A schematic view of the data acquisition system is shown in Fig. 1. Here S is the ... applied to the inversion of the measured data as briefly described below. .... below. The pipe has a radius of 57 mm and a nominal wall thickness ... image, which is observed in the pipe region bounded by the outer and inner tangents. Due.
TOMOGRAPHIC RECONSTRUCTION AND WALL THICKNESS MEASUREMENT WITH SPECIAL ATTENTION TO SCHATTERED RADIATION FOR IN SERVICE INSPECTION OF PIPES V.Vengrinovich1, Y.Denkevich1,S.Zolotarev1, G.-R.Tillack2, and U. Zscherpel2 1
Institute of Applied Physics, Akademicheskaya str.16, 220072, Minsk, Belarus.
2
Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, 12205,
Berlin, Germany.
1. Introduction An accurate assessment of the pipe wall thickness, decreased by metal loss due to corrosion, cavitation, erosion, etc., is an important task in various industrial sectors like chemical, petrochemical or power engineering to ensure the safe operation of the constructions and to prevent hazards or damages. Accordingly several non-destructive testing (NDT) techniques have been developed allowing wall thickness assessment and flaw detection over the years. The techniques that are in use for this purpose today, like eddy-current, magnetic, ultrasonic, and X-ray techniques, are summarized in Tab. 1. Table 1: NDT techniques for wall thickness assessment and flaw detection. No
Technique
1 2
Conventional Eddy-Current Rotating and Multi Coil Probes
3
Remote Field Eddy-Current Technique
Testable Materials Conductor Conductor
Refer. [1,2]
Detected Defects Pits, cracks, wall loss Cracks, surface breaking
[2-4]
4
Magnetic Flux Leakage
5
Ultrasonic IRIS
Thick Ferromagnetic a Thick Ferromagnetic a
Wall loss [2,4-6] [2]
Pits and wall loss
[2,4]
Large pits and wall loss
[2,4,7] [4]
Large pits and wall loss Inner diameter discontinuity. Tangential wall loss estimate Pits, cracks
6 7
Ultrasonic Lamb Waves Laser Optics
No limitations No limitations No limitations
8
Digital Radiography
No limitations
[8]
9
Planar Tomography
No limitations
[9]
1
All techniques reviewed in Tab. 1, except those based on radiography (8,9), apply sensors which either are inserted in the inspected pipe or require direct contact with inner or outer surface of the pipe (like 5 and 6). X-ray tomographic imaging of both the outer and the inner pipe surface could yield a non-contact and non-invasive solution of the stated problem, but regular CT is not applicable due to the restricted access to the pipe. Moreover, testing of insulated pipe components is possible today only with the help of radiographic techniques and in some cases by ultrasonic applications. For radiographic techniques film and digital detectors like imaging plates or flat panel detectors are available for image recording. For in-field applications gamma sources like Ir-192 are commonly in use for pipes exposure because of the limitations of X-ray energies and wall thickness range. The aim of this article is to discuss a new image reconstruction technique, based on the principles of the multi-step tomography (MST) [10-12] given a limited number of projections and restricted access for observation for both 2D and 3D applications. The MST as described in [10-12] cannot directly be applied for pipe wall thickness assessment because of the inadequate description of the X- ray propagation by the attenuation law. On one hand the range of penetrated wall thickness by X-rays or grays in the pipe in the primary beam direction is large requiring a large dynamic range of the detector system. As a result a strong beam hardening is observed. On the other hand a simple scattering model like the built-up factor model assuming a constant contribution of the scattered radiation to the projection image is not valid any more. A single humped distribution of Gaussian-type or Lorentz-type for the scattered radiation is found in this case. Additionally other concomitant factors influencing the quality of the projection data have to be considered like the edge and image blurring, and geometrical unsharpeness. Other restrictions appear from economic demands like minimizing the number of X-ray exposures within a restricted observation angle, which is usually less then 90°. A schematic view of the data acquisition system is shown in Fig. 1. Here S is the radiation source, D represents the detector, and P gives the inspected part of the pipe. T denotes the trajectory for the motion of the source.
2
Fig. 2 shows a 2D cross section of the 3D data acquisition system from Fig. 1.
The pipe under test contains two internal notches at the bottom (3x6 mm) and from the left side (1.5x6 mm). The intensity profile of the transmitted radiation in the detector plane is shown in Fig. 3a. The form of this profile is mainly given by the absorption of the radiation for unknown penetration length. Neglecting in a first step the contribution of scattered radiation the corresponding process model, applied to calculate that profile for pipe wall thickness estimation, is given by the linear absorption law: I a = I 0 exp[- m pd p ( x )]
(1)
where dp(x) gives the ray path in the pipe, x the coordinate transversal to pipe axis, I the radiation intensity, I0 the source intensity, and mp the attenuation coefficient.
3
The simple inversion of eq. (1) would allow the direct calculation of the ray path in the material if the attenuation coefficient mp, the intensity of the source I0, the noise level, and the influence of concomitant effects is known. In practice this information is not fully accessible. The techniques introduced in [13] can access the tangential wall thickness to the best accuracy of about 0.2 mm, excluding the range of double wall thickness path in the central part of the profile, where it is not possible to distinguishing between the two walls with this method from one exposure. Despite of this disadvantage the technique is widely used in the industry. Accuracy problems appear while applying this technique to access the tangential wall thickness with increasing wall thickness. They are related to the accuracy of the x-ray propagation model, approximated by eq. (1). It is a well-known fact that the chosen process model is the key point for data inversion. The real process model is much more complicated than the simplification made here. As a result the real measured projection data originate many uncertainties in the wall thickness reconstruction from fuzzy measured data. One possibility to overcome this problem is the application of conventional Computer Tomography to estimate the wall thickness from complete 3D reconstruction. For tangential wall thickness measurements a modified CT algorithm can be used [14]. This paper discusses a different technique for image restoration of pipes from a limited number of noisy radiographic projections for limited observation angle, used for wall thickness measurements. For this purpose the earlier developed method of Multi Step Image Reconstruction (MSR) [10-12] is applied to the data after smart raw data preprocessing. As the quality of the MSR technique strongly depends on the adequacy of the process model, an improved model is introduced considering effects concomitant to the absorption in the form of eq. (1). Those effects are discussed in more detail after the description of the mathematical concept for image restoration. 2. 2D Bayesian Pipe’s Image Reconstruction neglecting scattering effect The Bayesian image reconstruction technique with prior knowledge [10-12] is applied to the inversion of the measured data as briefly described below. The object space
4
is sampled on the uniform grid in polar coordinates yielding rectangular cells in this coordinate system. As the reconstruction problem is binary, i.e. only two substances have to be considered which are air and metal, each cell i holds only one of two possible values of the attenuation coefficient mi. The process model, given by the absorption law (1), can be represented in discrete form as follows
Pn,i = å m j ln,ij
(2)
i
where ln ,ij is the path length of the ray in the cell j, which is projected onto pixel i in projection n. Among many investigated the prior functional (see [10-12]), 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, is applied. This functional can be written as
B( m ) = åå ( m j - m j ,a ) 2 j
(3)
a
where the index a marks cells adjacent to the voxel j such that m j,a gives the attenuation coefficient in that voxel. Additional constraints are introduced into the prior restrictions in order to access the integral structure of the inner pipe surface and to allow internal cavities in the 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: 2 ìï æ ü ö 2ï ~ m m = arg min íå çç å m j ln,ij - pn ,i ÷÷ + a åå (m j - m j ,a ) ý . j a ø ïî n ,i è j ïþ
(4)
Applying the original algorithm based on the gradient descent carries out the minimization. The input data for the reconstruction procedure (3) are the ray sums pnm,i , which are obtained from the digitized data Dnm,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 assumed to be linear: D = A + BI ,
(5)
5
with I = I 0 exp( - mp ) . In general, the coefficients A and B can not be calculated theoretically because of the complexity of the physical processes taking place during Xray 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 pnm,i from the data, and finally the relation between D and pnm,i can be approximated with sufficient accuracy by:
Dn ,i = a + b exp (- pnm,i ) + c ( xn ,i - xc )
(6)
with xn,i giving the x-coordinate of pixel i in projection n, хс being the х-coordinate of the projection 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 eq. (6) is introduced to compensate the observed asymmetry of the X-ray image. While calculating the ray sums pnm,i the attenuation coefficient m is calculated taking into account the phenomenon of beam hardening. The parameters a, b, and c are determined from the mean square optimization of the experimental data Dn,i, provided by exposing the corresponding ideal pipe, the mean square difference between the measured D and the projection data obtained from eq. (5) is minimized. A has to be considered, if the dynamic range of the detector does not adapt to the large difference in ray path in metal for both the tangential and the central part of the inspected pipe. Usually these paths can differ with the factor of about 3 to 20 depending upon pipe diameter and wall thickness. It turns out that the coefficients can not be determined from the complete data set using eq. (6) because of data corruption within the tangential region in this case. This corruption can be observed as a strong edge smoothing of the profile as shown in fig. 3.
6
But the proposed technique works rather successful if this region is not taken into consideration for the determination of the coefficients in eq. (6). The described technique is illustrated by the following example. The geometrical setup of the data acquisition system, which is similar to the setup usually applied for pipe wall thickness evaluation, is shown in Figure 2. In our case the experimental data for the validation of the applied 2D image restoration technique have been measured under the conditions described below. The pipe has a radius of 57 mm and a nominal wall thickness of 6 mm. The machined notches 1 and 2 have a width of 6 mm and a depth of 3 mm and 1 mm, respectively (compare figs.2 and 4a). A X-ray tube with 0.8 mm focal spot, a voltage of 200 kV, and a current of 2 mA was used for the exposure to gain the 5 symmetrical arranged projections within an observation angle of 90°. The cone beam xray shots were taken with 1.5X magnification and 814 mm film detector distance (FFD). The Kodak MX film with lead screens was used as detector. A laser scanner with pixel size of 50 mm and a dynamic range of 12 bit was used to digitize these films. One of the received 2D cross sections extracted from the image is shown in Figure 3a. A large difference of ray path 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 the small tube voltage used for the exposure yield the bluring in the image, which is observed in the pipe region bounded by the outer and inner tangents. Due to this and the effect of X-ray scattering, the process model applied for the reconstruction task, given by the attenuation law, did not provide satisfactory agreement between the
7
Table 2. Mean squared errors of reconstruction for notch 1 (depth 3 mm, width 6 mm), notch 2 (depth 1 mm, width 6 mm), and the wall thickness. marginal reconstruction error [mm] 0.16
reconstruction error notch 1 [mm] 0.79
reconstruction error notch 2 [mm] 0.31
reconstruction error wall thickness [mm] 0.12
calculated and the measured data, which strongly influenced the image restoration quality. Hence the experimental data in the area of large disagreement between model and experiment were omitted and only the data set similar to that shown in fig. 3b were considered for the reconstruction. The optimal cut-off was estimated by minimizing the reconstruction error. It was estimated that a section of about 20 % of the total pipe projection should not be used for the reconstruction task. It reduces the amount of input data and essentially improves the quality of restoration. But also after this preprocessing significant discrepancy between the experimental and calculated data remain (compare Figure 4) due to many inherent uncertainties in the process model.
The result of image restoration is shown in fig. 4b, which can be compared with the original pipe image given in fig 4a. The absolute mean square errors of reconstruction are shown in Table 2.
8
3. Unconstrained 3D Image Reconstruction using a surface representation of the pipe Consider the nonlinear operator equation
A( R( M )) = F
(7)
where F is the set of measured data for all N projections: F = { f nm }; n = 1,L, N with the measured data f nm ( p ) in projection n, { p nj : j = 1, L, J n } the pixel position in projection n, J = å n J n the total number of pixels at all projections, R(M ) the radius vector of the current point of the inner surface, M ( x, y , z ) : {M i : i = 1,L, I } the location of inner surface grid knots. The operator A describes the correlation between the inner surface S and the optical density at the projection given by the attenuation law. The geometrical setup of the data acquisition system is shown in fig. 5.
Two hulls represent the reconstructed object: 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 S u : u = 1,L,U , S = U Su , Su1 I Su2 = 0, u1 ¹ u2 given by triangle with vertexes defining the nodes of a discrete grid. Let assign the projection of
9
the surface element S u in the n-th detector plane as s un . The partition of the surface is implemented such that the osculating sides of adjacent triangles and their common vertexes belong only to one of the vertexes. The direct operator sequentially searches through all inner hull elements for: (a) determination of the projection of triangle Su on the current detector plane (b) definition of triangle boundaries (c) selection of all pixels p nj : j = 1,L , J n with the center located inside the triangle s un or on its boundaries (d) search for the cross points of the rays, passing through the inner and boundary pixels of the triangle s un , with the inner and outer object surface. The ray sums f nc ( p ) for pixel p nj : j = 1,L , J n , defining the corresponding ray, are calculated as the distances between corresponding pairs of points. Hence the ray sums can be determined after calculating the forward operator A( R( M )) n = f nc ( p ), n = 1,L, N . The functional pJnn
pJnn
d n (R( M )) = å f ( p ) - f ( p ) / å f nm ( p ) c n
p1n
m n
(8)
p1n
represents the difference between the current inner surface S , which is given by a set of discrete grid knots M i , i = 1,L, I , and the inner surface, being the best fit to measured data. The iteration procedure for reconstructing the inner surface is implemented as follows: All N rays are developed through every knot M i , which has an intersection with the corresponding detector planes in the pixels p nj* , n = 1, L, N . For those pixels the differences between the measured and the calculated values of the ray sums
Df n ( p ) = f nc ( p ) - f nm ( p ) are calculated. From this result the weighed mean deviation S( M i ) is determined
S(M i ) = å Df n ( p ) ´ cos(y in ) / N
(9)
n
10
where y in represents the angle between the projection of the ray, intersecting the given knot M i , and the corresponding source position Jn with the radius vector R( M i ) . Finally the corresponding displacement of the node M i is given by
h (M i ) = l( 0) ´ S( M i ), 0 < l( 0) < 1,
(10)
where l( 0) is the parameter of relaxation for the first iteration. The displacement takes place along the direction perpendicular to the line connecting the node M i with the coordinate axis. The geometrical setup for the data acquisition, which is used for the validation of the described reconstruction technique, is shown in fig. 5. The specimen represents a complex pipe structure built from two half-shells with 2 horizontal and 5 vertical holes, one circumferencial notch and four axial wires with diameters from 0.8 to 1.6 mm. For the exposure an Ir-192 source with a focus of 3 mm is used. A total of five projections are recorded from the five source positions shown in fig. 5, which are located within an angle of 90° choosing steps of 22.5°. Imaging plates are used for image recording with pixel size of 56 mm and a dynamic range of 12 bit. The projections are similar to that shown in Figure 6.
The complete 3-D image of the object is reconstructed on standard PC hardware. The result of reconstruction is shown in fig. 7 together with the profiles marked in the image.
11
Investigating the results shown in the wall thickness profiles 1 and 2 (compare fig. 7b and c) it follows that the average pipe wall thickness is restored with high accuracy. The estimated error of restoration does not exceed 200 mm. This value holds for smooth wall thickness variations as known from corrosion and erosion. The accuracy of the reconstruction of the notches is smaller. The error is estimated at the level of about 500 mm. It is understandable that the proposed algorithm can not restore through-wall holes (see fig 7b) due to the applied surface representation in the reconstruction algorithm. But this was not the focus task of the investigation. Finally it turns out that the discussed reconstruction algorithm is capable to monitor wall thickness variations in pipes with accuracy sufficient to meet practical requirements. This technique overcomes the major restrictions of other radiographic methods, which usually allow only measurements at specific positions but do not access the minimal wall thickness on the circumference of the pipe.
12
4. 2D Bayesian Pipe’s Image Reconstruction considering scattering effect Consider a pipe under in service condition, i.e. the pipe is covered by insulation and filled with a work liquid. Supposing all absorbed radiation components are multiplicative for the filled and insulated pipe:
[
],
I a = I 0 exp - (m p l p + m i li + m l ll )
(11)
where Ia is the primary intensity, li(x) and ll(x) give the ray length in the insulation and liquid respectively. Additionally it is assumed that the absorbed, scattered and background contributions are additive. Then the process of radiation propagation is described by ì 3 ü I ( x ) = I 0 ( x) exp í-å mi (li ( x))li ( x ) ý + I s ( x) + I b ( x ) î i=0 þ
(12)
where I(x) is the total intensity in the detector plane (normalized dose), i.e. the sum of primary, scattered, and background radiation. Three substances have to be considered: insulation, steel, and liquid. 4.1. Scattering effect and background The contribution of scattered radiation to the total radiation in the detector plane for the case, investigated here, has been estimated applying standard Monte Carlo packages for photon transport [15]. The corresponding profiles for the primary and scattered radiation are shown in fig. 8.
13
The relative contribution of the scattered radiation differs as to the diameter of the pipe and the wall thickness. Additionally this contribution is very sensitive to the pipedetector distance, reaching the maximum value when the pipe is in contact with the detector. The idea, exploited in this elaboration in order to extract the scattered radiation from the total measured, assumes to approximate the profile of scattered radiation across the pipe axis by the Gaussian-type function. The parameters of the Gaussian distribution can be estimated from the measurements taken from a perfect pipe at the same experimental setup. The resulting profile signed as “Sc” in fig. 8, is extracted from the measured profile “Ab+Sc” to extract the profile “Ab”, described by eq. (1) æ x2 ö I s = a exp ç ÷ è b ø
(13)
where a and b are the unknown constants calculated using data from the perfect pipe. If a pipe with insulation and filling medium is assumed additional contributions to the scattered radiation have to be considered providing additive contributions from the three parts I s = I sp + I si + I sl
(14)
where Isp, Isi and Isl give the scattered radiation from the pipe, the insulation, and the filling medium, respectively. The background radiation is shown in fig. 8 as the difference between the “Ab” and “Ab+Sc” profiles taken outside the projection of the pipe itself. If flat panel detectors are used for image recording the contribution of the background can be calculated by extracting the “black” and “white” images from the raw data. But if using films or imaging plates the background is unknown. By analyzing multiple images of reference pipes the assumption of a linear approximation of the background radiation can be made. The two parameters of the linear function are taken from the profile under consideration by analytical continuation in the pipe projection area. I b = c( x + T ) ; - T £ x £ T
(15)
where c is a unknown constant calculated from the data of the current pipe, and Ib gives the background intensity.
14
4.2 Determination of the parameter I0 in equation (1) To determine I0 an appropriate signal transformation has to be implemented considering possible non-linearities of the detector and the beam hardening effects. The transfer function of the used fluorescence screen / film system is usually nonlinear but can be estimated experimentally. It can be represented as a plot of the optical density function versus the radiation dose (exposure time) for the corresponding screen/film system. As an example the transfer function D(t) is shown for RDF / FD8p system (AGFA) in fig. 9.
A simple procedure is proposed to handle the non-linearity of this function meeting reasonable calculation simplicity. For this only that part of the diagram in fig. 9 is used, which corresponds to the real range of measured densities for the phantom pipe. The D(t)function in this application range is approximated by a linear function. For the corresponding fit the following automatic procedure is implemented: ·
determination of the optical densities Dmin and Dmax from the given central cross section similar to that shown in fig. 9
·
determination of the radiation intensities Imin and Imax appropriate to the values Dmin and Dmax using the given diagram D(I).
·
calculation of the function l(x) with l being the length of the ray path in the metal from the geometry of the experimental setup
15
·
calculation of the function I(x) within the range between the two points A and B, which correspond to the projections of the internal and external tangents to the pipe surface, respectively
·
calculation of I0 using eq. (1) given the known beam hardening function (bhf) m(l) calculated from a known source spectrum [16] or measured experimentally [17] (both the theoretical and experimental functions are presented in fig. 10).
4.3 Extraction of the scattered radiation and background The extraction of the scattered radiation from the profile is provided by subtracting the calculated primary intensity Ia(x) from the experimental profile I(x). Finally the sum of two functions is evaluated: æ x2 I ' ( x ) = I s + I b = a expçç è b
ö ÷÷ + c( x + T ) ø
(16)
4.4 Calculation of the unknown coefficients a, b and c. In the next step the extraction of the scattered radiation and background is possible applying eq. (12). This step is necessary in order to interpret the experimental data after this extraction in the form of linear attenuation model (1).
16
Based on previous reasoning the unknown coefficients a, b and c can be calculated. For this purpose the following optimization procedure is applied, making use of the real measurements taken from a perfect pipe with:
{a~, b~, c~}= arg min ìíå [I î
'calc
i
' meas
- Ii
i
] : (a, b, c) Î R üýþ 2
n
(17)
where the superscripts at I’ point on calculated intensities using eq. (16) and measured values, respectively. Rn gives the admissible space for the unknown coefficients. In the following the measuring and calculation procedure is defined necessary to proceed with a perfect pipe: ·
take measurements with the phantom to receive the function I(x)
·
calculate– the primary intensity Ia in the region between the points A and B (compare fig. 2) using the procedure described in the paragraph 4.2
·
extract I0
·
calculate Ia(x) for the total pipe cross section
·
extract Ia from I(x) to receive I’(x)
·
calculate the coefficients a, b, and c using the eq.(17)
In order to use the calculated values of a, b, and c for the assessment of corroded pipes given I0 the following procedure is proposed ·
take in service measurements with the corroded pipe to receive function I(x)
·
extract the contribution from the insulation and filling if necessary
·
calculate the scattering contribution and the background from eq. (16) and pre calculated the coefficients a,b, and c
·
extract scattering and background contributions from I(x) to calculate Iameas(x) for the corroded pipe
·
reconstruct the pipe image with the procedure given by eq. (4) using the previously determined function Iameas(x)
·
use an appropriate procedure to measure the wall thickness in every part of the pipe.
17
4.5 Image reconstruction and wall thickness measurements of a pipe To illustrate the proposed technique the results of real x-ray exposures of the perfect pipes with known geometry as well as dual projections data made with the pipe phantom with a given defect1 are investigated. Figs. 11a through 13a show the x-ray profiles of 100x4,1 mm phantom pipe without insulation and filling. In all cases a Ir-192 gamma source and an RCF / FD8p screen/film system (AGFA) was used for exposures.
Profile 1 gives the measured and digitized profile after preprocessing. In spite of the perfect pipe geometry the profile is asymmetric due to unknown background and disruption of the acquisition geometry. Because it is not possible to provide white image correction operating with films, it is also impossible to predict the behavior of the background between the extreme points of the profile. Taking two points of the profile as the reference points it is possible to approximate the background by linear interpolation with eq. (15) and to calculate the coefficients c and d. The background profiles are shown by curves 2. By extracting the background profile from the measured data the corrected profile is received. This profile is symmetric with regard to the minimums, which approximately correspond to the tangential projection of the inner pipe wall (not shown).
1
The experimental exposures have been provided and the data delivered by the BASF AG, Ludwigshafen.
18
In the next step the primary radiation profile (fig. 13 a, profile 4) is simulated, which is a routine procedure if the dimensions of the pipe and the acquisition set up is known using eq. (1).
Extracting the primary radiation profile from the corrected profile yields the scattered radiation profile. An example is shown in fig. 13a, profile 5. The approximation of the scattering profile assumes a Gaussian-type function (13) with the coefficients a and b being calculated. The resulting simulated scattering profile is shown in figs. 11a through 13a, profiles 3. This approximation of the scattering profile received from the 19
phantom is used for the reconstruction of unknown pipe images. The use of this profile gains advantages for the 2D and 3D image reconstruction compared to exploiting the preprocessed profiles of type 1 or after additional background correction (type 4), as known from experience. Figs. 11a to 12a, profiles 4, show the primary profiles after extraction the scattered radiation, profile 3, and background, profile 2. While investigating unknown pipes, the proposed procedure provides (i) the measured profile after preprocessing (described in sections 4.2 and 4.3), (ii) the extraction of the background, (iii) the extraction of the scattered radiation, and (iv) the implementation of pipe image reconstruction according to the procedure described in sections 2 and 3. The entire procedure runs automatically. In Figs. 11b to 13b the results of the reconstruction of the pipe’s inner cross section using the original primary radiation profiles of type 4 is shown. Figs. 11 and 12 present the reconstruction result considering only one projection, which becomes possible using the precondition of the “perfect” pipe and the direct inversion of the eq. (1) with respect to the dp(x). The fluctuations of the restored inner diameter compared to the direct measurements do not exceed 1 pixel, i.e. 250 mm. Fig. 13 presents the 2D tomographic reconstruction result of the pipe image from two orthogonal projections, using the corrected profile, improved by extracting the scattered radiation. The error of reconstruction did not exceed the following values: averaged 0.29 mm ; maximum 0.36 mm, and minimum 0.13 mm, what is comparable to regular CT with 720 projections. The series of experiments and the results of the tomographic pipe image reconstruction, presented above, demonstrate the capability of the proposed technique to provide accurate wall thickness measurements having only two projections available. 5. Conclusions The proposed 2 and 3 dimensional tomographic reconstruction technique can be applied to the wall thickness determination for corroded pipes. For this purpose only a couple of projections within limited observation angle are sufficient to get accurate measures. The measurement error of less than 0.3 mm can be reached. The accuracy of the technique can be improved by extracting the contribution of concomitant effects, like 20
scattering and background radiation, from the exposed image. A corresponding procedure has been discussed in the paper. Finally the conclusion can be drawn that standard projection techniques using Xor Gamma rays in combination with X-ray film or imaging plates can be applied for the data acquisition to reconstruct wall thickness profiles in an in-field environment. This work was funded by the Federal Ministry of Economics of Germany and the National Academy of Sciences of Belarus.
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