Camera Calibration (Compute Camera Matrix P) 簡韶逸 Shao-Yi Chien Department of Electrical Engineering National Taiwan University Fall 2019 1
Outline • Camera calibration [Slides credit: Marc Pollefeys] 2
Resectioning X P ? x i i
Basic Equations x PX i i x i PX X i Ap 0
Basic Equations Ap 0 minimal solution P has 11 dof, 2 independent eq./points 5½ correspondences needed (say 6) Over-determined solution n 6 points Ap minimize subject to constraint p 1 ˆ 3 p 1 or P ˆ 3 p
Degenerate Configurations More complicate than 2D case (i) Camera and points on a twisted cubic (ii) Points lie on plane or single line passing through projection center
Data Normalization Less obvious (i) Simple, as before 3 2 (ii) Anisotropic scaling
Line Correspondences Extend DLT to lines P T l (back-project line) i T PX T PX l l (2 independent eq.) i 2 i i 1 i
Geometric Error
Gold Standard Algorithm Objective Given n≥6 2D to 2D point correspondences {X i ↔x i ’}, determine the Maximum Likelyhood Estimation of P Algorithm (i) Linear solution: ~ ~ (a) Normalization: x Tx X UX i i i i (b) DLT: (ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: ~ ~ ~ ~ -1 (iii) Denormalization: P T P U
Calibration Example (i) Canny edge detection (ii) Straight line fitting to the detected edges (iii) Intersecting the lines to obtain the images corners typically precision <1/10 (HZ rule of thumb: 5 n constraints for n unknowns
Errors in the World x P X i i Errors in the image and in the world
Geometric Interpretation of Algebraic error 2 ˆ ˆ w i d ( x , x ) i i i ˆ ˆ ˆ ˆ ˆ 3 w p depth(X; P) w x , y , 1 PX i i i i i 3 ˆ therefore, if p 1 then ˆ ˆ ˆ w d ( x , x ) ~ fd ( X , X ) i i i i i
Estimation of Affine Camera Last row = (0, 0, 0, 1) note that in this case algebraic error = geometric error
Gold Standard Algorithm Objective Given n≥4 2D to 2D point correspondences {X i ↔x i ’}, determine the Maximum Likelyhood Estimation of P (remember P 3T =(0,0,0,1)) Algorithm ~ ~ (i) Normalization: x Tx X UX i i i i (ii) For each correspondence A p b 8 8 (iii) solution is p A b 8 8 ~ -1 P T P U (iv) Denormalization:
Restricted Camera Estimation Find best fit that satisfies • skew s is zero • pixels are square • principal point is known • complete camera matrix K is known Minimize geometric error impose constraint through parametrization Image only 9 2n , otherwise 3n+9 5n Minimize algebraic error assume map from param q P=K[R|-RC], i.e. p=g(q) minimize ||Ag(q)||
Reduced Measurement Matrix One only has to work with 12x12 matrix, not 2nx12 ~ ^ T T Ap p A Ap A p
Restricted Camera Estimation Initialization • Use general DLT Clamp values to desired values, e.g. s=0, x = y • Note: can sometimes cause big jump in error Alternative initialization • Use general DLT • Impose soft constraints • gradually increase weights
Exterior Orientation Calibrated camera, position and orientation unkown Pose estimation 6 dof 3 points minimal (4 solutions in general)
Covariance Estimation ML residual error Example: n=197, =0.365, =0.37
Radial Distortion short and long focal length
𝑦, 𝑧 : non-distorted projection 𝑦 𝑒 , 𝑧 𝑒 : distorted projection
Correction of Distortion Choice of the distortion function and center : interior parameters Computing the parameters of the distortion function (i) Minimize with additional unknowns (ii) Straighten lines (iii) …
Correction of Distortion After radial correction
Another Method of Calibration • Notation K ≡ • Homography between the model plane and its image Ref: Zhengyou Zhang , “ Flexible camera calibration by viewing a plane from unknown orientations ,” ICCV1999 . 29
Another Method of Calibration • Constraints on the intrinsic parameters r 1 and r 2 are orthonormal 30
Another Method of Calibration • Close-form solution • Let 31
Another Method of Calibration • Close-form solution • From the two constraints on the intrinsic parameters • V is a 2n x 6 matrix, if 𝑜 ≥ 3 , we will have in general a unique solution b defined up to a scale factor. Once b is estimated, we can compute the camera intrinsic matrix A . 32
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples 33
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples 34
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples 35
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples • If the location of the corners are not correct adjust radial distortion manually 36
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples 37
Calibration Procedure http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples 38
Recommend
More recommend
