 500mm. It is clear that the proposed algorithm can not restore through-wall holes (see Figure 6b) due to the applied surface representation in the reconstruction algorithm. Finally it turns out that the discussed reconstruction algorithm is capable to monitor wall thickness variations in pipes with an accuracy sufficient to meet practical requirements. This technique overcomes the major restrictions of other radiographic methods which usually allow measurements only at specific positions.
The proposed 2 and 3 dimensional tomographic procedures can be applied to the wall thickness determination for pipes. For this purpose 3-5 projections within an observation angle of 90° are sufficient. The measurement accuracy that is reached in the presented examples can be interpreted as follows: regions with smooth and/or small wall thickness variation, as usual for corrosion and erosion, can be reconstructed with measurement errors less than about 400mm. In regions with sharp and/or large wall thickness variations the reconstruction error is increased to about 1 mm as shown for notches and flat bottom holes, which were used as phantoms for defects like corrosion pits. It is found that the reached accuracy of restoration is mainly influenced by the following factors:
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