Information Technology Reference
In-Depth Information
When the visual features are related to the projection of 3D points, the vector
s
is function of the 3D coordinates
X
=[
XYZ
]
of the 3D point
X
. In that case, the
interaction matrix related to
s
can be written as
L
=
∂
s
X
L
X
,
∂
J
s
=
∂
s
X
is the Jacobian matrix linking the variations of
s
and
X
,and
L
X
is the
interaction matrix related to the 3D point
∂
X
:
=
−
I
3
[
X
]
×
τ
,
X
=
L
X
τ
(16.10)
where [
a
]
×
is the anti-symmetric matrix of the vector
a
.
If one considers
n
visual features related to the same 3D point
X
, the global
s
n
]
can be written as
interaction matrix
L
for the features vector
s
=[
s
1
s
2
···
L
=
J
s
1
J
s
2
···
J
s
n
L
X
.
16.3.2
Interaction Matrix for 2D Point
with coordinates
X
=[
XYZ
]
with respect to the mirror
Consider a 3D point
X
frame
F
m
. Its central projection on the normalized image plane is obtained using
(16.1) and it is given by the point of homogeneous coordinates
x
=[
xy
1]
.Ifthe
visual feature is chosen as
s
=[
xy
]
, the interaction matrix
L
is
L
=
J
s
L
X
,
where
ρ
(
Y
2
+
Z
2
)
1
Z
+
ξ
−
ξ
XY
−
X
(
ρ
+
ξ
Z
)
J
s
=
.
(
X
2
+
Z
2
)
−
ξ
−
ρ
(
Z
+
ξρ
)
2
XY
ρ
Z
+
ξ
Y
(
ρ
+
ξ
Z
)
After few developments, the analytical expression of the interaction matrix
L
can
be written as
L
=
AB
(16.11)
where
ρ
−
1
⎛
⎞
⎠
,
ξ
(
x
2
+
y
2
)
1+
γ
+
x
2
⎝
−
+
ξ
ξ
xy
γ
x
ξγ
A
=
γ+ξ
(
x
2
+
y
2
)
1+ξγ
y
2
ξ
xy
−
+
ξ
γ
y
and
⎛
⎞
ξ
(
x
2
+
y
2
)
1+ξγ
γ
+
+
y
2
xy
−
γ
y
⎝
⎠
B
=
ξ
(
x
2
+
y
2
)
1+
γ
+
x
2
γ
ξγ
−
−
xy
−
x
=
1 +(1
2
)(
x
2
+
y
2
).
with
γ
−
ξ
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