Slide 1Table of ContentIntroductionIntroduction – Related WorksIntroduction – Related WorksIntroduction – Related WorksCamera ModelCalibration Concept – 2DCalibration Concept – 2DCalibration Concept – 2DCalibration Concept – 3DCalibration Concept – 3DCalibration Concept – 3DExperimental ResultExperimental ResultDiscussionConclusionFuture WorksA Generic Concept for Camera CalibrationPeter Sturm and Srikumar RamaligamSung HuhCPSC 643Individual Presentation 4April 15, 2009Table of ContentIntroductionCalibration◦2D Known and Unknown motion◦3D Known and Unknown motionDiscussionConclusionFuture WorkIntroductionDevelop a calibration method for any camera model◦Cameras w/o a single effective view pointGeneral model of camera adopted:◦Images consisting of pixels◦Each pixel captures light that travels along a ray in 3DCamera is fully described by:◦Coordinate Of rays◦Mapping b/w rays and pixelsIntroduction – Related WorksExisting Calibration methodsR.I. Hartley, A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, 2000C.C. Slama. Manual of Photogrammetry. Fourth Edition, ASPRS, 1980Calibration for the general imaging modelM.D. Grossberg, S.K. Naar. A general imaging model and a method for finding its parameters. ICC, 2001R. Swamminathan, M.D. Grossberg, and S.K. Nayar. Caustics of Catadioptric Cameras. ICCV, 2001Introduction – Related WorksEpipolar geometry estimation and modelingT. Pajdla. Stereo with oblique cameras. IJCV, 47(1), 2002.S. Seitz. The space of all stereo images. ICCV, 2001.Y. Wexler, A.W. Fitzgibbon, A. Zisserman. Learning epipolar geometryfrom image sequence. CVPR, 2003.Motion estimation for calibrated camerasJ. Neumann, C. Fermuller, Y. Aloimonons. Polydioptric Camera design and 3D Motion Estimation. CVPR, 2003.R. Pless. Using Many Cameras as One. CVPR, 2003Introduction – Related WorksThe special case of a linear calibration objectP. Sturms, S. Ramalingam. A Generic Calibration Concept: Theory and Algorithms. Research Report 5058, INRIA, France, 2003Camera ModelUse infinite extensions of half-ray (Camera Rays)Non-central Camera◦Camera rays correspond to different pixel does not intersectCentral Camera w/ optical center◦All camera rays intersect in a single pointCalibration Concept – 2DKnown motion◦Object’s motion b/w image is known◦Two object points can be mapped to a single coord. Frame◦Joining two points to compute pixel’s camera ray◦Knowledge of point position relative to coord. frames of object and the motion b/w the two coord. frameCalibration Concept – 2DUnknown Motion◦Estimate unknown motion ◦Let Q, Q’, Q” be the points on the calibration object◦Common frame = coord. frame associated w/ object’s first position (relative motions are given by rotation matrix and translation vector)QTR' ''1� �� �� �tQ0TR" ""1� �� �� �tQ0CollinearCalibration Concept – 2DCoefficients of a trilinear matching tensorDepends on the calibration tensorTotal 27 coefficients ( 8 are always zero, 6 pairs of identical ones)1310i iiC V==�Ci = Trilinear product of point coordinatesVi = Associated coefficients of the tensorCalibration Concept – 3DKnown Motion◦Two views are sufficient◦Equivalent procedure as 2D cameraUnknown Motion◦Start with Q, Q’, Q”◦Same coord. frame as in 2D◦Aligned points are collinearCalibration Concept – 3DRank of matrix must be less than 3◦All sub-determinants of size 3 x 3 vanishes (4 of them)◦Corresponds to a trilinear equation in point coord.34/64 coefficients are always zeroCalibration Concept – 3DEstimate tensors by solving linear equation systemLet Vi’=Vi, Wi’ = Wi, i = 1…37◦Estimate factors , ◦Compute and , i = 1…37◦Compute R’ and R”◦Compute t’ and t” by solving a straight forward linear least square problem'2 '2 '28 9 10λ V V V= + +'2 '2 '21 2 3μ W W W= + +'λiiVV ='μiiWW =15 16 1715 16 178 9 10R 'W W WV V VV V V- - -� �� �= - - -� �� �� �18 19 2018 19 2011 12 13R"W W WV V VV V V� �� �=� �� �- - -� �Experimental ResultResult from Central cameraEstimated motion parameters give rise to nearly perfectly collinear calibration pointsRadial distortion modeled correctlyExperimental ResultResult from fish-eye lensAligned calibration points are not always perfectly collinearOnly the central image region has been calibratedDiscussionAlgorithm for central cameras work fineNon-central catadioptric cameras give unsatisfying result◦Homography-based interpolation of calibration pointsGeneral algorithm does not work for perspective cameras, but for multi-stereo systems consisting of sufficiently many cmerasConclusionTheory and algorithms for a highly general calibration concept proposedDifferent cameras will likely require different designs of calibration objectFuture WorksDeveloping bundle adjustment procedures to calibrate from multiple imagesMotion and pose estimation and triangulation from perspective to the general imaging