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G
that transforms each pixel
P
i
of the template region into its corresponding pixel in
the current image
I
, i.e. finding the homography
G
such that
∀
i
∈{
1
,
2
, ...
m
}
:
P
i
)) =
I
∗
(
P
i
)
I
(
w
(
G
)(
(1)
P
i
=[
]
is the homogeneous image coordinate. Homography
G
is defined in the
Special Linear group
u
∗
v
∗
1
. The matrix
G
defines a projective transformation in the
image.
w
is a group action defined from
SL(
3
)
2
:
SL(
3
)
on
P
2
2
w
:
SL(
3
)
×
P
= P
(2)
2
atuomorphism:
Therefore, for all
G
∈
SL(
3
)
,
w
(
G
)
is a
P
2
2
w
(
G
)
:
P
→
P
(3)
P
∗
→
P
∗
)
P
=
w
(
G
)(
such that:
⎡
⎤
g
11
u
∗
+
g
12
v
∗
+
g
13
g
31
u
∗
+
g
32
v
∗
+
g
33
⎣
⎦
g
21
u
∗
+
g
22
v
∗
+
g
23
g
31
u
∗
+
g
32
v
∗
+
g
33
P
∗
)=
P
=
w
(
G
)(
(4)
1
G
of
G
, the problem consists in finding an
Suppose that we have an approximation
incremental transformation
G
, such that the difference between a region in current
image
I
(transformed from Template region by the composition
w
Δ
(
G
)
◦
w
(
Δ
G
)
)and
the corresponding region in reference image
I
∗
is null.
Homography
G
is in the
SL(
3
)
group which is a Lie group. The Lie algebra asso-
(
)
{
A
1
,
A
2
, ...,
A
8
}
(
)
ciated to this group is
SL
3
.Let
be a basis of the Lie algebra
SL
3
.
(
)
Then a matrix
A
x
can be expressed as follows:
8
i
=
1
x
i
A
i
A
(
x
)=
(5)
(
)
∈
SL(
)
A projective transformation
G
x
3
in the neighborhood of
I
can be parameter-
ized as follows:
∞
i
=
0
1
i
!
(
i
G
(
x
)=
ex p
(
A
(
x
)) =
A
(
x
))
(6)
As incremental transformation
Δ
G
also belongs to
SL(
3
)
, it can be expressed as
Δ
G
(
x
)
,
where
x
is a 8
×
1 vector. Therefore tracking consists in finding a vector
x
such that
∀
i
∈{
1
,
2
, ...
m
}
, the image difference
(
G
P
i
))
−
I
∗
(
P
i
)=
d
i
(
x
)=
I
((
w
)
◦
w
(
Δ
G
(
x
)))(
0
(7)
)]
be the
m
Let
d
1vector containing the image differ-
ences.Therefore, the problem consists in finding
x
(
x
)=[
d
1
(
x
)
,
d
2
(
x
)
, ...,
d
m
(
x
×
=
x
0
verifying:
d
(
x
0
)=
0
(8)
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