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where (w,h) denotes the image size in number of pixels, (Sw,Sh) the sensor size in millimeter, (v0,w0) the coordinates of the principle point, θ the shearing angle in the sensor coordinate system and Sx :=w/Sw, Sz :=h/Sh. To this end we obtain a mapping Θ :(X,Y,Z) 7→(x˜t,z˜t) (4) which depends in total on 17 parameters (6 extrinsic, 2 Scheimpflug angles, 4 intrinsic, 5 distortions coefficients). III. SCHEIMPFLUG CAMERA CALIBRATION A standard way for camera calibration in computer vision is the approach of minimizing a functional that measures to which extent the model (4) fits a given set of point-to-point correspondences resulting from a marker positions of a cali- bration plate. A familiar choice for the functional is the sum of squared projection errors. In particular, the estimate of the extrinsic parameters is not that easily performed. Therefore, usually simplified approximations are used as initial guess. For example, [2] starts from a distortion-free model and derives a first guess of the pinhole camera parameters as an approximation. It is then used as an initialization of a nonlinear bundle adjustment optimization that accounts for distortion and the 2-tilt Scheimpflug angels. In a similar way [4] starts with Zhang’s method [7] for estimating the Scheimpflug angels α, β. In a further step, α and β are kept fix and the remaining parameters are estimated, again by using Zhang’s method. This procedure is iterated until convergence. A. Approach for Embedded Laser-Camera Application The application scenario is about real-time affine recon- struction of geometric quantities by means of an embedded laser-camera system based on a DSP (TMS320DM6435, 700 MHz, 5600MIPS) and a hard-real time requirement of processing a measurement below 10ms. On such a platform the computational effort of trigonometric functions is about 20–40 times higher than standard vector operations. In our approach we exploit the fact that the laser projection plane and the plane of focus of the Scheimpflug camera are congruent. This setting allows a simplification of the general calibration procedure and gets along without the use of computational expensive functions. Since the mapping (4) reduces to Θ˜ : (X,Z) 7→ (x˜t,z˜t). Instead of solving the inverse problem of identifying the 17 parameters of the Scheimpflug camera model and tack- ling the problem from a global perspective, we consider the resulting geometric deformation as representation of a Riemannian manifold and exploit its local notions of angle and length of curves for accomplishing measurement tasks. In this view the measurement problem is solved by the following steps: (a) register point-to-point correspondences by means of a sufficiently dense grid of point markers on the plane of focus resulting from straight lines (geodesics in Euclidean geometry) and extraction of the point loca- tions in the image by image processing; (b) determine the neighboring deformed grid points to the sample point; (c) Fig. 3. Left: deformed regular grid of points by Scheimpflug camera and radial and tangential lens distortion:α=β=5◦, k1=−4.5e−3mm−2; right: angle reconstruction errors with 249 pairs orthogonal calibration lines and 286 pairs test lines with different inclined angles (left box: original lines with the same distortion as the grid, mean = 0.397◦, std = 0.431◦; right box: distortion rectified lines, mean = 0.082◦, std = 0.084◦.) apply 3-spline interpolation for approximate recovery of the corresponding geodesics in the resulting Riemannian manifold; (d) determine the Riemannian coordinates in the local coordinate system given by the geodesics; (e) compute the local inverse in order to obtain the Euclidean coordinates. In contrast to computing the full camera model which involves trigonometric functions and fractions, the outlined approach is also feasible on an embedded system as only polynomialsofmaximaldegree3have tobeevaluated.Fig.3 shows an example of a deformed regular grid of calibration points by a Scheimpflug camera and the result of angle measurement based on this approach. The result shows that the systematic angle reconstruction error resulting from non- linear Scheimpflug and lens distortion effects can be reduced substantially which meets the industrial requirements of the specific application. ACKNOWLEDGMENT Thisworkhas beenpartly funded by the AustrianCOMET Program. REFERENCES [1] I. Chavel, Riemannian geometry, a modern introduction. Cambridge University Press, 1993. [2] P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with Scheimpflug optical device,” in Proc. SPIE, 12th Int. Conference on Quality Control by Artificial Vision 2015, vol. 9534, 2015, pp. 953416–953416–7. [3] M. Iba´n˜ez-Ruiz, P. Beneyto-Martin, and M. Pe´rez-Martı´nez, “Lens density measurement with scheimpflug camera in vitrectomised eyes,” Archivos de la Sociedad Espan˜ola de Oftalmologı´a (English Edition), vol. 91, no. 8, pp. 385–390, 2016. [4] A.Legarda,A. Izaguirre,N.Arana,andA. Iturrospe,“Anewmethodfor Scheimpflug camera calibration,” in 10th Int. Workshop on Electronics, Control, Measurement and Signals, June 2011, pp. 1–5. [5] T. Scheimpflug, “Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes,” May 1904, GB 1196/1904. [6] R. Sidney, “The manual of photography: Photographic and digital imaging,” R. E. Jacobson, S. F. Ray, G. G. Atteridge, and N. R. Axford, Eds. Great Britain: Oxford: Focal Press, 2000, ch. The geometry of image formation, pp. 39–60. [7] Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 11, pp. 1330–1334, Nov. 2000. 114