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Moreover, for the rotation one has that (see for instance [12])
2 = tan 2
2
a
and for the translation
2 =(1 +
2 ) 2
2
z
a
t
.
Consequently, an upper bound of s r (
δ
) is given by
)=2arctan γ r
s r (
δ
(9.24)
and an upper bound of s t (
δ
) is given by
)= γ t .
s t (
δ
(9.25)
9.3.2
Lower Bounds
In Section 9.3.1 we have derived upper bounds of the worst-case errors s r (
δ
) and
s t (
). In this section we consider the computation of lower bounds of these errors.
The idea is to generate a sequence of camera poses ( R
δ
,
t ) such that:
p δ
1. the condition
p
holds true for all values of the sequence, i.e. each
t ) in the sequence determines an admissible image error and
hence a lower bound of the sought worst-case error;
2. the sequence approaches the sought worst-case error.
camera pose ( R
,
Let us start by defining the functions
ψ r = μ
(
Ω
( a )) if w ( a
,
t )
0
0
otherwise
t = ν
( t ) if w ( a
,
t )
0
ψ
0 t r
i e
where w ( a
,
t ) is given by
p .
w ( a
,
t )=
δ
p
(9.26)
Let us observe that w ( a
t ) is a barrier function, in particular it becomes negative
whenever the parameters a and t are not admissible, i.e.
,
p > δ
p
. Then, lower
bounds of s r (
δ
) and s t (
δ
) can be obtained as
s r (
δ
)=sup
a
t ψ r
,
(9.27)
s t (
δ
)=sup
a
t ψ t .
,
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