Description of the calibration parameters(2)

Let P be a point space of coordinate vector XX = [X;Y;Z] in the grid reference frame (reference frame shown on the previous figure).

Let XXc = [Xc;Yc;Zc] be the coordinate vector of P in the camera reference frame.

Then XX and XXc are related to each other through the following rigid motion equation:

XXc = Rc_1 * XX + Tc_1

In particular, the translation vector Tc_1 is the coordinate vector of the origin of the grid pattern (O) in the camera reference frame, and the thrid column of the matrix Rc_1 is the surface normal vector of the plane containing the planar grid in the camera reference frame.

The same relation holds for the remaining extrinsic parameters (Rc_2,Tc_2), (Rc_3,Tc_3), ... , (Rc_20,Tc_20).

Once the coordinates of a point is expressed in the camera reference frame, it may be projected on the image plane using the intrinsic camera parameters.

The vectors omc_1, omc_1, ... , omc_20 are the rotation vectors associated to the rotation matrices Rc_1, Rc_1, ... , Rc_20. The two are related through the rodrigues formula. For example, Rc_1 = rodrigues(omc_1).

Similarly to the intrinsic parameters, the uncertainties attached to the estimates of the extrinsic parameters omc_i, Tc_i (i=1,...,n_ima) are also computed by the toolbox. Those uncertainties are stored in the vectors omc_error_1,..., omc_error_20, Tc_error_1,..., Tc_error_20 (assuming n_ima = 20) and represent approximately three times the standard deviations of the errors of estimation.

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