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Aim of this lab:

The aim of this laboratory involves finding the relationship between 3D world coordinates and their 2D projected positions in the images.
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The two images below stereo1.jpg (Figure2) and stereo2.jpg (Figure3) are images with views of a calibration target. The XYZ coordinates of the target points are available in calibpts.m. These points are defined with respect to a coordinate frame having its origin at the base of the calibration target as shown in Figure1. This image also shows the numbering of the points.

   
Figure1: Image with order of points

   
Figure2: stereo2.jpg                                                                     Figure3: stereo2.jpg

We were asked to write a function to perform the camera calibration that is calibrate.m. We know that camera calibration involved finding the relationship between 3D world coordinates and their 2D projected positions in the image. To find the 2D image coordinates of these 3D world coordinates, we use the Matlab function ginput to digitise the points according to the order shown in Figure1. The result of this operation will return an array of the form [u, v] where u is the column and v is the row of each point in the image.
We then convert this array into a vector by using the Matlab's matrix concatenation syntax which is uv = [uv];

After that, we need to set up the matrix equation that needs to be solved. This is done in the calibrate.m. The equation set up will be in the form of :

                                                                                                BC = UV
where:
B is a 2n x 11 matrix of constraint coefficients
C is an 11 element column vector of calibration matrix coefficients to be solved, and
UV is a 2n column vector of the image coordinates of the target points.

We then need to solve the equation :
                                                                                               C = B \ UV

This matrix equation is overstrained, therefore Matlab will perform a QR decomposition on the matrix B in order to solve the least squares solution.

The function we have written that is calibrate.m will display the original image and plots the digitised point. It also projects the XYZ coordinates back into image coordinates using the calibration matrix. Besides that, the mean square error between the position of the uv coordinates and the projected XYZ coordinates for both images, as well as the error in satisfying the matrix calibration constraints will also be computed by running this function.

With the arbitrary scale of the camera calibration matrix, the main measure of the error in satisfying the camera matrix constraint is the value of :
                                                                                            (q₁ x q₃) . (q₂ x q₃)

This quantity represents the scaled cosine of the angle between the vectors (q₁ x q₃) and (q₂ x q₃), which should be perpendicular. To get a value that represents a pure (unscaled) cosine it should be divided by the magnitude of |q₁ x q₃| and the magnitude of |q₂ x q₃| in order get the dot product between two unit vectors, and hence the cosine.

The mean square error computed by the calibrate.m function here simply uses the value of the pure (unscaled) cosine and divided by the size of XYZ
                                                                           
                                                Mean Square Error  =   (q1^q3) . (q2^q3)  /  |q1^q3| * |q2^q3|    *    1 / size of XYZ

The results on executing the function are shown as follows:

MATLAB Command:
im1 = imread('stereo1.jpg');    %upload stereo1.jpg
im2 = imread('stereo2.jpg');    %upload stereo2.jpg

show(im1);
[u1 v1] = ginput(12);               %digitise 12 points in the image
uv1 = [u1 v1];                      %Matlab's concatenation syntax
C1 = calibrate(im1,calibPts,uv1)    %get the calibration of the image

show(im2);
[u2 v2] = ginput(12);
uv2 = [u2 v2];
C2 = calibrate(im2,calibPts,uv2)   

    
Figure4: stereo1.jpg after performing camera calibration                        Figure5: stereo2.jpg after performing camera calibration

Result on Stereo1:

1. The calibration matrix of stereo1.jpg computed from Matlab is :

        C1 =

             0.6764  -0.7124  -0.0357  363.2515
            -0.1870  -0.1191  -0.9485  345.2300
             0.0006   0.0004  -0.0002  1.0000

2. The mean squared error is : 9.808799e-004

3. The error in satisfying the camera calibration matrix constraint is : 5.922717e-009

4. (q1^q3).(q2^q3)/|q1^q3|*|q2^q3| = 1.177056e-002


Result on Stereo2:

1. The calibration matrix of stereo1.jpg computed from Matlab is :

        C2 =

              0.8825  -0.2267  -0.0265  347.8914
             -0.0881  -0.2265  -0.9133  339.0455
              0.0001   0.0004  -0.0002  1.0000

2. The mean squared error is : 1.536226e-004

3. The error in satisfying the camera calibration matrix constraint is : 3.515452e-010

4. (q1^q3).(q2^q3)/|q1^q3|*|q2^q3| = 1.843471e-003

After obtaining the results, we then save the result and store in a file called mydata.mat by using Matlab's save command as follows:

save mydata im1 im2 calibPts uv1 uv2 C1 C2

This will save the variables im1, im2, calibPts, uv1 , uv2 , C1 and C2 in a file called mydata.mat. These information will be needed for Lab 8.