1. What is the epipolar geometry?

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1 DGZfP Proceedings BB 67-CD Poster 7 Computerized Tomography for Industrial Applications and Image Processing in Radiology March - 7, 999, Berlin, Germany The Epipolar Geometry in Radioscopic Images D. Mery and D. Filbert, Technical University Berlin, Germany Abstract The epipolar geometry describes the correspondence between different projections of a rigid object. In this work, the epipolar geometry in radioscopic images is introduced. First, a didactical model with two cameras is shown. Since the x-ray images in the radioscopy are normally taken as monocular, the epipolar geometry in monocular images is given. A lineal model for the formation of the x-ray images is firstly used. However, the image intensifier of the radioscopic system presents a curvature, which deforms the x-ray image. A formation function for the image intensifier is modelled considering its input as a hyperbolic surface. The theory is demonstrated by an experiment with real data. Finally, a new approach to automated x-ray inspection of castings is proposed.. What is the epipolar geometry? The epipolar geometry is normally used in stereo vision to solve the correspondence problem in two cameras, i.e. for a given -D point in the first image, it would be found the corresponding -D point in the second one. The correspondence is usually utilised in stereo vision to reconstruct the -D coordinates of a number of points in a scene given several images obtained by cameras of known relative positions and orientations. x x P epipolar line image Q Q image Fig. : Epipolar geometry in two projections DGZfP Proceedings BB 67-CD 8 Poster 7

2 Computerized Tomography and Image Processing, 999 The correspondence between two images is considered in Fig. : for a point x = ( x, y) in image, we look for the point x = ( x, y ) in image it corresponds to. The points x and x are the projections of the same physical point P = ( X, Y, Z) in -D world coordinates. All possible physical points P that may have produced x lie on the line between x and Q, the optical center of image. As a consequence, all possible matches x of x are located on the projection of this line by the optical center Q in the image. The projected line is called the epipolar line. The epipolar constraint is that, for a given point x in the image, its possible matches in the image must all lie on the epipolar line of point x. The dimension of the search is then reduced from two dimensions to only one. The reader who is interested in a detailed description of the epipolar geometry may find it profitable to read [].. The epipolar geometry in monocular images In the previous section the correspondence problem with two cameras was described. However, in radioscopic images of castings the views are monocular since the images are generated by a x-ray tube on only one camera. The castings are inspected in different positions. A geometric model is shown in Fig. where the casting is located between the x-ray tube Q and the projection plane. The image is achieved via central projection. The point x represents the projection of a point P of the casting. x projection P casting Q beam Fig. : Projection of a point of the casting. The epipolar geometry in monocular images can be useful in finding correspondences between different projections of the casting, as will be explained in this section. The possible -D points P that may have caused the projected point x are on the beam of the x-ray tube (see Fig. ). If beam is considered like a part of the casting, it could be rotated together with the casting to the next position. The situation is shown in Fig., where the casting and the beam were rotated. Since DGZfP Proceedings BB 67-CD 8 Poster 7

3 Computerized Tomography and Image Processing, 999 the possible points P lie on beam, the corresponding points of x are located on the projection of this beam on the projection plane. The possible corresponding points of x define the epipolar line. epipolar line x projection beam P Q beam Fig. : Projection of a rotated casting. The epipolar line is the projection of rotated beam. The epipolar constraint remarks that x, i.e. the correspondence of x, must lie on the epipolar line of x. Fig. 4 illustrates the epipolar lines obtained with this model in a pair of radioscopic images. Fig. 4: Two x-ray images: in the left the points of interest are marked. On the right image the related epipolar lines are drawn. The lines pass through the corresponding points. Once the epipolar geometry between two x-ray images is established, the search for correspondences is nicely reduced to one degree of freedom problem. A mathematical approach of this problem is given in []. DGZfP Proceedings BB 67-CD 8 Poster 7

4 Computerized Tomography and Image Processing, 999. Consideration of the projective distortion in the image intensifier Due to the curvature of the image intensifier the approach mentioned above suffers from an inaccuracy in the computation of the epipolar lines, specially at the corners of the x-ray images, where the curvature is higher. A better solution should consider the projective distortion in the image intensifier. This can be attempted, however, by modelling the projection behaviour of the image intensifier for the entire visible -D scene adequately. The input surface of the image intensifier can be modelled as hyperbolic. The new model of the radioscopic system is schematically represented in Fig.. A cubic model could be used too []. to CCD-camera X-ray tube casting image intensifier Fig. : Distortion of a x-ray image in the image intensifier. A way to estimate this model is analysing the projective distortion of a calibration plate which is mounted in front of the image intensifier as shown in Fig. 6. x-ray tube image intensifier calibration plate Fig. 6: Plate to analyse the distortion in the image intensifier. The plate contains holes that have been placed in a regular grid manner. The distortion effect is illustrated in the x-ray image of the calibration plate, see Fig. 7. The regular grid distorted by this model is shown in Fig. 8. DGZfP Proceedings BB 67-CD 84 Poster 7

5 Computerized Tomography and Image Processing, 999 Fig. 7: X-ray of the calibration plate. Fig. 8: Hyperbolic modelled grid. Using this model it is clear that the epipolar lines are transformed now in epipolar curves. To illustrate this approach two consecutive x-ray images of an aluminium wheel are considered. In this experiment a standard video camera is used. It supplies an image with 76 lines, each having 768 pixels. The first image is shown in Fig. 9. The points of interest were selected manually (see blue marks) Fig. 9: X-ray image of a casting by. The second one is shown in Fig. 0a. The rotation of the casting in this experiment was 0. The corresponding epipolar curves are drawn in red. The results are acceptable, they corroborate the theory: the epipolar curves pass very close to their corresponding point. The inaccuracy is about three pixels (in 8 points the mean of the error was. pixels and the maximal error.77 pixels), what is sufficient to find correspondences along this curves. A detail is shown in Fig. 0b. Fig. illustrates the epipolar lines computed using the linear model. The results show that the distance between the corresponding points and the epipolar lines is considerable (in 8 points the mean of the error was.0 pixels and the maximal error. pixels). DGZfP Proceedings BB 67-CD 8 Poster 7

6 Computerized Tomography and Image Processing, (a) Fig. 0: (a) X-ray image by and the related epipolar curves using the hyperbolic model. (b) The inaccuracy in point is about 0. pixels ( 0. mm). (b) (a) Fig. : (a) X-ray image by and the related epipolar lines using the linear model. (b) The inaccuracy in point is about 0 pixels ( mm). (b) 4. Application of the epipolar geometry in the inspection of castings In this section a new approach to automated x-ray inspection of castings using the epipolar geometry is theoretically proposed. In the inspection a handling system to manipulate the castings is used. The epipolar geometry of the radioscopic system can be easily estimated since the position of the casting is available from the handling system, i.e. for each x-ray image the position of the casting in -D is known (examples were shown in the last section). To inspect a casting, a sequence of x-ray images in different positions are taken. Using classical methods for automated inspection [4-7], hypothetical flaws can DGZfP Proceedings BB 67-CD 86 Poster 7

7 Computerized Tomography and Image Processing, 999 be founded in each image. In order to keep low the amount of false positive detections, the next constraints must be satisfied: Correspondence: the epipolar constraint can be used to find the correspondences between the hypothetical flaws: if a detection has no correspondence in other image the hypothetical flaw is classified as false. It is clear that this correspondence problem involves the matching of features of the detected flaws. Elliptical trajectory: the tracking of a flaw using the epipolar geometry is simple: if two corresponding points x and x are founded in image and, the corresponding point x must belong to the epipolar curve of x in the third image and to the epipolar curve of x in the third image. Therefore, x belongs to the intersection of theses two curves, i.e. a point can be predicted in the third image as a function of its first two images. The tracking of a flaw must describe a elliptical trajectory since the movement of the casting is circular. Correctly localisation in the -D space: If the correspondence between two points x and x using the epipolar geometry is established, the localisation of the corresponding point in -D can be estimated by the estimation of the intersection between beam and beam, as shown in Fig.. The -D point must belong to the space occupied by the casting (its dimensions are normally known). Acknowledgements The authors are grateful to the German Academic Exchange Service, DAAD, and YXLON International X-Ray GmbH, Hamburg, for their support. We thank Dr.rer.nat U.Ewert and Prof.Dr.-Ing H. Wüstenberg from the Federal Institute for Materials Research and Testing, BAM, Berlin, for many interesting discussions on this subject. References [] Faugeras, O.: Three-Dimensional Computer Vision: A Geometric Viewpoint. The MIT Press, Cambridge MA, London 99. [] Mery, D.: Anwendung der Epipolargeometrie in der automatischen Gußfehlererkennung mit monokularen Aufnahmen. Jahresbericht 998 des Institutes für Meß- und Automatisierungstechnik, Technische Universität Berlin, 4-48, 998. [] Brack, C.; Götte, H.; Gossé, F.; Moctezuma, J.; Roth, M.; Schweikard, A.: Towards Accurate X-Ray Camera Calibration in Computer-Assisted Robotic Surgery. Proc. Int. Symposium Computer Assisted Radiology (CAR), Paris, 7-78, 996. [4] Purschke, M., Schulenburg, H.: Fortschritte der vollautomatischen Röntgenprüfung.Jahrestagung der Deutschen Gesellschaft für Zerstörungsfreien Prüfung e.v. (DGZfP), Bamberg September, 998. [] Heinrich, W.: Automatische Röntgenserienprüfung von Gußteilen.Dissertation am Institut für Allgemeine Elektrotechnik, Technische Universität Berlin, 988. [6] Hecker, H.: Ein neues Verfahren zur robusten Röntgenbildauswertung in der automatischen Gusßteilprüfung. Dissertation am Institut für Allgemeine Elektrotechnik, Technische Universität Berlin, 99. [7] Wenzel, T., Hanke, R.: Fast image processing on die castings. Anglo-German Conference on Non-Destructive Testing Imaging and Signal Processing, Oxford, 7-8 March, 998. DGZfP Proceedings BB 67-CD 87 Poster 7