Submitted version of the thesis - Airlab, the Artificial Intelligence ...

5.1. Camera Calibration 53
the principal points in terms of pixel dimension.
x = K[I|0]Xcam
where Xcam as (X,Y,Z,1) t to emphasize that the camera is assumed to be
located at the origin of a Euclidean coordinate system with the principal
axis of the camera point straight down the z-axis, and the point Xcam is
expressed in the camera coordinate system.
In general, points in the space will be expressed in terms of a different
Euclidean coordinate frame, known as the world coordinate frame. The two
frames are related via a rotation and a translation. The relation between
the two frames can be represented as follows:
x = KR[I|−C]X
where X is now in a world coordinate frame. This is the general mapping
given by a pinhole camera (In Figure 5.2). A general pinhole camera,
P = KR[I| − C], has a 9 degrees of freedom: 3 for K, 3 for R, 3 for C.
The parameters contained in K are called internal camera parameters. The
parameters of R and C which relate the camera orientation and position to
a world coordinate system are called the external parameters. In a compact
form, the camera matrix is
where t = −RC.
P = K[R|t]
The camera matrix, which consists of internal and external camera parameters,
is calculated by the camera calibration toolbox automatically.
Having the internal and external camera parameters from the toolbox, we
compute the homography H. Homography is an invertible transformation
from the real projective plane to the projective plane.
We developed two different approaches to determine the position of a
target according to the robot coordinates. In the first case, the target object
is a ball, and we are using the information coming from the diameter
of the ball. In this case, the transformation should be a 3D to 2D, since the
ball can be represented in world coordinates system with the X,Y,Z and in
the camera coordinates with two values X,Y. In the second case, we assume
that the target object is on the ground and we are using the information
coming from the intersection point of the object with the ground. This