Image Processing Reference
In-Depth Information
images.
11
The equation also represents the distance of a point to the hyperplane so
that the problem is a linear symmetry direction-fitting problem, see theorem 12.1, in
E
9
where one attempts to minimize
e
(
f
)=
2
d
q
=
f
T
(
Q
q
T
f
qq
T
h
(
q
))
d
qf
,
Q
|
|
with
f
=1
(13.126)
Here
h
is the strength of the correspondence
q
, which is equivalent to a probabil-
ity density in statistics, and a mass density in mechanics. The integral reduces to a
summation for a discrete set
q
j
, and
h
reduces to a discrete certainty on
correspondence, or a constant if it is not available. Note that the constraint
Q
=
{
}
=1
is deduced from the fact that we can determine
f
only up to a scale constant. This is
because (13.124) is homogeneous, i.e., it is satisfied by
λ
f
, where
λ
is a scalar, if it
is satisfied by
f
. For a discrete
f
Q
and no certainty data:
{
q
1
,
q
2
,
,
q
N
Q
=
···
}
(13.127)
the problem is confined to minimization of
⎛
⎝
⎞
⎠
q
1
T
q
2
T
.
q
N
T
f
T
(
j
q
j
q
j
T
)
f
=
f
T
Q
T
Qf
=0
,
with
Q
=
.
(13.128)
that is, the elements of
Q
are given by
⎛
⎝
⎞
⎠
c
R
1
c
L
1
,c
R
1
r
L
1
,c
R
1
,r
R
1
c
L
1
,r
R
1
r
L
1
,r
R
1
,c
L
1
,r
L
1
,
1
c
R
2
c
L
2
,c
R
2
r
L
2
,c
R
2
,r
R
2
c
L
2
,r
R
2
r
L
2
,r
R
2
,c
L
2
,r
L
2
,
1
. . . . . . . . .
c
R
N
c
L
N
,c
R
N
r
L
N
,c
R
N
,r
R
N
c
L
N
,r
R
N
r
L
N
,r
R
N
,c
L
N
,r
L
N
,
1
Q
=
(13.129)
9 matrix
Q
T
Q
as
λ
9
, and
its corresponding eigenvector as
v
9
, the solution of (13.128) is given by
f
=
v
9
.
Assuming that the thus obtained vector
f
has the elements
Designating the least significant eigenvalue of the 9
×
f
=(
f
1
,f
2
,f
3
,f
4
,f
5
,f
6
,f
7
,f
8
,f
9
)
T
(13.130)
and using (13.122) and (13.125), one can obtain
⎛
⎞
f
1
f
2
f
3
f
4
f
5
f
6
f
7
f
8
f
9
⎝
⎠
F
=
(13.131)
We recall from lemma 13.11 that if this
F
is to be a useful solution to our problem,
F
must be rank-deficient, i.e., its least eigenvalue must equal to zero. Ideally, if the
11
In principle, the correspondence set
Q
could be a dense set in this formalism.
