More Info at Open Access Database www.ndt.net/?id=18739 Frequency-based method to optimize the number of projections for industrial computed tomography Andrea Buratti 1, Soufian Ben Achour 1, Christopher Isenberg 1, Robert Schmitt 1 1 Laboratory for Machine Tools and Production Engineering WZL of RWTH Aachen University, Steinbachstraße 19, 52074 Aachen, Germany; e-mail: A.Buratti@wzl.rwth-aachen.de Abstract Industrial Computed Tomography (CT) has been regarded as a promising in the field of dimensional metrology due to its holistic approach to 3D coordinate measurements. Several CT scan parameters influence the measurement outcome and thus uncertainty of measurements. These quantities are the imaging parameters (tube voltage, tube current, prefilter, number of projections, etc.) and the manipulation parameters (source-object distance and workpiece orientation). The number of projections is an important factor to define the ability of the CT system to spatially resolve the geometrical features of a workpiece and defines the duration of a CT scan, as well. Low numbers of projections cause artifacts due to undersampling to occur which lower measurement accuracy, whereas acquiring too many projections involves high scanning times. CT users choose values for this parameter according to their experience. This often leads to high variability in measurement results, insufficient accuracy or unnecessarily long scanning time. Due to the complex transformations in the CT measurement chain, there is currently no analytical model that users can use to optimize the number of projections. In this study, we developed and evaluated a method to optimize the number of projections depending on the workpiece geometrical feature that has to be measured. Our focus is on dimensional metrology. The optimal value is the minimum number which ensures measurement accuracy fairly close to the theoretical maximum. First, the method considers a projection of the workpiece, which can be obtained by simulation or by a real scan. Then, the grey value distribution is considered and transformed into the Fourier domain. A radial filter which depends on the number of projections is applied to introduce the effect of undersampling on the projection. The filtered projection is then transformed back to the spatial domain and the contrast-to-noise ratio (CNR) is evaluated. The algorithm repeats this procedure by gradually decreasing the number of projections until the CNR decreases significantly. The obtained value represents the optimal number of projections. Finally, we applied this method to determine the optimal number of projections for the features of a test artifact and discussed the results. Keywords: Computed tomography, dimensional measurements, setup parameters, scan parameters, number of projections, prediction, geometrical feature 1 Introduction Industrial X-ray Computed Tomography (CT) is a multi-purpose non-destructive testing method. This technique allows reconstructing both the inner and the outer geometry of complex workpieces, without damaging or altering them. In the field of coordinate metrology, CT is a relatively new technology and represents an alternative to traditional coordinate measuring machines (CMM) [1]. Many factors influence the uncertainty of CT measurements. The user plays an important role by setting the CT scan parameters. These quantities are the imaging parameters (e.g. tube voltage, tube current, prefilter, number of projections) and the manipulation parameters (e.g. source-object distance and workpiece orientation). The number of projections is an important factor to define the ability of the CT system to spatially resolve the geometrical features of a workpiece and defines the duration of a CT scan, as well [2]. Low numbers of projections cause artifacts due to undersampling to occur which lower measurement accuracy, whereas acquiring too many projections involves high scanning times. CT users choose values for this parameter according to their experience. This often leads to high variability in measurement results, insufficient accuracy or unnecessarily long scanning time. Due to the complex transformations in the CT measurement chain, there is currently no analytical model that users can use to relate the number of projections to a given measurement task. 2 Method 2.1 Definition of the optimal number of projections First, we need to give a definition to the optimal number of projections. From a metrological point of view, the optimum is the number of projections that minimizes the influence of this parameter on the measurement uncertainty. We define it as the www.3dct.at 1
metrological optimum. Generally speaking, the required number of projection depends on the maximum spatial frequency of the projection images and on the extension of the projected reconstruction volume [3]. For a given number of projections N, we define the angular sampling interval Δθ as follows: = (1) The maximum spatial frequency max can be expressed as follows [3]: = (2) where R represents the radius of the reconstruction volume projected onto the detector in the spatial domain. The volume is here considered as a bounding sphere, as shown in Figure 1. Figure 1: Cone beam and reconstruction sphere From an industrial perspective, CT users set scan parameters, including the number of projections, so that they can achieve the desired accuracy in the shortest tomographic time. In this sense, minimizing the contribution of the number of projections on the measurement uncertainty means a virtually infinite tomographic time. The industrial optimum is defined as the minimum number of projections which allows to achieve the desired accuracy in the shortest tomographic time. Intuitively, an increase of the number of projections involves an increase of the measurement accuracy. We can show this by assessing CT measurement error with respect to calibrated values and representing it as a function of the number of projections. Figure 2 shows this behavior for diameter measurements on a hole plate. Figure 2: CT measurement error with respect to calibrated measurements for diameter measurements on a hole plate. www.3dct.at 2
In the left part of the graph, we can notice that the measurement error significantly decreases while N increases. Above a given value N*(800 1100 projections), further increasing N does not determine a significant decrease of the measurement error, meaning that the improvement is negligible. Hence, we can define the technical optimum N* as the number of projections corresponding to the transition value from the region where the measurement error strongly depends on N to the one where its dependency is slight. In other words, the technical optimum minimizes the scanning time while ensuring a value for the uncertainty contribution of N fairly close to the minimum one. The three definitions for the optimum are summarized in Figure 3. Figure 3: Definition of optimum number of projections. 2.2 Estimation of the optimal number of projections for dimensional metrology The metrological optimum depends on the maximum spatial frequency of the object, which in general tends to infinite, since the bandwidth of the spectral support related to the projections is not strictly limited [4]. Similarly to what we pointed out for the technical optimum, we can identify a frequency value in the spectral support which we define as the maximum useful frequency. Frequencies above this one do not provide relevant information for reconstructing the object. Let us consider a set P1 of N < N* projections (being N* the technical optimum) and the corresponding reconstruction volume. Since N < N*, the reconstruction volume will be relevantly affected by undersampling artifacts. Of course, this effect cannot be seen on the projection images, since it appears only after reconstruction. Moreover, let P2 be a fictitious set of N* projections related to the same object, which give the same reconstruction volume as that of P1. These projection images will be instead affected by the effects of undersampling (Figure 4). Figure 4: Real and fictitious projection sets Projections of set P2 are not real, but we can conceive a method to apply the effect of undersampling on them and identify N*. In general, the maximum useful frequency varies in different projection images of the same object. We consider then the www.3dct.at 3
projection with the maximum value among those available (e.g. obtained through simulation). In this way, we can consider one image and apply the effects of undersampling on it, so that it belongs to the set P2. By gradually decreasing the effects of undersampling, we can identify the minimum point at which image quality is approximately stable, and therefore compute N*. The algorithm to calculate N* consists of the following steps. First, we set an initial number of projection N = N 0, with N 0 certainly much higher than N*. In accordance with practical considerations, we set N 0 = 4000. Subsequently, we consider a projection image P(x,y) of the object. This can be obtained from a real scan or through simulation. We define a low-pass filter F in the frequency domain F( x, y). F is a radial filter with radial frequency w that can evaluated according to (1) and (2). = 4 (3) where R is the radius of the bounding sphere containing the workpiece in the spatial domain. The filter is then applied to the transformed projection image P. We therefore calculate the convolution l(x, y), which is a product in the frequency domain: We perform the inverse transformation of L and go back to the spatial domain., =,, (4),, (5) Now, we evaluate the image quality. In the field of dimensional metrology, we need to accurately detect the object surface and to reduce noise in order to get good measurement results. Hence, we choose the contrast-to-noise ratio (CNR) as an estimate for the image quality: = (6) where C is the image contrast and σ the standard deviation of the grey values. Since the chosen N = N 0 is higher than N*, we can consider the corresponding CNR as the maximum one: = (7) At this point, we set a new value for N by decreasing N 0. In general, at iteration i, the new number of projections Ni is given by: = (8) We repeat the procedure by evaluating the Fourier transform of the previously obtained filtered projection, by computing the new filter and applying it to the transformed projection. Finally, we perform the inverse trasformation of the convoluted projection and assess the new CNR. We iterate the algorithm as long as the CNR does not significantly decrease and its difference quotient with respect to N is much greater than zero. This condition at iteration i can be expressed as follows: < (9) where is a tolerance value which takes CNR random variability into account. The algorithm stops when (8) does not hold anymore. The corresponding number of projections represents the estimate for the technical optimum. Figure 5 and Figure 6 show a summary of the method. The algorithm was implemented into a MATLAB routine. www.3dct.at 4
Figure 5: Steps of the algorithm to optimize the number of projections Figure 6: Scheme of the algorithm to optimize the number of projections 3 Description of the investigation In order to study the influence of undersampling on the measurement accuracy, we designed a test artifact. This workpiece is a disk made of POM (Figure 7a) and features several through holes with different diameters (Figure 7b). The holes were arranged along the disk radius to analyze the difference in the undersampling effect: we can expect that holes farther from the center will be more affected by these artifacts. www.3dct.at 5
Figure 7: Test artefact, 3D model (a) and draft (b). To get reference measurements, the hole diameters were calibrated with a Zeiss Prismo tactile CMM. The diameters were measured at a height of 4 mm from the top surface. Subsequently, CT scans were performed with a Zeiss Metrotom 1500. Table 1 shows the used scan parameters. Material POM SOD [mm] 425 Tube Voltage [kv] 85 Tube Current [ma] 530 Integration time [ms] 1500 Table 1: Scan parameters for the test artifact. By applying the algorithm to optimize the number of projections, we get N* = 900. Figure 8 shows the behavior of the CNR related to the projections as a function of the number of projections. Figure 8: CNR as a function of the number of projections. www.3dct.at 6
The effects of undersampling are shown in Figure 9. We can notice that for N = 900 there is no relevant undersampling artefact. Figure 9: Effects of undersampling on reconstruction for 10 (a), 100 (b), and 900 (c) projections For values of the number of projections between 25 and 1600, CT measurements were performed and all diameters were measured. Volume data were analysed using VG StudioMax software. The relative error with respect to CMM measurements is calculated as follows: = (10) where is the CT measurement of the diameter and is the corresponding calibrated measurement. Figure 10 shows the relative measurement error for diameter D16 (Figure 7b) as a function of the number of projections. We can notice that the measurement error is constant after 900 projections. This means that N = 900 is the technical optimum for feature D16. Figure 10: Relative measurement error of feature D16 as a function of the number of the number of projections. 4 Conclusions and future prospects In this study, we presented a method to optimize the number of projections for performing metrological tasks. The method identifies a technical optimum which guarantees high measurement accuracy while minimizing the tomographic time. It can be applied to both the whole object volume or to a region of interest. Experimental data are consistent with the predicted results, since the prediction of the technical optimum corresponds to the empirically determined one. Future work at the Laboratory for Machine Tools and Production Engineering (WZL) will deal with determining the industrial optimum as well as investigating the influence of different types of measurement task on the optimum. Acknowledgements The work has been funded by the European Union 7th Framework Programme under Grant Agreement 607817. The authors would like to thank Evelina Ametova (KU Leuven) for the fruitful discussions. www.3dct.at 7
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