A11 - Camera Calibration

In this activity we model the geometric aspects of image formation to recover information that was lost in the process of projecting brightness values from surfaces existing in 3D world space to 2D sensor space.

To do this, we first take a picture of a checkerboard pattern known as Tsai Grid.

Using SciLab's locate function, we selected 25 points on this image to get the pixel value of these points. We also take note of the real world coordinates of these points by letting the left side of the board to be the x-axis, the right side as the y-axis and the vertical as the z-axis. Each square has a side length of one inch. The origin is shown in pink below.

The green dots are the 25 points we've selected for our calibration. Next, we set up the matrix Q shown by equation below for values of i from 1 to 25. (i stands for image coordinate, o for real world object coordinate)

We then solve for a using equation below.




The resulting values of matrix a can then be used in the following equations to get the resulting 2D image coordinate by knowing the real world coordinates of the point. (a_34 is set to 1)

Implementing this method:

Real world coordinates of the green dots:
1. (8,0,12)
2. (6,0,10)
3. (2,0,10)
4. (4,0,9)
5. (6,0,8)
6. (6,0,3)
7. (4,0,2)
8. (2,0,3)
9. (6,0,5)
10. (4,0,3)
11. (0,8,12)
12. (0,5,10)
13. (0,2,10)
14. (0,5,8)
15. (0,3,7)
16. (0,5,4)
17. (0,7,2)
18. (0,2,1)
19. (0,3,3)
20. (0,5,1)
21. (0,0,1)
22. (0,0,3)
23. (0,0,5)
24. (0,0,6)
25. (0,0,11)

Corresponding Image Coordinates (Pixel Value)

point	y_image	z_image
1	23.214286	235.11905
2	53.571429	200
3	108.33333	205.35714
4	82.738095	185.11905
5	54.761905	163.09524
6	57.738095	73.214286
7	85.714286	63.095238
8	110.11905	86.904762
9	55.952381	108.92857
10	85.119048	80.357143
11	246.42857	236.90476
12	201.19048	202.38095
13	159.52381	205.95238
14	201.19048	166.66667
15	172.61905	152.97619
16	199.40476	80.357143
17	227.97619	55.952381
18	159.52381	55.357143
19	172.02381	85.119048
20	198.80952	45.238095
21	133.92857	60.714286
22	134.52381	92.261905
23	133.92857	125
24	133.33333	141.07143
25	132.7381	224.40476

Resulting matrix a

-13.2561
9.736856
-0.88994
134.7117
-4.07642
-4.2728
15.07773
45.71427
-0.01371
-0.01504
-0.00559

New image coordinates after using the matrix a

point	y_new	z_new
1	21.842763	235.67576
2	53.69017	199.59722
3	108.32033	205.44708
4	82.330771	184.49906
5	55.041758	162.50365
6	58.274085	73.794279
7	85.553674	63.772888
8	110.40641	86.620738
9	57.005504	108.60976
10	85.109881	80.396867
11	248.48566	236.84126
12	200.81792	201.5441
13	158.94669	205.61779
14	200.29104	164.72204
15	172.19655	151.17308
16	199.27642	93.813046
17	227.597	52.018523
18	158.96577	54.178965
19	171.88968	83.28296
20	198.54783	42.893636
21	134.57351	61.133497
22	134.29241	92.497568
23	134.00484	124.58259
24	133.85857	140.90327
25	133.10108	225.42079

Getting the difference between these computed values with the actual values gives the following mean values for each coordinate:
y: 0.474705
z: 1.365954

-o0o-
Collaborator: Cole Fabros

-o0o-
Grade: 10/10 since I implemented the camera calibration well, and that the mean difference for each axis is within one pixel. :)