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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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