Page - 113 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

Riemannian Manifold Approach to Scheimpflug Camera Calibration for Embedded Laser-Camera Application Xiaoying Tan1, Volkmar Wieser1, Stefan Lustig2 and Bernhard A. Moser1 Abstract—This industrial spotlight paper outlines a Rie- mannian geometry inspired approach to measure geometric quantities in the plane of focus of a Scheimpflug camera in the presence of nonlinear distortions caused by the Scheimpflug model and non-linear lens distortion. I. INTRODUCTION For the standard pinhole camera model, the image sensor is parallel to the lens plane and perpendicular to the optical axis. For this type of camera, the points on a plane surface parallel to the lens can be focused sharply on the sensor plane. However, for some specific application scenarios, the surface of interest is oblique to the lens plane. For example, to capture most parts of a tall building facade into the camera view, the camera needs to be tilted upwards with respect to the building facade. In this case, the standard camera is only able to project a narrow line region of the building facade on sharp focus. It is interesting that the Gaussian focus equation remains valid under the condition that the sensor plane, the lens plane and the object plane intersect in a common line [5]. The Scheimpflug model is encountered in various fields of applications, e.g., architectural photography [6] or in ophthalmology for measuring the thickness of the cornea [3]. In this industrial spotlight paper we address the prob- lem of accurately measuring geometric quantities in the Scheimpflug plane in the presence of non-linear lens dis- tortion effects by following a Riemannian geometry ap- proach [1]. In contrast to state-of-the-art approaches the outlinedapproach is feasibleonembeddedplatformsandgets along without guessing initial values and iterative optimiza- tion steps. Rather, it models the image formation mapping from the Scheimpflug plane to the image plane directly by exploiting point-to-point correspondences and interpolation. In section II we recall the Scheimpflug model and cali- bration approaches from literature. Section III-A outlines our parameter-free approach together with experimental results. II. SCHEIMPFLUG CAMERA In contrast to the standard pinhole camera, in the Scheimpflug camera model the sensor plane and the lens plane are no longer parallel. See Fig. 1 of a schematic view of the Scheimpflug model. The mathematical model of its image formation mapping can be derived from decomposing the mapping from world coordinates (X,Y,Z) to image pixel coordinates (x˜t,z˜t) into a concatenation of mappings as 1 X. Tan, V. Wieser and B. Moser are with the Software Competence Center Hagenberg (SCCH),xiaoying.tan@scch.at 2S. Lustig is associated with SCCHstefan.lustig@scch.at Fig. 1. Scheimpflug camera model: the sensor plane Ptilt and lens plane Plens are no longer parallel. The image formation mapping is modeled by means of the virtual parallel plane Pperp. Fig.2. Theprocessof imageformationof theScheimpflugmodelaccording to (1), (2) and (3) indicated in Fig. 2. First of all, the mapping from (X,Y,Z) to a virtual parallel sensor plane (x′,z′) models the familiar pinhole camera. By taking non-linear radial and tangential lens distortion effects into account, due to suboptimal shape and mounting of lens, and modeling these effects by means of polynomial functions we obtain( x˜′ z˜′ ) := ( x′ z′ ) + ( ∆x(k1r2+k2r4+k3r6) ∆z(k1r2+k2r4+k3r6) ) + ( 2t1x′z′+t2(r2+2x′2) 2t2x′z′+t1(r2+2z′2) ) , (1) where r2 :=∆x2+∆z2,∆x :=x′−x0,∆z :=z′−z0, (x0,z0)are the coordinates of the optical axis on Pperp, k1,k2,k3 are radial and t1,t2 are tangential distortion parameters. The mapping from (x˜′,z˜′) to (xt,zt)models the proper Scheimpflug effect by taking the tilt of the sensor plane into account. Let us denote byα the angle between z˜′ and zt and byβ the angle between x˜′ and xt, then due to [4] we obtain( xt zt ) :=λ · ( x˜′/cosβ+ z˜′ tanα tanβ z˜′/cosα ) (2) whereλ := f/(f−x˜′ tanβ−z˜′ tanαcosβ)and f is the focal length. Finally, we obtain the image pixel coordinates( x˜t z˜t ) := ( Sx −Szcotθ 0 Sz/sinθ ) · ( xt zt ) + ( v0 w0 ) (3) 113