# F06 Camera Calibration

I mentioned in class that our HiBall tracker has 6 sensors and 6 lenses (it is a dodecahedron), which surprisingly (to the designers) produces the equivalent of 26 usable cameras. We experimentally determined the view parameters for each of these cameras by rotating the HiBall on a precision goniometer while observing a single fixed LED. This gave us a set of U,V sensor outputs and corresponding X,Y,Z LED coordinates. Of course there are errors in the U,V readings resulting from noise and non-linearity and errors in the X,Y,Z coordinates of the LED resulting from errors in the goniometer and our ability to accurately mount the LED. Using some interesting math, we determined a 3×4 view matrix for each of the 26 views from this noisy data using least-squares estimation.

Here is the actual view matrix for the primary view out of the top lens on a real HiBall.

$ M = [ [ -1.941433, 17.673074, 0.034771, -0.000622 ],[ -16.878952, -1.848124, -0.186179, 0.003329 ],[ -0.038476, -0.018944, -1.000000, 0.017880 ] ] $

This gets used to predict where the image of an LED will appear on the image plane like this:

$ [ [ U t ], [ V t ], [ t ] ] = M [ [ X ], [ Y ], [ Z ], [ 1 ] ] $

Then we divide through by $t$ to get the value of $U$ and $V$.

The origin of the image plane is in its center. The range of U,V is -1 to 1. The width of the sensor is 4 millimeters (this implies that the A and B vectors should be about 2 millimeters long).

What can you tell me about this camera? What is the coordinate of its center of projection? What is the distance from the center of projection to the image plane? What is the field of view in degrees? Are the A and B vectors perpendicular? Is the O vector perpendicular to the image plane? Are the A and B vectors the same length?

Now here is the view matrix for one of the off axis views looking from the center sensor out of one of the side lenses.

$ [[ -1.304973, 11.151752, 7.827527, 0.111088 ],[-12.094732, -1.552276, 1.266075, 0.025642],[ 0.000000, 0.000000, -1.000000, 0.008316]] $

Answer the questions above for this camera.

I recommend you use Matlab or some other aid for calculation.

Write up your answers email them to me. Explain how you calculate your answers so I can give you partial credit.