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In-Depth Information
Once the matrix
ω
∗
is known, the matrix
A
is readily obtained based on Cholesky
factorisation (Press et al.,
2007
). In terms of the intrinsic camera parameters them-
selves, the DIAC
ω
∗
=
AA
T
can be expressed as
⎡
⎤
α
u
+
α
u
cot
2
θ
+
u
0
α
u
α
v
cos
θ/
sin
2
θ
+
u
0
v
0
u
0
ω
∗
=
⎣
⎦
α
u
α
v
cos
θ/
sin
2
θ
α
v
/
sin
2
θ
v
0
(1.66)
+
u
0
v
0
+
v
0
u
0
v
0
1
when
A
is defined according to Birchfield (
1998
)(cf.(
1.11
)). Written in terms of the
IACs
ω
i
and the DIACs
ω
i
, the basic equations of self-calibration (
1.63
) become
ω
i
=
B
i
−
b
i
p
T
ω
1
B
i
−
b
i
p
T
T
(1.67)
ω
i
=
B
i
−
b
i
p
T
−
T
ω
1
B
i
−
b
i
p
T
−
1
.
Hartley and Zisserman (
2003
) show that the image of the absolute dual quadric
Q
∗
∞
is identical to the DIAC and can be expressed as
ω
∗
=
PQ
∗
∞
P
T
(1.68)
with
P
as the projection matrix of the camera, where (
1.68
) is shown to be equiv-
alent to the basic equations of self-calibration (
1.67
). At this point, metric self-
calibration based on the absolute dual quadric according to Hartley and Zisserman
(
2003
) proceeds as follows:
1. Determine
Q
∗
∞
based on known elements, especially zero-valued elements, of
ω
∗
and the known projection matrix
P
using (
1.68
). As an example, for a known
principal point
(u
0
,v
0
)
the sensor coordinate system can be translated such that
u
0
=
v
0
=
0, leading to
ω
13
=
ω
23
=
ω
31
=
ω
32
=
0. In addition, a zero skew
90
◦
, yields
ω
12
=
ω
21
=
0.
2. Determine the transformation
H
based on an eigenvalue decomposition of the
absolute dual quadric
Q
∗
∞
angle, i.e.
θ
=
H IH
T
I
according to
Q
∗
∞
=
with
=
diag
(
1
,
1
,
1
,
0
)
.
3. Determine the metric camera projection matrix
P
(M)
=
PH
and the metric scene
x
(M)
i
point coordinates
W
H
−
1
W
=
x
i
.
x
(M)
Since the solution for
P
(M)
W
i
is obtained based on a merely algebraic
rather than a physically meaningful error term, it is advantageous to perform a re-
finement by a full bundle adjustment step. A summary of the scene reconstruction
and self-calibration methods based on projective geometry described so far in this
section is given in Fig.
1.4
.
An alternative approach to self-calibration is based on the Kruppa equations. Ac-
cording to Hartley and Zisserman (
2003
), they were originally introduced by Kruppa
(
1913
) and modified by Hartley (
1997
) to establish a relation based on an SVD of
the fundamental matrix
F
which yields quadratic equations in the parameters of
the DIACs. The main advantage of this technique is that it does not require a projec-
tive reconstruction. Another approach, termed 'stratified self-calibration' by Hartley
and Zisserman (
2003
), is to proceed stepwise from projective to affine and finally to
metric reconstruction. A detailed description of these methods is beyond the scope
of this section.
and
˜
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