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Fig. 12.1
Projective relationships of images formed at different times by moving object
known, four noncollinear feature points in a plane
4
can be used along with the re-
lationships in (12.5) and (12.6) to develop a set of linear equations in terms of the
image feedback that can be solved to compute and decompose
H
(
t
). Specifically,
the homography matrix can be decomposed to recover the rotation
R
(
t
) between
F
c
and
x
f
(
t
)
d
∗
F
c
, the normal vector
n
∗
, the scaled translation
(although the individ-
3
and
d
∗
are generally unknown), and depth ratio
ual components
x
f
(
t
)
α
j
(
t
)
using standard techniques [13, 44]. The expression in (12.6) can be used to recover
m
j
(
t
), which can be used to partially reconstruct the state
y
(
t
) so that the first two
components of
y
(
t
) can be determined.
∈
R
Assumption 12.1.
The relative Euclidean distance
x
3
j
(
t
) between the camera and
the feature points observed on the target is upper and lower bounded by some known
positive constants (
i.e.
, the object remains within some finite distance away from the
camera). Therefore, the definition in (12.4) can be used to assume that
y
3
≥
y
3
j
(
t
)
≥
y
3
(12.7)
where
y
3
,
y
3
∈
R
denote known positive bounding constants. Let us define a convex
hypercube
Ω
in
R
,as
y
3
}.
Likewise, since the image coordinates are constrained (
i.e.
, the target remains in the
camera field of view) the relationships in (12.3), (12.4), and (12.6) along with the
fact that
A
is invertible can be used to conclude that
4
Ω
=
{
y
3
|
y
3
≤
y
3
≤
The homography can also be computed with 8 noncoplanar and noncollinear feature points
using the “virtual parallax” algorithm.
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