·Table of Contents ·Methods and Instrumentation | A New Method for Radiographic Image Evaluation for Pipe Wall Thickness MeasurementJ. Belenkij, C. Müller (Nockemann), M. ScharmachBundesanstalt für Materialforschung und Prüfung (BAM), Unter den Eichen 87, 12205, Berlin, Germany. V. Vengrinovich, Institute of Applied Physics, Akademicheskaya str.16, 220072, Minsk, Belarus. Contact |
Fig 1: |
f | - film-focus-distance |
r | - outside-radius of the tube |
r2 | - inside-radius of the tube |
R | - outside-radius of the isolation |
w | - wall-thickness |
QA und QB | - tangents at the outer as well as inner circle Cross-section, the tube, |
O'a1, O'a2, O'ax | - is normal to the tangents as well as to the sinews |
Qx1 und Qx2 | - radiations with projection-positions in the
coordinates
x1 area 1, where once the ray the tube cut through, and x2 area 2, where the ray the tube twice cut through |
cx, qx und dx | - the outer and inner local radiuses of the tube |
Ox | - the coordinate-axis in the film-level, (x-coordinate) |
Q | - the punctual source |
A, B | - are border-points on the projection (film-level): from the area 1, where the rays cross only the wall. |
A', B' | - are border-points on the projection (film-level): from the area 3, where the rays cross only the wall. |
Route BB' | - projections of the rays, passing twice through both external and internal tube surfaces. |
(1) |
where I_{0} and I - are X-ray the respective intensities of the source itself and of that part which was not absorbed in the material, m is an absorption coefficient. Then taking the logarithm and direct inversion allows calculation of the function l(x), followed by simple geometric transformations to calculate the inner radius of a perfect tube, given the external tube radius is known.
From this information we see that the central problem as also disussed in [4], arises from the function I(x), calculated from expression (1) with like projections of ray sums on the X-axis, is essentially distinguished from the real experimental function I, and hence the direct inversion of (1) is not-efficient enough for the tube geometry estimate. In other words the model of the process in the form of (1) is inadequate. In this parametric space, the methods of 2D and 3D pipe image restoration from a limited number of projections could become a solution of the problem, but they need to be exposed to the complication of data acquisition procedure, which is not always possible.
The other possible solution of the problem for direct inner radius restoration could result from more accurate simulation of the X-ray - metal interactions, but at present this work is not complete. In literature we find several discussions regarding absorption phenomena which accompany absorption affecting these measurements. First of all there is Compton scattering, but its influence is mostly spread on the central part of the diagram, while main deflections appear in the outer regions. Other possible reasons for image diffusion are back scattering, edge-effect, blurring, unsharpness, beam hardening, source non monoenergeticity, film granularity, digitization errors, source point spread function, etc. These effects "corrupt" the data with respect to the absorption law adapted process model. They introduce uncertainties into the applied measuring procedure which manifests an error in the X-ray data inversion when calculating the dimensions from the data. This so called corruption is caused in the majority by the large beam passing length gradients and the sharp edges of the object which are in its turn most important factors for the measurement results of the dimensions of the object. To avoid the uncertainties it is necessary to achieve a highly accurate knowledge of the process model that describes the actual transfer function of the complete measuring system. But our conventional knowledge of those factors do not permit us to estimate their influence quantitatively.
In the research discussed herein, an effort is made to develop the practical method for the estimate of inner tube radius, given tube X-ray projection. The paper will be organized as follows. First the new calibration procedure for tube radius estimate is proposed. After this the appropriate geometrical formula for inner and outer radii and wall thickness calculation are given. Then the X-ray cross-section diagram is partitioned and those parts are classified as to the discrete estimated errors of inner and outer radius determination using the proposed technique. Finally, the example for inner and outer radius determination, based on the results of X-ray exposure of a real tube, is shown and some initial conclusions will be made.
In order to determine the radii profile of the tube, the through calibration used in the proposed approach is provided like a "black box" with respect to the limited part of the tube located in section AB of the X-ray profile, given the known outer tube radius r. This corresponds to the calibration of a solid cylinder with a known radius. The prior knowledge about the cylinder radius is also an important issue as it allows restoration of the assembly source-object-detector geometrical arrangement if it was unknown before starting the calibration procedure. For this it is also supposed that we deal with an ideal cylinder and that in the calibration section the absorption low is clearly identified and valid.
The next step is to find the regression function F between experimentally determined gray levels G and those theoretically calculated for the ideal cylinder by eq.(1), for that X-ray profile's section which was taken for the calibration. Finally, one can transform the G-data with the formula:
(2) |
where S - is calculated data.
Such a calibration helps to consider the complementary effects which accompany the absorption effect discussed above. Having the mapped data S and inserting in the absorption equation (1), one can directly invert this equation with respect to relative (relatively to the cylinder radius r) semi-span s of X-ray distance passed through the metal:
(3) |
where X - current coordinate in the X-ray profile (after discretization let's consider below under coordinate X the number of corresponding pixel).
(4) |
where r - known tube outer radius, s and n are defined above.
It is clear that expression (4) is valid for the X-range AB (fig.1) under the condition that the deflection of the radius r1 value comparing to the radius value r, used for calibration, is the same for both upper and lower cross points of the ray under consideration and the cylinder. Close to points A and B the accuracy of radius estimation appears worse due to the edge effects.
The interior tube radius r2 can be estimated from geometrical considerations under the condition that one deals with a tube with a perfect outer surface, having radius r and that the deflection of the radius r2 value comparing to the radius value r, used for calibration, is the same for both upper and lower cross points of the ray under consideration and the internal cylinder's surface. Then the following formula is valid:
(5) |
Expression (5) is valid for the X-range OB (fig.1) except in the area close to the inner surface tangential line due to the edge effect and except the central part of the tube due to the large scattering effect in this zone.
The local wall thickness is determined as a difference between the outer radius mean value and inner local radius value:
(6) |
The wall thickness determined by (6) can be averaged over some pixels length.
The resulting gray value profiles (fig. 2) were analyzed. Five different spatial domains were classified according to the actual dominant X-ray interaction mechanism with the object. A calibration procedure is proposed which considers the distinction between the contributions from these domains and the shape of the formula.
Fig.2 shows an X-ray profile for the corresponding specimen. Y-axis represents the gray level after scanning by the scanner of the type proposed.
Fig 2: |
Fig. 3 illustrates a typical case depicting the diagrams of the inner and outer radii variation over the X-axis (in pixels where the size was 50 s as calculated with the proposed technique). The tube to be investigated has a perfect geometry with outer and inner radii 22,125mm and 19,38mm respectively. Two diagrams show both radii in mm calculated by expressions (4) and (5).
Fig 3: |
The vertical lines separate five classified zones which have different scales along X-axis. The readings start (zero X-value) from the center of the pipe. Upper and lower horizontal lines correspond to the outer and inner radii respectively.
The left profile parts represent the radii profiles more accurately, which is quite natural, as the calibration was done using the data taken from the left part as well, while some observed asymmetry usually exists for both sides.
The specific zones are:
It is seen that the calculated inner radius properly corresponds to its real value in the zone B and B' while the best fit of the outer radius estimate is achieved in the zone D and D'. In the other zones the application of recommended formula leads to the errors due to the visible influence of concomitant mechanisms.
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