3D camera axes have the origin in center of projection of the camera.
Using center manifold projection and normal form analyses, we are able to predict all nonlinear dynamics in the vicinity of these singularities.
This point will be located where a line joining the point Q with the center of projection C meets the image line.
These axes are defined as shown in Figure 2. The 3D camera coordinate system has its origin in the center of projection of the camera.
By analyzing bifurcations from the marginal gain settings of a nonlinear reactor under PI-control, several characteristics of the closed-loop reactor dynamics are revealed via the center manifold projection and normal form techniques of dynamic singularity theory.
Let the center of projection be the point C=(X C,Z C,1 T, expressed in homogeneous coordinates, and let l= a,b,c)T be the homogeneous vector representing the image line.
The line from the center of projection, normal to the image line, is called the principal axis of the camera, and the cross point between the image line and the principal axis is called the principal point.
where K is the 2×2 camera internal parameter matrix, R is the 2×2 camera rotation matrix, I is the 2×2 identity matrix, and (widetilde {mathbf {C}}={X_{mathrm {C}}, Z_{mathrm {C}}}) is the center of projection expressed in inhomogeneous coordinates.
In order to simplify the problem, suppose that the camera is placed at a canonical position, that is, suppose that the center of projection is the origin of the coordinate system, C= 0,0,1 T, and that the principal axis is the Z axis (α=0), as in Fig. 6.
In Section 3.1, we chose the camera to have its center of projection at the origin of the system of coordinates, with its principal axis the Z axis and with focal length equal to 1. Since this is the camera employed in the reconstruction, it will be the one to project the interpolated coordinates to the image plane.
