Robotics 2 Camera Calibration. Barbara Frank, Cyrill Stachniss, Giorgio Grisetti, Kai Arras, Wolfram Burgard
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1 Robotics 2 Camera Calibration Barbara Frank, Cyrill Stachniss, Giorgio Grisetti, Kai Arras, Wolfram Burgard
2 What is Camera Calibration? A camera projects 3D world-points onto the 2D image plane Calibration: Finding the quantities internal to the camera that affect this imaging process Image center Focal length Lens distortion parameters
3 Motivation Camera production errors Cheap lenses Precise calibration is required for 3D interpretation of images Reconstruction of world models Robot interaction with the world (Hand-eye coordination)
4 Projective Geometry Extension of Euclidean coordinates towards points at infinity Here, equivalence is defined up to scale: Special case: Projective Plane A linear transformation within is called a Homography
5 Homography Homography has 9-1(scale invariance)=8 DoF A pair of points gives us 2 equations Therefore, we need at least 4 point correspondences for calculating a Homography
6 Pinhole Camera Model Perspective transformation using homogeneous coordinates: Intrinsic camera parameters Extrinsic camera parameters
7 Pinhole Camera Model
8 Pinhole Camera Model Perspective transformation using homogeneous coordinates: world/scene coordinate system
9 Pinhole Camera Model Perspective transformation using homogeneous coordinates: camera coordinate system
10 Pinhole Camera Model Perspective transformation using homogeneous coordinates: image coordinate system
11 Pinhole Camera Model Interpretation of intrinsic camera parameters:
12 Pinhole Camera Model Interpretation of intrinsic camera parameters: focal length x-offset y-offset
13 Lens distortion Non-linear effects: Radial distortion Tangential distortion Compute corrected image point: where : radial distortion coefficients : tangential distortion coefficients
14 Camera Calibration Calculate intrinsic parameters and lens distortion from a series of images 2D camera calibration 3D camera calibration Self calibration
15 Camera Calibration Calculate intrinsic parameters and lens distortion from a series of images 2D camera calibration 3D camera calibration self calibration need external pattern
16 Camera Calibration Calculate intrinsic parameters and lens distortion from a series of images 2D camera calibration 3D camera calibration self calibration
17 2D Camera Calibration Use a 2D pattern (e.g., a checkerboard) Trick: set the world coordinate system to the corner of the checkerboard
18 2D Camera Calibration Use a 2D pattern (e.g., a checkerboard) Trick: set the world coordinate system to the corner of the checkerboard Now: All points on the checkerboard lie in one plane!
19 2D Camera Calibration Since all points lie in a plane, their component is 0 in world coordinates
20 2D Camera Calibration Since all points lie in a plane, their component is 0 in world coordinates
21 2D Camera Calibration Since all points lie in a plane, their component is 0 in world coordinates Thus, we can delete the 3 rd column of the Extrinsic parameter matrix
22 2D Camera Calibration Since all points lie in a plane, their component is 0 in world coordinates Thus, we can delete the 3 rd column of the Extrinsic parameter matrix
23 2D Camera Calibration Homography Since all points lie in a plane, their component is 0 in world coordinates Thus, we can delete the 3 rd column of the Extrinsic parameter matrix
24 2D Camera Calibration
25 2D Camera Calibration
26 2D Camera Calibration Note that form an orthonormal basis, thus:
27 2D Camera Calibration
28 2D Camera Calibration
29 2D Camera Calibration
30 2D Camera Calibration
31 2D Camera Calibration is symmetric and positive definite
32 2D Camera Calibration is symmetric and positive definite Thus: Note: K can be calculated from B using Cholesky factorization
33 2D Camera Calibration is symmetric and positive definite Thus: Note: K can be calculated from B using Cholesky factorization define:
34 2D Camera Calibration is symmetric and positive definite Thus: Note: K can be calculated from B using Cholesky factorization define: Reordering of leads to the system of the final equations:
35 Direct Linear Transformation Each plane gives us two equations Since has 6 degrees of freedom, we need at least 3 different views of a plane We need at least 4 points per plane
36 Direct Linear Transformation Real measurements are corrupted with noise Find a solution that minimizes the least-squares error
37 Non-Linear Optimization Lens distortion can be calculated by minimizing a non-linear function Estimation of using non-linear optimization techniques (e.g. Levenberg-Marquardt) The parameters obtained by the linear function are used as starting values
38 Results: Webcam Before calibration: After calibration:
39 Results: ToF-Camera Before calibration: After calibration:
40 Summary Pinhole Camera Model Non-linear model for lens distortion Approach to 2D Calibration that accurately determines the model parameters and is easy to realize
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