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where (
x
s
h
,
z
s
h
) are the coordinates of a 3D point. In our application, these
coordinates are nothing but the coordinates of a point projected onto the unit sphere.
They can be computed from the projection of a point onto the image plane and the
inverse transform (13.2).
y
s
h
,
13.3.2.2
Interaction Matrix
In the case of moments computed from a discrete set of points, the derivative of
(13.7) with respect to time is given by
N
h
=0
(
ix
i
−
1
y
s
h
z
s
h
x
s
h
+
jx
s
h
y
j
−
1
s
h
z
s
h
y
s
h
+
kx
s
h
y
s
h
z
k
−
1
s
h
m
i
,
j
,
k
=
z
s
h
)
.
(13.8)
s
h
For any set of points (coplanar or noncoplanar), the interaction matrix related to
L
m
i
,
j
,
k
can thus be obtained by combining (13.8) with the well known interaction
matrix
L
X
s
˙
of a point
X
s
on the unit sphere (defined such that
X
s
=
L
X
s
V
)[13,
28, 11]:
L
X
s
=
−
s
]
×
1
r
I
3
+
r
X
X
s
[
X
(13.9)
s
where
r
is the distance of the 3D point to the sphere center. In the particular case
of a coplanar set of points, the interaction matrix related to
m
i
,
j
,
k
can be determined
[28]:
L
m
i
,
j
,
k
=
m
vx
m
vy
m
vz
m
wx
m
wy
m
wz
(13.10)
where
⎧
⎨
β
d
m
i
+2
,
j
,
k
−
m
vx
=
A
(
im
i
,
j
,
k
)
+
B
(
β
d
m
i
+1
,
j
+1
,
k
−
im
i
−
1
,
j
+1
,
k
)
+
C
(
β
d
m
i
+1
,
j
,
k
+1
−
im
i
−
1
,
j
,
k
+1
)
,
m
vy
=
A
(
β
d
m
i
+1
,
j
+1
,
k
−
jm
i
+1
,
j
−
1
,
k
)
+
B
(
β
d
m
i
,
j
+2
,
k
−
jm
i
,
j
,
k
)
+
C
(
β
d
m
i
,
j
+1
,
k
+1
−
jm
i
,
j
−
1
,
k
+1
)
⎩
m
vz
=
A
(
β
d
m
i
+1
,
j
,
k
+1
−
km
i
+1
,
j
,
k
−
1
)
+
B
(
β
d
m
i
,
j
+1
,
k
+1
−
km
i
,
j
+1
,
k
−
1
)
+
C
(
β
d
m
i
,
j
,
k
+2
−
km
i
,
j
,
k
)
,
m
wx
=
jm
i
,
j
−
1
,
k
+1
−
km
i
,
j
+1
,
k
−
1
,
m
wy
=
km
i
+1
,
j
,
k
−
1
−
im
i
−
1
,
j
,
k
+1
,
m
wz
=
im
i
−
1
,
j
+1
,
k
−
jm
i
+1
,
j
−
1
,
k
with
β
d
=
i
+
j
+
k
and (
A
,
B
,
C
) are the parameters defining the object plane in the
camera frame:
1
r
α
X
=
s
=
Ax
s
+
By
s
+
Cz
s
.
(13.11)
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