Single Camera Structure and Motion Estimation - Visual Servoing via Advanced Numerical Methods - page 206

y 1 j ( t ) ≥ y 2 j ( t ) ≥

y 1 ≥ y 1 y 2 ≥ y 2

∈ R

where y 1 , y 2 , y 1 , y 2 denote known positive bounding constants.

Assumption 12.2. The motion of the camera is assumed to be smooth such that the
velocities are assumed to be bounded by a constant. Thus, y j ( t ) belongs to class C 1 ,
which also implies that the first derivatives of y j ( t ) is bounded by a constant. For
the remainder of this chapter, the feature point subscript j is omitted to streamline
the notation.

Assumption 12.3. The subsequent development is based on the assumption that
both p ( t ) and its time derivative p ( t ) are available, where p ( t ) ( i.e. , the optic flow) is
available through numerical differentiation of the image coordinates. Assuming that
p ( t ) and p ( t ) are available, then (12.3), (12.4), and (12.6) can be used to conclude
that y 1 ( t ) , y 2 ( t ) , y 1 ( t ) , y 2 ( t ) can be computed.

12.3 Perspective Camera Motion Model

At some spatiotemporal instant, the camera views a point q on the object. As seen
from Figure 12.2, the point q can be expressed in the coordinate system F c as

m = x f + Rx o q (12.8)

Fig. 12.2 Coordinate relationship describing the position of a point on object as seen from
the camera