Activity 11: Camera Calibration

The task: to calibrate our cameras by mapping the real-world 3D coordinates of an object unto its image 2D coordinates.

From the lecture, this mapping is algebraically represented by the following equations where the 2D coordinates (yi,zi) are solved for using the 3D coordinates (x0,y0,z0).

In matrix form, this is written as:
and can be rewritten as: Q = ad
The transformation matrix, a, can then be solved for using:
Activity procedure:

1. take a picture of the provided 3D calibration checkerboard and pick out an origin and 20 edge points.3D calibration checkerboard with chosen points marked with X's
   

Real-world coordinates of chosen points:

x0	y0	z0
8	0	12
4	0	12
0	0	12
0	4	12
0	8	12
4	0	9
0	4	9
8	0	6
4	0	6
0	0	6
0	4	6
0	8	6
4	0	3
0	4	3
8	0	0
4	0	0
0	0	0
0	4	0
0	8	0
0	0	1

2. Use scilab to process the image and use the locate() function to find the image coordinates of the chosen points.
Image coordinates of the chosen points:

yi	zi
17.714286	429.85714
137.71429	411.85714
234.57143	399
324.57143	418.71429
436.85714	443.57143
139.42857	319.28571
323.71429	321.85714
23.714286	219
140.28571	222.42857
237.14286	227.57143
324.57143	224.14286
439.42857	221.57143
142.85714	129
324.57143	130.71429
29.714286	8.1428571
143.71429	35.571429
238	60.428571
326.28571	36.428571
437.71429	6.4285714
238	87

Code:
stacksize(4e7);
im=imread('C:\Documents and Settings\Endura\My Documents\186\DSC04829.jpg');
imshow(im);
x=locate()

3. Use scilab to input the chosen points into the above equation (13), and use the above
equation (15) to solve for the transformation matrix a.
Code:
xo=[8 4 0 0 0 4 0 8 4 0 0 0 4 0 8 4 0 0 0 0];
yo=[0 0 0 4 8 0 4 0 0 0 4 8 0 4 0 0 0 4 8 0];
zo=[12 12 12 12 12 9 9 6 6 6 6 6 3 3 0 0 0 0 0 1];
yi=[18 138 235 325 437 139 324 24 140 237 325 439 143 325 30 144 238 326 438 238];
zi=[430 412 399 419 444 319 322 219 222 228 224 222 129 131 8 36 60 36 6 87];

for i = 1:length(xo)
Q((2*i)-1,:) = [xo(i) yo(i) zo(i) 1 0 0 0 0 -(yi(i)*xo(i)) -(yi(i)*yo(i)) -(yi(i)*zo(i))];
Q(2*i,:) = [0 0 0 0 xo(i) yo(i) zo(i) 1 -(zi(i)*xo(i)) -(zi(i)*yo(i)) -(zi(i)*zo(i))];
d((2*i)-1,:) = yi(i);
d(2*i,:) = zi(i);
end

a = inv(Q'*Q)*Q'*d;

The following values for were obtained:
a =

- 26.918641
12.900713
- 0.7063340
238.39153
- 6.5092147
- 6.8013387
27.705465
59.226318
- 0.0239055
- 0.0275689
- 0.0016180

Testing this calibration using 3 test points, there was a discrepancy of a few pixels. The error average at less than 5% though, and this to me is an acceptable range given that there are many factors that could contribute to the distortion of the image.

i give myself a grade of 10 for this activity. i accomplished the desired calibration with minimal error.

thank you to cole for the help.