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T
=
I
∗
(
p
∗
)
I
(
w
(
G
)
◦
w
(
G
(
x
))(
p
∗
)
I
(
p
∗
)
G
(
x
)
e
+
−
Warp
Min
G
GG
(
x
)
←
Fig. 6.7
ESM framework
6.4.2
Algorithm
The problem is an old but an important subject in computer vision. Many algorithms
to solve this have already proposed [5, 39, 22, 24, 25]. Here we used the Lie algebra
to express the homography matrix [7]. This parametrization is useful because it has
not singular points.
Let the parametrization of
G
be
8
i
=1
x
i
A
i
G
(
x
)=
ex p
(
A
(
x
))
where
A
(
x
)=
(6.13)
and
A
i
(
i
= 1
8) are a set of base matrices of
sl
(3) [29].
Based on this parametrization the minimization function is
,...,
f
(
x
)=
y
T
(
x
)
y
(
x
)
(6.14)
where
y
(
x
) is
q
-dimensional vector obtained by stacking the pixel brightness
difference
y
(
x
)=
I
(
w
(
G
(
x
)
p
∗
))
I
∗
(
p
∗
)
−
.
(6.15)
The minimization algorithm can be a gradient descent, Gauss-Newton or
Levenberg-Marquardt method. We proposed an efficient second-order minimiza-
tion algorithm [7]. It approximates the Hessian matrix with small number of com-
putation. Note that we can have a Taylor series expression of
y
(
x
) as
y
(
x
)=
y
(0)+
1
2
(
J
(0)+
J
(
x
))
x
+
O
(
x
3
)
.
(6.16)
Then the approximation at
x
=
x
0
is
y
(
x
0
)=
y
(0)+
1
2
(
J
(0)+
J
(
x
0
))
x
0
.
(6.17)
Define the matrix
J
esm
=
1
2
(
J
(0)+
J
(
x
0
))
(6.18)
and the minimization algorithm will be
x
0
=
J
esm
y
(0)
(6.19)
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