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A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation Outline • • • • • Introduction Model and Approach Further Discussion Experiments and Results Conclusions Introduction • Lens distortion usually can be classified into three types : – radial distortion (predominant) – decentering distortion – thin prism distortion Wang, J., Shi, F., Zhang, J., Liu, Y.: A new calibration model and method of camera lens distortion. Introduction • Method of obtaining the parameters of the radial distortion function and correcting the images. These previous works can be divided roughly into two strategic approaches – multiple views method – Single view method Introduction Correct the radial distortion • Former approach – based on the collinearity of undistorted points. – Need the camera intrinsic parameters and 3D-point correspondences. • This paper – based on single images and the conclusion that distorted points are concyclic and uses directly the distorted points. – uses the constraint, that straight lines in the 3D world project to circular arcs in the image plane, under the single parameter Division Model Model and Approach • Radial Distortion Models – PM、DM • Distorted straight line is a circle • calibration procedure to estimate the center and the parameter of the radial distortion – Circle fitting : LS、LM Radial Distortion Models , ∶ distorted image point , ∶ undistorted image point • The Polynomial Model (PM) that describe radial distortion : = 1 + λ1 2 + λ2 4 + ⋯ (1) ∶ distances of , to the distorted centre ∶ distances of , to the distorted centre λ ∶ radial distortion parameter Radial Distortion Models • The Division Model (DM) that describe radial distortion : = 1+λ1 2 +λ2 4 +⋯ (2) • we use single parameter Division Model as our distortion model : = 1+λ 2 (3) Radial Distortion Models • To simplify equation, we suppose distorted center is the origin image coordinates system, thus : P (0,0) = 1+λ 2 , = 1+λ 2 where 2 = 2 + 2 (4) The Figure of Distorted Straight Line • We consider collinear points and their distorted images. • Let = + straight line equation from (4) We have 1+λ 2 = 1+λ 2 + (5) = + + λ 2 + 2 (6) λ (7) 2 2 + + − 1 λ 1 λ + =0 The Figure of Distorted Straight Line 2 + 2 + λ − 1 λ 1 + λ =0 (7) • The graphics of distorted “straight line” is a circle under the condition of model (3) we use single parameter Division Model as our distortion model : = (3) 2 1+λ Estimate the and λ • Let 0 , 0 be the coordinates of the distorted center . From (7) , we have 2 − 0 1 − λ 2 + + − 0 2 + 1 λ λ − 0 − 0 + = 0 2 2 (8) 1 + − 20 + − − 20 λ λ 2 +0 + 0 − λ 0 + 1 λ 0 1 λ + =0 (9) Estimate the and λ 2 + 2 1 + − 20 + − − 20 λ λ 2 2 +0 + 0 − Let A= λ λ 0 + 1 λ 0 − 20 , = 2 2 = 0 + 0 − 1 λ + =0 1 − λ 0 λ + (9) − 20 1 0 λ 1 + λ 2 + 2 + + + = 0 , we have (10) Estimate the and λ Let 1 A = − 20 , = − − 20 λ λ 1 1 2 2 = 0 + 0 − 0 + 0 + λ λ λ , we have 2 + 2 + + + = 0 (10) Base on the relation of , , and , we have (圓方程式參數A、B、C 與 radial distortion 參數 P、λ 的關係式) 2 2 1 λ 0 + 0 + 0 + 0 + − = 0 (11) Estimate the and λ 2 + 2 + + + = 0 2 2 0 + 0 + 0 + 0 + 1 − λ =0 (10) (11) • Obtain 0 , 0 of distorted center – Extract three “straight line” from image , we can get , , =1,2,3 by circle fitting from (10) according to (11) , we have 1 − 2 0 + 1 − 2 0 + 1 − 2 = 0 1 − 3 0 + 1 − 3 0 + 1 − 3 = 0 2 − 3 0 + 2 − 3 0 + 2 − 3 = 0 (12) • Obtain the parameter of radial distorted λ – substitute 0 , 0 obtained from 12 to (11) 1 λ = 0 2 + 0 2 + 0 + 0 + (13) Sum up whole algorithm • Extract ( ≥ 3) “straight line” = 1,2, ⋯ , from the image • Determine parameter , , =1,2,⋯, by fitting every “straight line” with a circle according to (10) 2 + 2 + + + = 0 • Calculate the center 0 , 0 of the radial distortion according to (12) 1 − 2 0 + 1 − 2 0 + 1 − 2 = 0 1 − 3 0 + 1 − 3 0 + 1 − 3 = 0 2 − 3 0 + 2 − 3 0 + 2 − 3 = 0 • Compute the parameter λ of radial distortion according to (13). 1 λ = 0 2 + 0 2 + 0 + 0 + Circle fitting • It is a very important step to fit circle above algorithm. – data extracted from image are only short arcs, it is hard to reconstruct a circle from the incomplete data. Distorted “straight line” • Method Circle to fit Distorted center – Direct Least Squares Method of Circle Fitting (LS) – Levenberg-Marquardt Method of Circle Fitting (LM) Circle fitting - LS 2 + 2 + + + = 0 (10) • For each point , on the “straight line”, (10) gives , , 1 = − 2 + 2 (14) • Stacking equations from N points together gives M =b (15) Where M is N×3, b is N×1 matrix Circle fitting - LS • Directly using linear least squares fit method, we can get , , T T = M M −1 MT b (16) Circle fitting - LM Main ideas • Let the equation of a circle be 1 2 + 2 + 2 + 3 + 4 = 0 Subject to the constraint : 2 2 + 3 2 − 41 4 = 1 • (17) (18) The distance from a point , to the circle = 2 1+ 1+41 Where = 1 2 + 2 + 2 + 3 + 4 (19) (20) Circle fitting - LM • From (18) 2 2 + 3 2 − 41 4 = 1 , we can define an angular coordinate by 2 = 1 + 41 4 cos , 3 = 1 + 41 4 sin (21) • Apply the standard Levenberg-Marquardt scheme to minimize the sum of squared distance ℱ= 2 in the three dimensional parameter space 1 , 4 , Further Discussion • In Algorithm1, we must have ( ≥ 3) “straight lines”, we relax this constrain and discuss the conditions of – Only one straight line (L1) – Only two straight lines (L2) – Non-square pixels Only One Straight Line (L1) • Suppose the distortion center is the image center and calculate the distortion parameter λ by (13) 1 λ = 0 2 + 0 2 + 0 + 0 + (13) Only Two Straight Lines (L2) • Extract “straight line” 1 、2 from the image; • Determine parameter , , =1,2 by fitting the “straight line” 1 、2 with a circle according to (10); 2 + 2 + + + = 0 (10) (12) become 1 − 2 0 + 1 − 2 0 + 1 − 2 = 0 (22) Only Two Straight Lines (L2) • Select a suitable interval = − 30, + 30 is suggested, for any 0 ∈ , calculating 0 according to (22); 1 − 2 0 + 1 − 2 0 + 1 − 2 = 0 ∈ (22) 0 → 0 ( , ) ( − 30, ) 0 , 0 = ( + 30, ) Only Two Straight Lines (L2) • Calculate the distortion parameter λ = (13), for any = 0 , 0 ; 1 λ = 0 2 + 0 2 + 1 0 + 1 0 + 1 λ = 0 2 + 0 2 + 2 0 + 2 0 + (13) 1 2 λ 1 + λ 2 according to 0 → 0 0 , 0 = ∈ ( , ) ( − 30, ) ( + 30, ) λ 1 λ 2 1 1 λ = λ + λ 2 2 Only Two Straight Lines (L2) • Calculate the corresponding corrected points 1 , 1 ( 2 , 2 ), for any λ , and all distorted points 1 , 1 1 ( 2 , 2 2 ) according to (4); = 1+λ 2 1+λ 2 2 + 2 , = where 2 = (4) • Let [d, k] = min = min 1 + 2 , then obtain the optimal estimation and λ . 1 2 1 2 Non-square Pixels • Let 0 , 0 : the coordinates of the distorted centre : pixel aspect radio The distorted radius is given by 2 = − 0 2 + 2 − 0 • From (8) we have 2 − 0 − λ 2 + − 0 2 2 1 λ + λ − 0 − 0 + = 0 + + − 20 λ +0 2 + 2 0 2 − 0 λ 2 2 2 (23) + − − 2 2 0 λ 1 + 0 + = 0 λ λ (24) Non-square Pixels 2 + 2 2 + − λ +0 2 + 2 0 2 − − 2 2 0 λ 1 0 + = 0 λ λ 20 + − λ 0 + (24) • Equation (24) shows the graphics of distorted “straight line” is an ellipse under the condition of model (3). • Similarly let A = λ − 20 , = − − 2 2 0 λ = 0 2 + 2 0 2 − λ 0 + λ 0 2 + 2 2 + + + = 0 + 1 λ , we have (25) and 1 λ 0 2 + 2 0 2 + 0 + 0 + − = 0 (26) Experiments and Results Experiments and Results Experiments and Results Experiments and Results Experiments and Results Experiments and Results Conclusions • Advantage – Neither information about the intrinsic camera parameters nor 3D-point correspondences are required. – based on single image and uses the distorted positions of collinear points. – Algorithm is simple, robust and non-iterative. • Disadvantage – It needs straight lines are available in the scene.