A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation

1. A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation

2. Outline • Introduction • Model and Approach • Further Discussion • Experiments and Results • Conclusions

3. 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.

4. 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

5. 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

6. 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

7. Radial Distortion Models • The Polynomial Model (PM) that describe radial distortion : (1)

8. Radial Distortion Models • The Division Model (DM) that describe radial distortion : (2) • we use single parameter Division Model as our distortion model : (3)

9. Radial Distortion Models • To simplify equation, we suppose distorted center is the origin image coordinates system, thus : , (4) P (0,0)

10. The Figure of Distorted Straight Line • We consider collinear points and their distorted images. • Let straight line equation from (4) We have (5) (6) (7)

11. The Figure of Distorted Straight Line (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)

12. Estimate the and • Let be the coordinates of the distorted center . From (7) , we have (8) (9)

13. Estimate the and (9) Let , we have (10)

14. Estimate the and Let , we have (10) Base on the relation of , we have (圓方程式參數A、B、C 與 radial distortion 參數 P、 的關係式) (11)

15. Estimate the and (10) (11) • Obtain of distorted center • Extract three “straight line” from image , we can get by circle fitting from (10) according to (11) , we have (12) (13)

16. Sum up whole algorithm • Extract “straight line” from the image • Determine parameter by fitting every “straight line” with a circle according to (10) • Calculate the center of the radial distortion according to (12) • Compute the parameter λ of radial distortion according to (13).

17. 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. • Method • Direct Least Squares Method of Circle Fitting (LS) • Levenberg-Marquardt Method of Circle Fitting (LM) Distorted “straight line” Circle to fit Distorted center

18. Circle fitting - LS (10) • For each point on the “straight line”, (10) gives (14) • Stacking equations from N points together gives b (15) Where M is N3, b is N1 matrix

19. Circle fitting - LS • Directly using linear least squares fit method, we can get (16)

20. Circle fitting - LM Main ideas • Let the equation of a circle be (17) Subject to the constraint : (18) • The distance from a point to the circle (19) Where (20)

21. Circle fitting - LM • From (18) , we can define an angular coordinate by , (21) • Apply the standard Levenberg-Marquardt scheme to minimize the sum of squared distance in the three dimensional parameter space

22. Further Discussion • In Algorithm1, we must have “straight lines”, we relax this constrain and discuss the conditions of • Only one straight line (L1) • Only two straight lines (L2) • Non-square pixels

23. Only One Straight Line (L1) • Suppose the distortion center is the image center and calculate the distortion parameter by (13) (13)

24. Only Two Straight Lines (L2) • Extract “straight line” from the image; • Determine parameter by fittingthe “straight line” with a circle according to(10); (10) (12) become (22)

25. Only Two Straight Lines (L2) • Select a suitable interval is suggested, for any , calculating according to (22); (22)

26. Only Two Straight Lines (L2) • Calculate the distortion parameter according to (13), for any ; (13)

27. Only Two Straight Lines (L2) • Calculate the corresponding corrected points , for any , and all distorted points according to (4); , (4) • Let [d, k] = min= min, then obtain the optimal estimation and .

28. Non-square Pixels • Let : the coordinates of the distorted centre : pixel aspect radio The distorted radius is given by • From (8) we have (23) (24)

29. Non-square Pixels (24) • Equation (24) shows the graphics of distorted “straight line” is an ellipse under the condition of model (3). • Similarly let , we have (25) and (26)

30. Experiments and Results

31. Experiments and Results

32. Experiments and Results

33. Experiments and Results

34. Experiments and Results

36. Experiments and Results

37. 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.