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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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