Page - 165 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

Line Processes for Highly Accurate Geometric Camera Calibration Manfred Klopschitz, Niko Benjamin Huber, Gerald Lodron and Gerhard Paar Abstract—The availability of highly accurate geometric camera calibration is an implicit assumption for many 3D computer vision algorithms. Single-camera applications like structure frommotion or rigid multi-camera systems that use stereo matching algorithms depend on calibration accuracy. We present an approach that has proven to deliver accurate geometric information in a reliable, repeatable manner for many industrial applications. The major limitation in typical cameracalibrationmethods is theprintingaccuracyof theused target. We address this problem by modeling the calibration target uncertainty as a line process and incorporate a lifted cost function into a bundle adjustment formulation. The regu- larized targetdeformation is incorporateddirectly into thenon- linear least-squares estimation and is solved in a non-iterative, principled framework. I. INTRODUCTION Geometric camera calibration defines the mapping be- tween points in world coordinates and their corresponding image locations. These parameters model imperfections of the camera optics, i.e. lens distortion, intrinsic parameters of the idealized pinhole camera and extrinsic parameters like absolutecameraorientationandrelativeorientation formulti- camera setups. Most calibration methods assume known 3D world points and minimize a reprojection error of the known 3D structure into detected image correspondences. The resulting error is a result of model imperfections, target imperfections and feature point localization inaccuracies. Impressive reprojection errors have been shown in [5] by estimating feature points and 3D structure in an iterative procedure. We argue, like [2], [4], that the most important aspect for many applications is printing accuracy, but present a non-iterative calibration formulation that estimates and cor- rects for target uncertainty within a single bundle adjustment minimization. The geometric camera calibration process estimates the mapping between points in world coordinates and their cor- responding image locations. We define the image projection using standard notation, for the pinhole model xp=KR[I|−C˜]X=PX ∣∣∣∣∣ K=   f cxf cy 1   R and C˜model the location of the camera in space and K defines the intrinsics. Lens distortion is added to the pinhole Joanneum Research Forschungsgesellschaft mbH, Steyrergasse 17, 8010 Graz, Austriafirstname.lastname@joanneum.at This work was supported by the K-Project Vision+ which is funded in the context of COMET - Competence Centers for Excellent Technologies by BMVIT, BMWFJ, Styrian Business Promotion Agency (SFG), Vienna Business Agency, Province of Styria Government of Styria and FFG under the contract 838299 HiTES3D. The programme COMET is conducted by the FFG. projection, for example using this popular model: xd=xp+FD(xp,δ) FD(xp,δ)= [ x1p(k1r2p+k2r 4 p)+2p1x1px2p+p2(r 2 p+2x 2 1p) x2p(k1r2p+k2r 4 p)+p1(r 2 p+2x 2 2p)+2p2x1px2p ] with xp = (x1p,x2p)T, rp = √ x21p+x 2 2p and δ = (k1,k2,p1,p2)T. k1,k2 are the radial distortion coefficients and p1,p2 the tangential distortion coefficients. II. A LIFTED STRUCTURE ADJUSTMENT FORMULATION Bundle adjustment (BA) minimizes the sum of the ge- ometric distances of all image measurements xij and their corresponding projected 3D pointsPiXj in image space: min Pi,δ,Xj ∑C(xij,FD(PiXj,δ)) where Pi is the pinhole camera model, δ the distortion parameters and C is the reprojection error, for example with a quadratic error Cs(x,xp)= ∥∥x−xp∥∥2 for classical BA. Optimizing all BA parameters with all pinhole terms, distortion terms and the structure Xj simultaneously is ill- conditioned. Therefore, related work that also adjusts the calibration target updates the structureXj in an iterative way by using heuristics of multiple BA runs [2] or use minimal structure constraints [4] and suffer from convergence issues and limitations in possible distortion models. We want to limit the adjustment of the calibration target as far as possible and only adjust the structure if the observed error cannot be explained by other parameters of our model. Suppose we have a scalar error e and rewrite the error as a robust kernelψ(e) by introducing an additional variablew, i.e. a line process [3] ψ(e)=min w ( 2w2e2+(1−w2)2)|w∈ [0,1]. For small errorsw→1 and for large errorsw vanishes and ψ(e)becomesconstant, see [7] foran intuitiveexplanation in thecontextofoutlier estimation (the samekernel isusedhere for simplicity) and [6] for a recent application to robust BA. We apply this concept to camera calibration and introduce variables to represent the correctness of the calibration target and therefore 3D structure. Adding the lifted cost function to represent structure imperfections leads to this extended calibration formulation: min Pi,δ,Xj,wj { ∑C(xij,FD(PiXj,δ))+α∑ j ψ( ∥∥Xj−Xjc∥∥)} = min Pi,δ,Xj,wj { ∑ ij C(xij,FD(PiXj,δ)) +2α∑ j w2j ∥∥Xj−Xjc∥∥+α∑ j (1−w2j)2 } 165