A cone-beam CT geometry correction method based on intentional misalignments to render the projection images correctable Felix Meli, Benjamin A. Bircher, Sarah Blankenberger, Alain Küng, Rudolf Thalmann More info about this article: http://www.ndt.net/?id=21918 Abstract Federal Institute of Metrology METAS, Laboratory for Length, Nano- and Microtechnology Lindenweg 50, 3003 Bern-Wabern, Switzerland e-mails: felix.meli@metas.ch, benjamin.bircher@metas.ch CT systems, in particular industrial CTs used for dimensional measurements, require an excellent geometrical accuracy. The geometry of a cone-beam CT system is characterised by 9 parameters that must meet specific values. Therefore, CT systems consist of several stages, which allow adjusting some of these parameters, while other parameters are expected to remain fixed. However, stages suffer from geometrical imperfections, such as positioning and guideway errors, which cause distorted CT projections. If misalignment errors are known, e.g. by calibration, some of them can be corrected in the projections by image processing. Nevertheless, geometry errors that alter the ray directions through the investigated object cannot be corrected in the projection images, even if the corresponding geometrical parameters are known. We propose a method to convert CT geometry errors into image correctable ones, by repositioning some of the available stages. Repositioning is required in all cases where the x-rays would not cross the object in nominal directions. The resulting geometry is still misaligned but the projections can now be fully corrected using image processing. The such obtained corrected projection image set can then be reconstructed using efficient standard algorithms. The method was evaluated by simulations and enables recovering dimensional measurements with sub-voxel accuracy. Keywords: Industrial computed tomography, CT image correction, CT geometry correction, CT metrology 1 Introduction CT is increasingly used for dimensional measurements [1]. Among others, the CT geometry can be a major influence factor on the measurement uncertainty [2]. Especially computationally efficient reconstruction algorithms, such as FDK [3] or Katsevich [4], rely on geometrically accurate data [5]. Advanced methods can process projections taken at an arbitrary but known geometry, however, the computing effort of such techniques is much higher [6,7]. For this reason, intensive efforts are made to accurately align CT systems [2]. However, residual geometry errors might still deteriorate the data quality. Therefore, a variety of correction methods applied to projection images [8] and reconstruction algorithms [9] were suggested. Here, we present a CT geometry correction method for systems with a limited number of motion axes and a calibrated or measured geometry: First, the erroneous CT geometry is mechanically repositioned using some of the available stages to obtain the correct X-ray source position relative to the object. Second, all remaining errors are corrected in the projections by image transformations resulting in ideal projection images. The image processing step is straightforward to implement and does not require access to the reconstruction algorithm, which is usually limited. 2 Correction methods 2.1 CT geometry errors and corrections For accurate CT measurements the CT geometry must be known and stable over the entire scan time, which is often hours; or preferably over weeks to extend calibration intervals. Stability issues are a major factor since systems have dimensions in the metre range and incorporate heat sources, such as the X-ray tube, the detector, motors, pumps, controllers and further support systems dissipating heat up to several hundred watts. In order to overcome these issues, the CT geometry correction method presented in the following requires real-time geometry measurements or knowledge from previous stage calibrations to produce an image set representing an almost perfect CT geometry, even though the used displacement stages are non-ideal. It requires a repositioning of three axes of the sample stage during the measurement and additional projection image processing. First, the degrees of freedom (DoF) of the cone-beam CT geometry are analysed. In Figure 1 left a coordinate system that is fixed to the sample is defined. In this coordinate system detector and X-ray source rotate around the sample. The X-ray source is assumed to have a point shape and as such its angular orientation is not defined. Thus, its position is defined by 3 parameters while the position and orientation of the detector require 6 parameters. In this coordinate system the projection geometry can be described by 9 parameters corresponding to the 9 DoF of the cone-beam CT system. Another definition where the www.3dct.at 1
coordinate system is fixed to the flat panel detector is shown in Figure 1 right. 6 parameters are required for the object under investigation, i.e. 3 translations and 3 angular orientations, as well as 3 additional parameters for the position of the X-ray source spot. Both coordinate systems describe the same physical system. But the coordinate system on the left, which seems less straightforward, enables separating two different types of possible projection errors: The errors due to a source-sample misalignment and the errors due to a sample-detector misalignment. Figure 1: Sample based coordinate system (left) and detector based coordinate system (right). 9 parameters are required to explicitly define the cone-beam CT geometry. 2.2 Source-sample misalignment: not correctable in the projection image The first group of geometry errors are not correctable on the projection image. Using the coordinate system defined in Figure 1 left, position deviations of the point source cannot be corrected by image processing. In particular, the position coordinates (x s, y s, z s ) have to be equal to the nominal value required for a specific projection. If this is not met, the X-ray paths through the object do not correspond with the required projective directions. The resulting faulty projection could possibly be corrected if the sample geometry was known, but at this stage of the measurement this is not the case and therefore no image corrections can be applied. The only solution is to physically adjust the source position with respect to the sample as required for the specific projection. Because the source is constrained in many industrial CT systems, the sample has to be repositioned even though this will increase the deviations in the projected image. Subsequently, the resulting projection image errors can then be fully corrected if the CT geometry is known. A typical CT system features several stages to adjust its geometry, but usually there are less than 9 DoF available. Therefore, some of the 9 geometry parameters are required to remain in their nominal position. In order to fulfil the requirements of the correction method proposed, a CT system requires a sample stage with at least 3 DoF and the stage resolutions should be better than the highest achievable voxel resolution. First, the sample stage is assumed to enable 3 translational motions with high resolution. In this case, it is straight forward to correct the position of the sample in such a way that the relative source coordinates attain the required values. The correction of the sample position will then result in magnified image errors on the detector that can be corrected. Instead of 3 translational shifts of the sample it is also possible to adjust the correct source coordinates by one angular and two translational shifts or by two angular and one translational shift. This is relevant, because the sample stage of an industrial CT always consists of a rotational stage. Therefore, only two additional DoF are required. Furthermore, most systems can change the source-object distance to adjust the magnification and systems capable of performing helical scans include a further linear stage to translate the sample parallel to the rotation axis. In such systems, a sufficient number of DoF are available to correct the relative sourcesample position. 2.3 Sample-detector misalignment: correctable in the projection image The second group of errors are fully image correctable projection errors. Using again the above defined sample coordinate system (see Figure 1 left) all deviations of the 6 detector parameters can be corrected in the image. In particular, the lateral position can be corrected with an image shift and a distance deviation by scaling the image (magnification). The rotation around the magnification axis (detector roll) can be corrected by an image rotation while the two remaining angular errors (detector pitch and yaw) can be corrected approximately using a trapezoidal image transformation, i.e. a homographic transformation [10]. Additionally, the image correction procedure typically includes bilinear or higher order pixel interpolation in order to deal with fractions of pixel coordinates. The above mentioned image corrections can be performed using a single multiplication with a 3x3 transformation matrix [10]: www.3dct.at 2
[ ] [ ] = [ ] and [ ] = [ /w ], (1) / h 1 where x and y are the input and x and y the transformed pixel coordinates. Using the above defined homographic image transformation (Equation 1) not all image corrections are exact. While the lateral position, the magnification and the detector roll can be exactly corrected with homographic transformations, the correction is only a first order approximation for angular detector tilts (rot y, rot z ) [8]. Figure 2 left shows the exact image distortions with a detector pitch and yaw of 10 each, calculated using a geometrical projection model. The image distortions are cone angle dependent (source-detector distance, SDD) but are independent of the source-object distance (SOD) for a given detector size. Figure 2 centre shows the optimal homographic image correction, when applied to the exact distortions (Figure 2 left), the pixel shifts are reduced from 30 mm to 10 mm but not completely eliminated (Figure 2 right). Figure 3 shows the same distortions as in figure 2 but for smaller, more realistic detector pitch and yaw values of 0.1 each. The homographic transformation is also for small distortions only an approximation. Therefore, an exact cone-beam image correction method is applied in the following as explained in section 3.1. Figure 2: Tilted 400 mm x 400 mm, detector pitch 10, detector yaw 10 (SDD 1000 mm). Left: The colour scale indicates the pixel shift in millimetres with respect to the ideal geometry. Centre: Homographic approximation of the pixel shift. Right: Remaining pixel shift in millimetres after a homographic image correction. Figure 3: Tilted 400 mm x 400 mm, detector pitch 0.1, detector yaw 0.1 (SDD 1000 mm). Left: The colour scale indicates the pixel shift in millimetres with respect to the ideal geometry. Centre: Homographic approximation of the pixel shift. Right: Remaining pixel shift in millimetres after a homographic image correction. www.3dct.at 3
3 Results 3.1 Detector image correction using a geometrical forward projection model The implemented image corrections are based on a geometrical projection model using all 9 DoF of the CT geometry. A grid having the same number of points as there are pixels on the detector is generated and projected onto the detector for the nominal and then for the actual CT geometry. The difference serves as correction map for the pixel shifts. For the nominal CT geometry, the 2D grid is placed in the centre of the object rotation axis in a plane parallel to the detector. The grid increments correspond to the detector pixel pitch divided by the magnification. Next, the grid is used in the actual CT geometry to be corrected. The resulting locations of the geometrical projection of the grid points serve as pixel correction map. Bilinear interpolation was used, in order to deal with non-integer pixel shifts. The CT geometrical projection model and the corresponding image correction were programmed in LabVIEW using standard functions. On a typical office PC, the calculation of the correction map followed by bilinear image interpolation takes 0.3 s for a 2k projection image (Figure 3). This procedure enables correcting each projection image with its individual CT geometry deviation. The undefined border pixels of corrected images have to be neglected in the subsequent volume reconstruction to avoid artefacts. Alternatively, the border pixels could be padded with air intensities. In order not to lose too much of the image area, the correction method is intended to be used for small CT geometry deviations only. Figure 3: Image correction example, detector 400 mm x 400 mm, SDD 1000 mm. Left: projection image with nominal CT geometry. Right: CT geometry with tilted rotational axis rot x : 10, rot y : 20, rot z : -4 (coordinate system Figure 1 right) derived from the left image by applying an image transformation based on the forward projection model and bilinear pixel interpolation. It is emphasised that the misalignment is exaggerated for demonstration purposes. 3.2 Evaluation of the correction method by simulation In the following, the developed correction method is evaluated using simulations performed with the artist simulation package (BAM, version 2.8.3). To exclude any influence originating from the X-ray source and detector, a monochromatic point source with 55 kv and an ideal detector consisting of 2000 x 2000 pixels with 0.2 mm pitch, were simulated. The sourceobject distance was 90 mm and the source-detector distance was 1200 mm, resulting in a 13.333-fold magnification and a voxel size of 15 µm. A set of 3000 simulated X-ray projections of a multi-sphere standard, consisting of three rings of eight Ø 1 mm steel spheres (see Figure 4), was computed. The simulations require the object geometry in the STL format that describes a triangulated surface. Therefore, the CAD (STEP) file was converted into an STL mesh using VG Studio MAX 3.0 with the mesh accuracy setting extra high, resulting in a surface consisting of 88320 triangles per sphere. Three different cases were simulated as shown in Figure 5: (a) An ideal CT geometry, (b) a rotary axis tilt of 1 along the magnification axis, and (c) a case equal to (b) with two additional linear sample shifts, y and z, to render the errors correctable in the projection images. The misaligned projective geometry in Figure 5b can easily be identified by comparing the positions of the spheres in the centre of the projection to the ones in Figure 5a and 5c. VG Studio MAX 3.0 was used for reconstruction and analysis of the sphere locations, diameters and form deviations. The projections from the correctable case (Figure 5c) were transformed according to the forward projection model, described in section 3.1, using bilinear interpolation. Subsequently, all three cases were reconstructed using a standard FDK algorithm with the ideal CT geometry as input. A local threshold method with an ISO50 starting contour was used to determine the object surface with sub-voxel accuracy. To locate the spheres, geometrical primitives were fitted with a Gaussian least squares www.3dct.at 4
method, using 9943 fit points, distributed with the auto fit points option and disabled fit point filter. The fitted spheres were used to determine the position, diameter and form. Using the 24 sphere positions (Figure 4), all 276 possible sphere centre-tocentre distance deviations were calculated. The sphere form error was defined as the maximum peak-to-valley distance between 9943 used surface points and the fitted sphere. Figure 4: CAD of the multi-sphere standard used for the simulations. It consists of three rings of eight Ø 1 mm spheres each, with a total size of 20 x 20 mm. Figure 5: Simulated geometries and corresponding projection images: (a) Ideal CT geometry, (b) rotation axis pitch of 1, and (c) equal to (b) with two linear shifts, y and z, to render the projection images correctable. The angles and displacements are not to scale. www.3dct.at 5
Figure 6: Deviations between CAD reference data and the STL mesh, used as input for the simulations, as well as the reconstructions with a 15 µm voxel size, employing an ideal CT geometry, a rotary axis pitch of 1, and a rotary axis pitch of 1 with two additional linear shifts and subsequent image correction. Shown are volume slices, sphere centre-to-centre distance deviations, form deviation and diameter deviation. Note the different axis scaling. Figure 6 shows volume slices of sphere 1 (see Figure 4), the sphere centre-to-centre distance deviations, form deviations, and diameter deviations for the input STL mesh and the three reconstructed simulated cases in respect to the CAD model. The analysis of the STL mesh shows limitations due to the surface triangulation that introduced a maximum centre-to-centre distance deviation of 0.6 µm, a form deviation of (5.4 ± 0.0) µm (mean ± ), as well as a diameter deviation of (-2.0 ± 0.0) µm. These values slightly increased to 0.6 µm maximum distance deviation, (8.2 ± 0.7) µm form deviation, and (-2.5 ± 0.1) µm diameter deviation for the reconstructed simulation employing an ideal CT geometry. These errors are assigned to the limited resolution of the simulated detector (voxel size 15 µm), the employed surface determination method and the inexact FDK reconstruction algorithm, which introduces cone-beam artefacts. However, the deviations remained below a fraction of the voxel size due to the employed sub-voxel surface determination method. For a rotary axis pitch of 1, the reconstruction of the spheres became highly distorted as shown in the corresponding volume slice (see Figure 6, 3 rd column). The maximum deviation of the sphere centre-to-centre distances increased to 22.7 µm. The average form and diameter deviations were (214.4 ± 0.9) µm and (37.1 ± 1.6) µm caused by the strong distortion of the sphere surfaces. After applying the developed correction method, i.e. inducing a sample shift and post-processing the images, the obtained result was very close to the ideal scan geometry. The maximum sphere centre-to-centre distance deviation was 1.2 µm, which www.3dct.at 6
corresponds to a 15-fold reduction compared to the uncorrected case. The maximum deviation was only 0.6 µm larger than for the ideal case, which corresponds to a 1/25 pixel. This residual error is assigned to the employed bilinear interpolation method during image-processing that reduces the accuracy of the sub-voxel surface determination. Form and diameter deviations, respectively, averaged to (7.5 ± 0.8) µm and (-4.2 ± 0.3) µm. The complete recovery of the form error might be due the lowpass filtering caused by the linear interpolation, whereas the diameter determination is again very sensitive to the interpolation and surface determination. In summary, the correction method recovers all dimensional measurements with sub-voxel accuracy. It remains some sensitivity to grey value deviations, caused by the employed interpolation step during image processing. It is emphasised that all influencing factors, such as the STL mesh used as an input for the simulations, must be considered when dealing with subvoxel accuracies. 4 Conclusions and outlook A CT geometry correction method was proposed that produces a set of projection images corresponding to an ideal CT geometry. It is based on repositioning the CT system with some of the available stages, in order to obtain the correct relative source-sample position. The CT geometry changed in such a way produces projections which can be fully corrected by an additional image-processing step. Simulations showed that the method works, indeed, almost recovering the ideal case. Limitations include the voxel resolution and the bilinear interpolation step during image processing. Higher order interpolation methods might improve the accuracy, however, they are computationally more intensive. The obtained corrected projection image set can be reconstructed using efficient standard algorithms. No access to specific parameters of the reconstruction is required. An optimal metrology CT system features an independent metrology system capable of measuring the actual values of all 9 geometry parameters defining each projection in situ. Such a system is currently under development at METAS [11 13]. It is intended to implement the described correction procedure with in situ CT geometry data. Future work will include evaluation of the correction method for fluctuating CT geometries and with experimental projection data. References [1] De Chiffre L, Carmignato S, Kruth J P, Schmitt R and Weckenmann A 2014 Industrial applications of computed tomography CIRP Ann. - Manuf. Technol. 63 655 77 [2] Ferrucci M, Leach R K, Giusca C, Carmignato S and Dewulf W 2015 Towards geometrical calibration of x-ray computed tomography systems a review Meas. Sci. Technol. 26 92003 [3] Feldkamp L a., Davis L C and Kress J W 1984 Practical cone-beam algorithm J. Opt. Soc. Am. A 1 612 [4] Katsevich A 2004 An improved exact filtered backprojection algorithm for spiral computed tomography Adv. Appl. Math. 32 681 97 [5] Kruth J P, Bartscher M, Carmignato S, Schmitt R, De Chiffre L and Weckenmann A 2011 Computed tomography for dimensional metrology CIRP Ann. - Manuf. Technol. 60 821 42 [6] Gordon R, Bender R and Herman G T 1970 Algebraic reconstruction techniques (ART) for three-dimensional microscopy and X-ray photography J Theor Biol 29 471 [7] Deffise M and Clack R 1994 A Cone-Beam Reconstruction Algorithm Using Shift-Variant Filtering and Cone-Beam Backprojection IEEE Trans. Med. Imaging 13 186 95 [8] Ferrucci M, Ametova E, Carmignato S and Dewulf W 2015 Evaluating the effects of detector angular misalignments on simulated computed tomography data Precis. Eng. 45 230 41 [9] Kyriakou Y, Lapp R M, Hillebrand L, Ertel D and Kalender W A 2008 Simultaneous misalignment correction for approximate circular cone-beam computed tomography Phys. Med. Biol. 53 6267 89 [10] Hughes J F, Dam A Van, McGuire M, Sklar D F, Foley J D, Feiner S K and Akeley K 2014 Computer Graphics: Principles and Practice (Addison-Wesley) [11] Bircher B A, Meli F, Küng A and Thalmann R 2017 Towards metrological computed tomography at METAS euspen s 17th International Conference & Exhibition [12] Bircher B A, Meli F, Küng A and Thalmann R 2018 A geometry measurement system for a dimensional cone-beam CT submitted to 8th Conference on Industrial Computed Tomography, Wels, Austria (ict 2018) [13] Bircher B A, Meli F, Küng A and Thalmann R 2017 Characterising the Positioning System of a Dimensional Computed Tomograph (CT) Submitted to MacroScale - Recent developments in traceable dimensional measurements www.3dct.at 7