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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