Visual Servoing and Pose Estimation with Cameras Obeying the Unified Model - Visual Servoing via Advanced Numerical Methods - page 232

N

h =1 x s h ∼

1
Z 2 , m 020 =

N

h =1 y s h ∼

1
Z 2 , m 110 =

N
h =1 x s h y s h ∼

1
Z 2 , m 101 =

that: m 200 = ∑ ∑ ∑

N
h =1 x s h z s h ∼

1
Z , m 011 =

N
h =1 y s h z s h ∼

1
Z

N

h =1 z s h ∼

∑ ∑ and m 002 = ∑ N . By neglecting

1
Z 4 when the depth increases enough, the polynomial can be

the term depending on

approximated as

m 100 − m 010 , I 1 ≈ N ( m 200 + m 020 ) − (13.24)

1
Z 2 and

∼

where N is the number of points. Therefore, it can be obtained that I 1

1
√ I 1 ∼

s I = Z . Note that if the set of points is centered with respect to the optical

axis ( i.e. m 100 = m 010 = 0), we have

I 1 ≈ N ( m 200 + m 020 ) . (13.25)

In this case, note the similarity between s I = 1

√ I 1 and the features given by (13.22).
In geometrical terms, if the set of points is centered with respect to the optical axis,
the projection onto unit sphere and the projection onto a classical perspective behave
in the same way when the depth increases. Besides, an example of interaction matrix
variations with respect to depth distributions is given in Section 13.5.1.

13.4.3 Features Selection

We could consider the center of gravity of the object's projection onto the unit sphere
to control the rotational DOF:
x s g = x s g , z s g = m 100

m 000

m 010 m 001

m 000 , m 000 ,

y s g , .

However, only two coordinates of x s g are useful for the control since the point pro-
jection belongs to the unit sphere making one coordinate dependent of the others.
That is why in order to control rotation around the optical axis, the mean orientation
of all segments in the image is used as a feature. Each segment is built using two
different points in an image obtained by re-projection to a conventional perspective
plane.
Finally, as mentioned previously, the invariants to 3D rotation s I = 1

√ I 1 are con-
sidered to control the translation. In practice, three separate set of points such that
their centers are noncollinear can be enough to control the 3 translational DOF. In
order to ensure the nonsingularity of the interaction matrix, the set of points is di-
vided in four subsets (each subset has to encompass at least 3 points). This allows
us to obtain four different features to control the 3 translational DOF.

13.5 Results

In this section, an example of interaction matrix variations with respect to depth
distribution is given. Thereby, several results of pose estimation and visual servoing
are presented.