Efficient 3D Crease Point Extraction from 2D Crease Pixels of Sinogram

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1 Efficient 3D Crease Point Extraction from 2D Crease Pixels of Sinogram Ryo Jinnouchi 1, Yutaka Ohtake 1, Hiromasa Suzuki 1, Yukie Nagai 1 1 The University of Tokyo, Japan, {jinnouchi, yu-ohtake, suzuki, nagai}@den.rcast.utokyo.ac.jp Abstract In this study, we extract 3D crease points on the surface of an object from its X-ray CT projection images (sinogram) without reconstructing the surface mesh. The idea behind this is that most of the 3D creases are captured as crease pixels in the 2D projection images, so the creases may be reconstructed from the 2D crease pixels. So we first extract the crease pixels in each of the 2D projection images and apply reconstruction of the crease pixels to obtain 3D crease points. Connecting these crease points, a wireframe model is created. This method requires a small number of projection images, and thus makes it possible to reduce computational time. The error of the reconstructed wireframe model falls within half of the scaled pixel size. Keywords: CT scanner, creases on surface, CT reconstruction 1 Introduction One major use of X-ray CT scanners is in creating 3D surface mesh models [1]. These mesh models are generated by using an iso-surfacing method, such as Marching Cubes [2], from the CT volume that is reconstructed from the sinogram (projection images). For reverse engineering, 3D creases are useful to generate a CAD model, and surface mesh models are often used to extract creases (the flow shown by blue arrows in Figure 1). However, the quality of these creases depends on that of the surface mesh models, but surface mesh models extracted from CT volume scans often fail to preserve sharp creases. In this study, we propose a method to extract the 3D creases without reconstructing the surface mesh directly from the sinogram (the flow shown by red arrows in Figure 1). The idea behind this is that Figure 1. Data flow of the proposed method (red arrows) compared with that of the conventional method (blue arrows). ict Conference

2 most of these 3D creases are captured as crease pixels in the 2D projection image so that 3D crease points are reconstructed from those crease pixels. Hence, we first extract the crease pixels in each of the 2D projection images and apply reconstruction to these to obtain 3D crease points. A benefit of this method is that only a small number of projection images are needed for reconstruction. Additionally this method is robust against inaccuracies of projection values, because this algorithm uses the binarized feature images computed from a sinogram and does not use the projection values themselves. 2 Algorithm 2.1 Crease detection from projection images Crease pixels in projection images are comprised of pixels in areas where the gradient sharply changes. To detect these crease pixels, we use normal tensor voting [3]. The advantage of using normal tensor voting is that it is robust against noise, so it is suitable to detect sharp edges. For each pixels, a 3 3 matrix V of Normal tensor voting is given by,, /,, /, 1,, /, / 1 where i identifies a pixel in a kernel, is a normal vector, k is the kernel size, and, is the projection value of the i-th pixel. Images of the normals are created beforehand from the projectionimages by computing the gradient of the projection values (Figure 2 (b)). To distinguish if the pixel of interest is in a crease, three eigenvalues,,and of are checked. If minimum eigenvalue is much smaller than the other two, the pixel has potential as a crease pixel, and the crease direction is specified by the eigenvector corresponding to. If the pixel has crease potential, is saved as a crease plausibility image (Figure 2 (c)). This value varies from 0 to 1, and if this value is close to 1, it suggests that the crease is crossing the center of the kernel. Applying non-maximum suppression to this image, we obtain the detected crease image (Figure 2 (d)). Figure 2. (a) Original projection image, (b) Normal image, (c) Crease plausibility image, (d) Crease image. 2.2 CT reconstruction The crease images obtained in Section 2.1 (Figure 3) are used as the sinogram for CT reconstruction with the Feldkamp method [4] using the Shepp Logan filter [5]. In this step, only a small number of images are needed. Figure 4 (a) and (b) show slices of reconstructed CT volumes from 50 and 100 crease images, respectively. Creases are seen as white pixels in the images. If you use only 50 projection images in ordinary CT reconstruction, as shown in Figure 4 (c) and (d), the extracted surface mesh from the CT volume fails to reconstruct the original model. 440

3 Figure 3. Projection images (above) and detected crease images of Fandisk model (below). Figure 4. Slice of CT volume of (a) 100 crease images, (b) 50 crease images, and (c) 50 original projection images. (d) Extracted surface mesh from CT volume of (c). 2.3 Crease points extraction Crease points are extracted from the CT volume constructed and discussed in the previous section as follows. In the discussion, the Z-axis is defined as the rotation axis. As shown in Figure 5, in each Z-axis slice of the CT volume, we selects the voxels that have higher values than a user-defined threshold and their 8-neighbors. Next, a position and an extending direction of each selected voxel are computed. The vector represents the direction in which the crease is extending from the voxel, as shown in Figure 6. The position is interpolated from the voxel position and is defined by,,, where,, are voxel position indexes, is the voxel value, and is the voxel size. For determining, a variance covariance matrix of position weighted by the voxel value is defined by,,,,, where i identifies a voxel in the kernel,,, are voxel position indexes. The averaged position,, z is computed from the voxel positions in the kernel with weighting their voxel values. The eigenvector of the maximum eigenvalue of the matrix becomes the extending direction of crease. Figure 6 shows Fandisk example of extracted points and extending directions. ict Conference

4 Figure 5. Extraction of points from the CT volume. Figure 6. Diagram of points and extending directions using an example of Fandisk. 2.4 Wireframe model generation By connecting crease points, a wireframe model is generated. To decide the combination of points that should be connected, we compute an evaluation value, for all combinations of two points, and. The smaller the value of,, the higher the evaluation (indicating the points should be connected)., 1 2 Above, and are user-defined weighting coefficients (we set 1 and 5 in our experiments). The first term of the above formula represents the distance between two points, and the second term indicates similarity of the extending directions of the two points. The third term represents the similarity between the connecting direction and extending directions. The following shows the crease points connection algorithm when applying the evaluation value,. 1. Compute evaluation value, for all combinations of two crease points. 2. In ascending order of evaluation value,, if its combination satisfies the below conditions, connect the two points, and. A) Evaluation value, is lower than the user-defined threshold. B) Both points have fewer than two connections. C) Connecting the two points does not create a smaller angle than a user-defined threshold angle. The resulting wireframe model of Fandisk example is shown in the left block of Figure

5 3 Results and discussion 3.1 Simulation data We used simulation projection images of Fandisk shown in Figure 3 for error checking and comparison with the dihedral-thresholding method. The dihedral-thresholding method extracts creases by thresholding dihedral angles of a mesh model that is extracted from a CT volume reconstructed from projection images. For this comparison, we used the simulation data of Fandisk model whose resolution is pixels. The number of projections used for our proposed method was only 50, while 700 projection images were used for the dihedral-thresholding method. The detected crease images are shown in Figure 3, and the reconstructed wireframe model is shown in the left block of Figure 7. As shown, more shallow creases are reconstructed in the left block versus the right block. However, some creases vertical to the rotation axis are missing. This is a limitation of our method Although some creases are missing, our method produced better results than the dihedral-thresholding method. The right block of Figure 7 shows the results of extraction of creases by the dihedralthresholding method and in close-up, the dihedral-thresholding-method-extracted creases can be Figure 7. Left: Wireframe model reconstructed by our method, Right: Extracted edges by the dihedralthresholding method. Figure 8. Fandisk error histogram of our method and the dihedral-thresholding method. ict Conference

6 observed in zig-zag lines. This was because the mesh extracted from the CT volume by Marching Cubes has difficulty preserving sharp edges. Compared with the dihedral-thresholding method, our method extracted crease edges cleanly. Figure 8 shows an error histogram for this example that will be discussed later. Figure 9 shows another example of the reconstructed wireframe results from simulation projection images of a mechanical part. Both the outside and inside crease lines of the object are reconstructed with our method. Figure 8 and Figure 10 show error histograms of points that constitute Fandisk, and mechanical part wireframe models and color maps that represent the errors for each point. The error is defined as the distance between a point and the nearest edge of the ideal wireframe model created from a CAD model. In the color maps, most of the creases have a low value of error except for overdetected points and corner points. Using our method, 76.8% of Fandisk crease points and 84.2% of the mechanical part crease points have an error value that is lower than half of the scaled pixel size. On the other hand, for the dihedral-thresholding method, only 30.0% of Fandisk crease points and 18.4% of the mechanical part crease points meet this criterion. We used a standard laptop PC equipped with a Core i7-3540m 3.00GHz CPU and an NVIDIA GeForce GT 640M LE GPU. The computation of CT reconstruction was accelerated by the GPU. The computation time of each procedure is summarized in Table 1. Figure 9. Projection images (above), detected crease images (below), and reconstructed wireframe model from simulation data of a mechanichal part (right). Figure 10. Mechanical parts error histogram of our method and the dihedral-thresholding method. 444

7 Crease detection [sec.] CT reconstruction [sec.] Crease points extraction [sec.] Wireframe model generation [sec.] Fandisk model Mechanical part The metal plate The step cylinder Table 1. Computation time at each procedure. 3.2 Real scan data Figure 11 and Figure 12 show results for real scan data of a metal plate and step cylinder. The data were scaned by a Carl Zeiss METROTOM 800 system. The resolution of the projection images are pixels and pixels, and the numbers of views are 50 and 100, respectively. Computation time is shown in Table 1. To evaluate accuracy, we measured two dimensions of the metal plate, the width of the plate and diameter of a circle shown by the green arrows in Figure 11. As shown in the table on the right in Figure 11, error of each parameter is much smaller than the scaled pixel size of the CT machine (0.138 mm). In the case of the step cylinder, errors of the reconstructed circles have almost the same value as the scaled pixel size of mm (the table on the right in Figure 12). Figure 11. Projection images (above), detected crease images (below), reconstructed wireframe model from real scan data of metal plate, and comparison of reconstructed points and original model (right) using our method. Figure12. Projection images (above), detected crease images (below), reconstructed wireframe model from real scan data of step cylinder, and comparison of reconstructed points and original model (right) using our method. ict Conference

8 3.3 Discussion Our proposed method may be applied to detection of features other than 3D creases if they are captured in the projection images. One example of such features is fillet borders. Fillet borders are captured as curvature discontinuity lines in projection images, and they may be detected by thresholding the deviation of curvature images. Figure 13 shows the results of fillet border detection and reconstruction of the wireframe model with our method. All fillet borders are reconstructed. From a theoretical point of view, the problem setting of this method is closely related to compressed sensing [6] except that both the signals and their measurements are sparse in our method. We will apply the reconstruction techniques used in compressed sensing to our problem to achieve more efficient computation in the future. 4 Conclusion and future work We proposed a method to obtain 3D crease points and reconstruct a wireframe model from 2D crease images without using a surface mesh model. 2D crease pixels are detected using normal tensor voting from projection images. Using only images, we reconstructed the CT volume and extracted 3D crease points. A wireframe model was created by connecting pixels according to evaluation values computed from position and extending direction of all points. Reconstructed wireframe models had better accuracy than the dihedral-thresholding method. For simulation data, 75% of reconstructed crease points had an error value smaller than half of the scaled pixel size. For real data, the error was about three times smaller than the scaled pixel size. In future work, we will focus on other part features such as fillet borders and will improve our method by finding a solution to extract the feature lines that are vertical to the rotation axis. Figure 13. Examples of fillet border reconstruction. Pink objects are 3D model and blue lines are fillet borders. References [1] Kruth, J.P., Bartscher, M., Carmignato, S., Schmitt, R., De Chiffre, L., Weckenmann, A., Computed tomography for dimensional metrology, CIRP Annals - Manufacturing Technology, Vol 60, Issue 2, pp , 2011 [2] Lorensen, W. E., and Cline, H. E., Marching cubes: A high resolution 3D surface construction algorithm, Computer Graphics, Vol. 21, No. 4, pp , [3] Kim, H. S., Choi, H. K., and Lee, K. H., Feature detection of triangular meshes based on tensor voting theory, Computer-Aided Design, Vol. 41, Issue 1, pp.47 58, 2009 [4] Feldkamp, L. A., Davis, L. C., and Kress, J. W., Practical cone-beam algorithm, Journal of the Optical Society of America A: Optics, Image Science, and Vision, Vol. 1, Issue 6, pp , 1984 [5] Shepp, L. A., and Logan, B. F., The Fourier reconstruction of a head section, IEEE Transactions on Nuclear Science, Vol. 21, No. 3, pp.21 34, 1974 [6] Donoho, D.L., Compressed sensing, IEEE Transactions on Information Theory, Vol. 52, No. 4, pp ,