Computer Vision. Geometric Camera Calibration. Samer M Abdallah, PhD

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1 Computer Vision Samer M Abdallah, PhD Faculty of Engineering and Architecture American University of Beirut Beirut, Lebanon Geometric Camera Calibration September 2, Computer Vision Geometric Camera Calibration 1

2 Series outline Cameras and lenses Geometric camera models Geometric camera calibration Stereopsis 2 Computer Vision Geometric Camera Calibration 2

3 Lecture outline The calibration problem Least-square technique Calibration from points Radial distortion A note on calibration patterns 3 Computer Vision Geometric Camera Calibration 3

4 Camera calibration Camera calibration is determining the intrinsic and extrinsic parameters of the camera. The are three coordinate systems involved: image, camera, and world. Key idea: to write the projection equations linking the known coordinates of a set of 3-D points and their projections, and solve for the camera parameters. 4 Computer Vision Geometric Camera Calibration 4

5 Projection matrix M is only defined up to scale in this setting. 5 Computer Vision Geometric Camera Calibration 5

6 The calibration problem 6 Computer Vision Geometric Camera Calibration 6

7 Linear systems A x b = Square system: Unique solution Gaussian elimination A x b = Rectangular system: underconstrained: Infinity of solutions Overconstrained: no solution Minimize Ax-b 2 7 Computer Vision Geometric Camera Calibration 7

8 How do you solve overconstrained linear equations? 8 Computer Vision Geometric Camera Calibration 8

9 Homogeneous linear equations A x 0 = Square system: Unique solution = 0 Unless Det(A) = 0 A x 0 = Rectangular system: 0 is always a solution Minimize Ax 2 under the constraint x 2 = 1 9 Computer Vision Geometric Camera Calibration 9

10 How do you solve overconstrained homogeneous linear equations? The solution is the eigenvector e 1 with least eigenvalue of U T U. 10 Computer Vision Geometric Camera Calibration 10

11 Example: Line fitting Problem: minimize with respect to (a,b,d). Minimize E with respect to d: Minimize E with respect to a,b: where and 11 Computer Vision Geometric Camera Calibration 11

12 Estimation of the projection matrix The constraints associated with the n points yield a system of 2n homogeneous linear equations in the 12 coefficients of the matrix M, When n 6, homogeneous linear least-square can be used to compute the value of the unit vector m (hence the matrix M) that minimizes Pm 2 as the solution of an eigenvalue problem. The solution is the eigenvector with least eigenvalue of P T P. 12 Computer Vision Geometric Camera Calibration 12

13 Estimation of the intrinsic and extrinsic parameters Once M is known, you still got to recover the intrinsic and extrinsic parameters! This is a decomposition problem, NOT an estimation problem. ρ Intrinsic parameters Extrinsic parameters 13 Computer Vision Geometric Camera Calibration 13

14 Estimation of the intrinsic and extrinsic parameters Write M = (A, b), therefore Using the fact that the rows of a rotation matrix have unit length and are perpendicular to each other yields and 14 Computer Vision Geometric Camera Calibration 14

15 Estimation of the intrinsic and extrinsic parameters 15 Computer Vision Geometric Camera Calibration 15

16 Taking radial distortion into account Assuming that the image centre is known (u 0 = v 0 = 0), model the projection process as: 1 λ 1 p = 0 z λ M 1 P where λ is a polynomial function of the squared distance d 2 between the image centre and the image point p. It is sufficient to use low-degree polynomial: λ = 1+ κ pd q p= 1 2 p, with q 3 and the distortion coefficients κ p ( p = 1, K, q) ˆ 2 2 d = u + vˆ 2 This yields highly nonlinear constraints on the q + 11 camera parameters. 16 Computer Vision Geometric Camera Calibration 16

17 Calibration pattern The accuracy of the calibration depends on the accuracy of the measurements of the calibration pattern. 17 Computer Vision Geometric Camera Calibration 17

18 Line intersection and point sorting Extract and link edges using Canny; Fit lines to edges using orthogonal regression; Intersect lines. 18 Computer Vision Geometric Camera Calibration 18

19 References Computer Vision: A Modern Approach. D. Forsyth and J. Ponce, Prentice Hall, 2003 Introductory Techniques for 3-D Computer Vision. E. Trucco and A. Verri, Prentice Hall, 2000 Geometric Frame Work for Vision Lecture Notes. A. Zisserman, University of Oxford 19 Computer Vision Geometric Camera Calibration 19

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