Image Processing Reference
In-Depth Information
)
, β
i
β
i
(
X
where
β
i
β
i
(
X
) and:
⎛
⎝
⎞
⎠
cos
β
1
sin
β
1
cos
β
1
sin
β
1
r
1
−
−
r
1
r
1
r
1
H
X
=
.
.
.
.
cos
β
p
sin
β
p
p
cos
β
p
p
sin
β
p
[6.22]
r
p
−
−
r
p
r
p
r
p
r
i
=
r
x,i
X
)
1
/
2
,H
∗
X
+
r
y,i
X
: a 4
×
p
size matrix
.
)
∗
H
∗
(
X
We then have to calculate the line matrix (
X
−
X
) whose
k
th
element,
denoted by
I
k
, has the following form:
r
k
W
k
X
−
X
,
1
I
k
=
[6.23]
where
W
k
r
x,k
)
∗
.
(
r
y,k
,
−
r
x,k
,k
r
y,k
,
−
k
We can then easily prove the following results [LEC 99]:
⎧
⎨
W
k
X
=0
,k
=1
,...,p,
[6.24]
sin
β
k
−
β
k
.
1
=
r
k
⎩
r
k
W
k
X
r
k
We will give a few elements of proof of this property and it will then be easy to
imagine extensions. To do this, first, the vectors
and
X
X
are partitioned in sub-vectors
with positions (
R
) and speeds (
V
):
,
R
V
,
R
V
X
X
we then have:
=
R
∗
,
V
∗
J J
kJ
R
V
,
W
k
X
k
2
J
[6.25]
=
R
+
k
V
∗
J
(
R
V
+
k
)
,
where
J
=
01
−
10
.
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