Game Development Reference
In-Depth Information
(
)
(
)
1
3
f
x
=
q
'
−
x
q
'
t
=
−
ε
q
'
2
4
−
y
t
/
f
1
4
,
,
t
=
−
ε
q
'
−
x
t
/
f
,
(24)
0
y
z
0
z
y
x
z
0
z
x
where
1) is related to the so-called
oriented projective geometry
(Stolfi,
1991). In the current case,
ε
z
(=
±
ε
z
can be determined by judging if the CS's origin lies
in front of the camera (
t
z
>
0).
However, due to the influence of noise and camera distortion, from equation 24
it is impossible to guarantee that the recovered matrix
0) or behind it (
t
z
<
w
c
R
is orthonormal, which
w
c
is a requirement that must be met for
R
to be a rotation matrix. The closest
w
R
can be found by employing one of the methods
provided in Weng et al. (1992) or Horn (1987). However, by doing so, the
resulting parameters may not fulfill the linear projection model
optimally
anymore. That is why a geometrically valid DLT is needed.
orthonormal matrix to
Geometrically Valid DLT
Employing the DLT method described, one can recover 11 independent elements
of the matrix
P
. However, according to a previous section in this chapter,
P
has
only 10 DOFs. This means that the recovered 11 independent elements may not
be geometrically valid. In other words, certain geometric constraints may not be
fulfilled by the 10 intrinsic and extrinsic parameters of a camera recovered from
the reconstructed
P
of the simple DLT. For this problem, there are three
solutions.
Camera geometry promotion
In order to match the 11 DOFs of the projection matrix
P
, one could add one
more DOFs to the camera parameter space by taking into account the skew
factor
u
. By this change, substituting equation 6 into equation 5 yields (ref.
equation 17):
()
T
( )
T
()
T
−
f
r
+
u
r
+
x
r
−
f t
+
ut
+
x t
pppp
pppp
1
1
1
1
x
1
2
0
3
x
x
y
0
z
1
2
3
4
()
T
()
T
2
2
2
2
≅
−
r
+
r
−
+
f
y
f t
yt
1
2
3
4
y
2
0
3
y
y
0
z
,
(25)
3
3
3
3
pppp
()
T
r
t
1
2
3
4
3
z
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