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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