Information Technology Reference
In-Depth Information
˜
˜
To seek a fast solution to this maximisation problem, we can differentiate
Q
φ
t
−
1
)
in terms of the six transformation parameters, and set them to be zero individually.
Therefore, the optimal solutions for the six DOFs can be obtained by solving these
numerical equations:
(
φ
t
|
˜
˜
∂
Q
(
φ
t
|
φ
t
−
1
)
=
0
,
(21)
∂
T
d
which is available, and
˜
˜
˜
˜
∂
Q
(
φ
t
|
φ
t
−
1
)
∂θ
=
∂
Q
(
φ
t
|
φ
t
−
1
)
∂
R
∂
q
∂θ
=
0
,
(22)
∂
R
∂
q
where the rotational vector
θ
has three Euler angles,
(
θ
x
,
θ
y
,
θ
z
)
.Further,
˜
˜
∂
Q
(
φ
t
|
φ
t
−
1
)
˜
˜
˜
=
−
∑
i
∑
j
p
(
α
˜
|
β
,
φ
)(
β
−
γ
)
f
˜
α
/
α
˜
,
(23)
t
j
t
i
t
−
1
t
i
t
j
xt
zt
∂
R
The rotation matrix can be formulated using a unit
quaternion
that can be expressed
as
q
=(
q
0
,
q
1
,
q
2
,
q
3
)
:
⎡
⎤
q
0
2
q
1
2
q
2
2
q
3
2
+
−
−
2
(
q
1
q
2
−
q
0
q
3
)
2
(
q
1
q
3
+
q
0
q
2
)
⎣
⎦
.
q
0
2
q
1
2
q
2
2
q
3
2
R
=
2
(
q
1
q
2
+
q
0
q
3
)
−
+
−
2
(
q
2
q
3
+
q
0
q
1
)
(24)
q
0
2
q
1
2
q
2
2
q
3
2
2
(
q
1
q
3
−
q
0
q
2
)
2
(
q
2
q
3
+
q
0
q
1
)
−
−
+
The Levenberg-Marquardt (L-M) technique [23] is applied to search for the optimal
solutions, as this is a non-linear optimisation problem, solved by combining gradi-
ent descent and Gauss-Newton iteration. The variation of the rotation matrix from
frame to frame can be derived using a linear optimisation [3], where the incremental
rotation quaternion is computed at each frame with
Δ
1
q
=(
−
ζ
,
θ
x
/
2
,
θ
y
/
2
,
θ
z
/
2
)
,
(25)
ζ
=(
θ
x
2
+
θ
y
2
+
θ
z
2
)
/
4
.
Assuming a vector
v
, then the derivatives of the rotation matrix with
respect to the quaternion can be obtained
=[
v
1
,
v
2
,
v
3
]
⎡
⎤
c
1
c
4
−
c
3
c
2
R
−
1
∂
∂
⎣
⎦
,
q
=
c
2
c
3
c
4
−
c
1
(26)
c
3
−
c
2
c
1
c
4
where
⎧
⎨
c
1
=
q
1
v
1
+
q
4
v
2
−
q
2
v
3
,
c
2
=
−
q
3
v
1
+
q
0
v
2
+
q
1
v
3
,
(27)
c
3
=
q
2
v
1
−
q
1
v
2
+
q
0
v
3
,
⎩
c
4
=
q
1
v
1
+
q
2
v
2
+
q
3
v
3
.
Search WWH ::
Custom Search