Information Technology Reference
In-Depth Information
will be shown in the following. It will be also shown that the feature choice
s
I
=
1
√
I
1
allow obtaining interaction matrices almost constant with respect to the depth dis-
tribution.
13.4.2.1
Variation with respect to Rotational Motion
Let us consider two frames
F
2
related to the unit sphere with different
orientations (
1
R
2
is the rotation matrix between the two frames) but with the same
center. In this case, the value of
I
t
isthesameforthetwoframes,sinceitisinvariant
to rotational motions. Let
F
1
and
X
=
2
R
1
X
X
and
be the coordinates of a projected
point in the frame
F
1
and
F
2
respectively. Let us consider a function invariant to
rotations
f
(
X
1
,...,X
N
) that can be computed from the coordinates of
N
points
onto the unit sphere (such as the invariants computed from the projection onto the
unit sphere). The invariance condition between the frames
F
1
and
F
2
can thus be
written as
X
1
, ...,X
N
)=
f
(
2
R
1
X
1
, ...,
2
R
1
X
N
)=
f
(
f
(
X
1
, ...,X
N
)
.
(13.15)
The interaction matrix that links the variation of the function
f
with respect to trans-
lational velocities can be obtained as
L
f
v
=
∂
f
(
X
s
1
+
T
,...,X
N
+
T
)
∂
,
(13.16)
T
where
T
is a small translational motion vector. Let us now apply this formula for the
camera position defined by the frame
F
2
:
X
1
+
T
,...,X
N
+
T
)
∂
f
(
2
R
1
X
1
+
T
2
R
1
X
N
+
T
))
L
f
v
=
∂
T
=
∂
f
f
(
=
∂
,...,
.
(13.17)
∂
T
∂
T
From (13.17), it can be obtained that
f
2
R
1
(
X
N
+
1
R
2
T
)
X
1
+
1
R
2
T
)
2
R
1
(
L
f
v
=
∂
,...,
.
(13.18)
∂
T
Combining this equation with the invariance to rotations condition (13.15), we get
X
1
+
1
R
2
T
,...,X
N
+
1
R
2
T
)
∂
L
f
v
=
∂
f
(
.
(13.19)
T
From this, we easily obtain
X
1
+
T
,...,X
N
+
T
)
∂
T
L
f
v
=
∂
f
(
∂
(13.20)
T
∂
T
where
T
=
1
R
2
T
. Finally, combining (13.20) with (13.16), we obtain
L
f
v
=
L
f
v
1
R
2
.
(13.21)
Search WWH ::
Custom Search