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Differentiating the above equation yields
˙
ξ
=
J
(
q
)
q
(19.10)
where
J
(
q
)=
∂α
(
q
)
.
(19.11)
∂
q
The matrix function
J
defined by (19.11) is called the image Jacobian for
ξ
. Simi-
larly, we define the image Jacobian for
ξ
I
by
J
I
(
q
)=
∂α
I
(
q
)
∂
(19.12)
q
where
α
I
(
q
)=
(
q
)
α
σ
1
(
q
)
α
σ
2
(
q
)
...
α
σ
|
I
|
.
(19.13)
19.5
Image Feature Estimation and Selection
This section discusses image feature estimation and selection. Section 19.5.1 pro-
vides a method to obtain an estimate of
ξ
I
. Section 19.5.2 presents a selec-
tion method to obtain a set of correctly extracted image features.
ξ
from
19.5.1
Image Feature Estimation
We first describe a reconstruction algorithm of generalized coordinates
q
by using
the image Jacobian matrices. Let two vectors
o
and
q
o
which satisfy
ξ
o
=
(
q
o
)
ξ
α
(19.14)
be given. A linear approximation of (19.9) near
q
o
is given by
o
J
(
q
o
)(
q
o
)=
q
ξ
−
ξ
−
(19.15)
or
o
)+
q
o
q
=
J
+
(
q
o
)(
ξ
−
ξ
(19.16)
q
is an approximate value of
q
,and
J
+
(
q
o
) the Moore-Penrose inverse of
J
at
q
o
. An approximate value of
q
is also obtained by
where
o
I
J
I
(
q
o
)(
q
o
)=
q
−
ξ
I
−
ξ
(19.17)
or
q
=
J
+
I
(
q
o
)(
o
I
)+
q
o
ξ
I
−
ξ
(19.18)
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