Thickness Measurement of Metal Plate Using CT Projection Images and Nominal Shape

X-ray computed tomography (CT) is important in industrial process control because it allows non-destructive inspection of a wide range of components. CT scanning requires thousands of projection images to be captured, however, which makes this inspection very time-consuming process. This study establishes a new method for measuring the thickness of parts with shortened scanning and computation times by using only 10 to 30 projection images and predefined nominal shape of the part. The nominal shape is aligned with each projection of the part’s image, and the dimensions of the part are then computed on the basis of the projection images and the nominal shape. In preliminary tests, the approach proved effectiveness for measuring the thickness of a metal bracket and a metal plate.


Background
In recent years, interest in the use of X-ray computed tomography (CT) for industrial non-destructive measurements has increased.There is a demand for metal parts inspection with X-ray CT measurements.In general, CT reconstruction is conducted after recording a series of projection images, from different angles around the target object.Then, dimensions measurements can be computed from the surface mesh that is tomographically reconstructed from the projection images.However, this approach requires thousands of projection images, which leads to very long scanning and computation times [1].In addition, for components made of heavy metals such as copper or steel, dimensional measurements are difficult to compute from CT volumes [2,3].For example, Figure 1 shows thickness measurement results of a heavy metal bracket, whose nominal thickness is 6.0 mm.The thickness measured from the CT volume is greatly differed from the actual value obtained by Coordinate Measuring Machine (CMM).CT reconstruction is also sensitive to beam hardening, which is often difficult to correct for.To address these challenges, this study developes a novel algorithm that allows precise thickness measurements of metal parts, using only 10 to 30 projection images and the predefined nominal shape of the part.The small number of projection images allows short scanning and computation times.Choosing projection images for which transmission length is short and metal artifacts are low returns very precise measurements.We tested the effectiveness of the novel algorithm in measurements of the thickness of metal parts made from two grades of steel and in two different shapes.

Algorithm
Thickness measurements are calculated from k projection images (k = 10 to 30) and the nominal shape, which would be drawn from the plans for the part on a practical manufacturing line.We want to use as few projection images as possible to minimize scanning and computation time but this offers much less information about the dimensions of the part.Thus, the algorithm also includes the nominal shape of the part, which can be used to align the projection images with six degrees of freedom.This alignment determines the position in which the object was scanned and its orientation.Alignment of the projection images with the nominal shape proceeds in three steps: (1) compute the position of the centroid of the object [4,5], (2) calculate the attitude of the object using a matching algorithm, and (3) optimize the alignment in all six degrees of freedom using the Nelder-Mead method [6].Once each projection image is aligned to the nominal shape, the dimensions of the object can be measured from the resulting nominal shape and the projection images.

Alignment 2.1.1 Centroid
First, the 3D centroid of the object is estimated from the 2D centroid of each projection image (Figure 2a).The projection value p and transmission length L are directly proportional (Equation 1).
where p is the negative logarithm of the X-ray transmission value, L is the transmission length and µ is the attenuation coefficient.This relationship is useful for the estimation of the centroid.The vector from the X-ray source to the centroid of each projection image, c c c, is calculated from the weighted mean of the vectors pointing from the source to the pixels on the detector, where the weights are computed from the projection values on the M × N pixel images (Equation 2).
θ i (i = 0, 1, . . ., k) is the angle by which the X-ray source (or the part) has been rotated between each scan of a projection image.p(u, v) is the projection value and v v v(u, v) is the vector pointing from the source to pixel (u, v) on the detector.The resultant vector c c c(θ i ) represents the centroid of the image of the scanned object, with some errors.Thus, the point at which all 2D centroid vectors cross is used to identify the object's 3D centroid using the least-squares method (Figure 2b).

Matching algorithm
Pairs of projection images are used in each matching step.Virtual projection images are created from the nominal shape of the object that is rotated at set intervals around three axis in 3D space.These virtual images are then used to automatically rotate the recorded projection images into desired orientations.The overlapping rate is used to indicate validity of the matching; it is defined as the overlap between the virtual projection of the object A and the projection of the object B, area S(A ∩ B), divided by S(A ∪ B), which is the whole area that includes all points in both A and B when the virtual image is compared against the scanned R is the overlapping rate that indicates matching accuracy.Alignment matching initially proceeds in rough steps of about 10 • , which is called rough matching, then is fine tuned steps of about 1 • , which is called fine matching, to minimize computation time.The accuracy of the matching depends on the rotation steps.Therefore, the alignment accuracy is optimized in the next part of the algorithm.Figure 3b shows a series of alignment steps over which the overlap is maximized, where the blue image is a scanned projection image and the orange image is a virtual projection image created from the nominal shape of the object.

Optimization
Error may have been introduced when identifying the 3D centroid of the object, and the alignment matching algorithm proceeds in discretized rotation steps.Optimization is applied to minimize these errors and align the selected projection images precisely with each other with reference to the nominal shape.The Nelder-Mead method [6] is used for this optimization, with the Zeromean Normalized Cross-Correlation (ZNCC) [7] (Equation 4).The ZNCC, R ZNCC , is a pattern-matching parameter, that is robust against brightness errors and image features.
I(i, j) and T (i, j) are the projection values of two images.The projection values of the images must be considered to ensure accurate alignment.Using ZNCC, the projection images can be aligned to the nominal shape based on the image pattern.The ZNCC is determined by the position and orientation of the two images to be compared, and maximized ZNCC results in optimized alignment.The Nelder-Mead method is generally used to find the minimum or maximum of an objective function in multi-dimensional space.This numerical method uses the concept of a simplex without the need for derivatives.In our algorithm, ZNCC is used as the object function for the Nelder-Mead method and is maximized to optimize alignment.

Thickness measurement 2.2.1 Transmission length estimation
Equation 1 gives the basic relationship between transmission length and projection value.However, the relationship is also affected by beam hardening effects [8].Thus, the length of the X-ray transmission should be verified for each projection value.
Using the transmission lengths from the aligned nominal shape and the projection values from the projection images, the relationship between the transmission length, L, and the projection value p can be approximated using a polynomial function [9].L and p the data collected from all projection images, where r × r divided projection areas have high ZNCC.
w i is the weight that needs to be estimated.No constant term appears because a projection value of 0 would correlate with 0 transmission length.This approximation function is used in the next step of the thickness-measurement process.

Thickness measurement
The thickness T at the measurement point on the surface can then be determined from L, the normal unit vector n n n at that point , and the unit vector a a a that points from the source to the pixel on the detector (Figure 4a).n n n is calculated from the nominal shape.Using these parameters, the thickness is computed according to Equation 6.
The projection images, for which |a a a • n n n| is high are selected for thickness measurements (Figure 4b).This parameter indicates that the X-ray transmission was almost parallel with the thickness direction.

Experiments
The proposed algorithm was applied to measure the thickness of a uniform-thickness bracket made from SPCC and a plate made from SUS304 (Figure 5).Twenty projection images of 2048 × 2048 pixels with a pixel size of 0.2 mm were used.They were recorded with METROTOM 1500, from Carl Zeiss AG.Table 1 lists the scanning parameters.The projection images do not need to be scanned at regular rotation intervals.Thus, the angles of these projection images were selected to give short transmission lengths through the part.
Rough matching was conducted at a pitch of 10 • and fine matching was conducted at a pitch of 1 • .In the optimization process,

Results and discussion
Figure 6 compares the aligned nominal shape and the projection images.The comparison was conducted by overlapping the virtual projection images from the aligned nominal shape with the projection images taken from several angles.No difference in the shape was apparent.Computation of this alignment took 729 seconds, and the rough matching process took up 71 % of this time (Table 2).The computation time depends on the amount of data required for the nominal shape.Though the overlapping rate was decreased between the fine matching and optimization steps, the ZNCC increased (Table 2).This indicates that the pattern matching affected the precision of the alignment process.Figure 7a shows the relationship between the transmission length and the projection value.The set of points in the graph indicates the correspondence between the expected transmission length through the nominal shape and the recorded projection values, and the blue curve shows the fitting result of Tukey's biweight method.Figure 7b and Table 3 list the thickness results from selected points on the nominal shape.Figure 8 and 9 show the same results for tests with the flat plate.They follow the same trends as the results for the bracket.In addition, the thickness of the actual bracket was 1.597-1.603mm and that of the actual plate was 1.007-1.012mm, as measured with a micrometer at points labeled A-D in Figure 7b ans Figure 9b.Even so, the thickness measurements returned by the proposed method are clearly more precise than those returned by a conventional CT scan.
Thickness is difficult and time-consuming to measure over a large area with a micrometer.The proposed method can measure a large area if X-rays pass through the material without too much attenuation and scattering.Figure 10 shows the thickness results using a color map, where red indicates points that were thicker and blue indicates regions that were thinner than the nominal shape.The standard deviation in the estimated bracket thickness is ±50 µm and that of the plate is ±30 µm.These results show that this method can return the thickness measurements within 3% of the actual value.
Considering that the alignment was successful, this deviation in the thickness results was likely caused by errors in the projection values.As shown in Figures 7a and 9a, the points spread widely, likely due to radiation scattering.This error affects the transmission length approximation and the thickness calculations.As Figure 11

Summary
This paper has proposed an algorithm to measure the dimensions of metal parts, using much fewer projection images than are usually needed for CT inspection in conjunction with the predefined nominal shape of the part.So, a few projection images can be recorded because our algorithm automatically aligns the projection images to a model of the part to give a robust estimation of the part dimensions.The alignment process requires identification of the object's centroid, matching the orientation of various projection images, and optimization of this matching.We tested the imaging process with a metal bracket and a metal plate.The alignments were successful.Being limited to the quality of the projection images caused by scattering, this measurement accuracy is within 3% of the actual thickness of the part.In addition, this measurements can be applied over wider areas than can manual measurement with a micrometer.More research is needed to increase the quality of projection images recorded by projecting X-rays through metal parts.In addition, further study is needed so that we can apply this alignment algorithm to measurements of other dimensions.

Figure 1 :
Figure 1: Thickness measurements of a heavy metal component

Figure 2 :
Figure 2: Estimation of the centroid

Figure 3 :
Figure 3: Alignment of projection images to nominal shape

Figure 6 :Figure 7 :
Figure 6: Comparison of the aligned nominal shape and the scanned projection image of the bracket

Figure 8 :
Figure 8: Comparison of the aligned nominal shape and the scanned projection image of the plate

Figure 9 :
Figure 9: Experimental results for the plate measurements

Figure 10 :
Figure7ashows the relationship between the transmission length and the projection value.The set of points in the graph indicates the correspondence between the expected transmission length through the nominal shape and the recorded projection values, and the blue curve shows the fitting result of Tukey's biweight method.Figure7band Table3list the thickness results from selected points on the nominal shape.Figure8 and 9show the same results for tests with the flat plate.They follow the same trends as the results for the bracket.In addition, the thickness of the actual bracket was 1.597-1.603mm and that of the actual plate was 1.007-1.012mm, as measured with a micrometer at points labeled A-D in Figure7bans Figure9b.Even so, the thickness measurements returned by the proposed method are clearly more precise than those returned by a conventional CT scan.Thickness is difficult and time-consuming to measure over a large area with a micrometer.The proposed method can measure a large area if X-rays pass through the material without too much attenuation and scattering.Figure10shows the thickness results using a color map, where red indicates points that were thicker and blue indicates regions that were thinner than the nominal shape.The standard deviation in the estimated bracket thickness is ±50 µm and that of the plate is ±30 µm.These results show that this method can return the thickness measurements within 3% of the actual value.Considering that the alignment was successful, this deviation in the thickness results was likely caused by errors in the projection values.As shown in Figures7a and 9a, the points spread widely, likely due to radiation scattering.This error affects the transmission length approximation and the thickness calculations.As Figure11shows, the projection values significantly changed at

Table 2 :
Computation time, R and R ZNCC for the bracket

Table 4 :
Computation time, R and R ZNCC for the metal plate