Image Processing Reference
In-Depth Information
measurement noise of the data is zero, then
F
has an eigenvalue that is zero. How-
ever, there is no guarantee for this to happen automatically in practice because the
measurements on which one bases the calculation of
f
are not noise-free. Accord-
ingly, a numerical correction method, guaranteeing the singularity of
F
, is applied.
12
Naturally, the epipolar line represented by (
−−−−→
O
L
E
L
)
LD
is given by the last row
of
V
, whereas (
−−−−→
O
R
E
R
)
RD
is given by the last row of
U
. The epipoles in (homoge-
neous) analog image coordinates
x, y,
1 are given by:
(
−−−−→
O
L
E
L
)
L
=
M
I
−
1
(
−−−−→
O
L
E
L
)
LD
(
−−−−→
O
R
E
R
)
R
=
M
I
−
1
(
−−−−→
O
R
E
R
)
RD
when the intrinsic matrices
M
I
and
M
I
are known for both cameras. In that case,
E
can be found too (up to a scale factor):
T
E
=
M
I
FM
I
=
TR
(13.133)
Here, the matrix
T
encodes the displacement between the stereo cameras, whereas
R
represents their relative rotation, lemma 13.11.
13.7 Further Reading
An alternative introduction to camera calibration and stereo vision of world geometry
can be found [219]. In [70] a detailed discussion on projective geometry tools and
geometry reconstruction by images can be found, including reconstruction by use
of more than two views. Modern developments in the theory and practice of scene
reconstruction are described in detail and comprehensive background material is pro-
vided in the monograph of [98]. In [102] the essentials of camera calibration, includ-
ing an extended pinhole camera model with a nonlinear lens correction, is discussed.
Epipolar constraints are covered in detail in [8]. Studies where wide-baseline stereo
issues are discussed include [211, 215]. Even without camera calibration, scene un-
derstanding is possible from image views for numerous applications [14, 106, 170].
Robots controlled by active vision [50] or stereo from motion [194] are significant
application fields of geometry studies in computer vision. Another significant do-
main is architectural or similar scenes, where there is little or no motion [55, 86].
12
This can be achieved by the singular value decomposition of
F
that we will discuss in Sect.
15.3. The decomposition will yield:
F
=
UΣV
T
F
=
UΣ
V
T
F
←
F
⇒
⇒
(13.132)
where
Σ
is the same (diagonal) matrix as
Σ
except that the least significant (diagonal)
element,
Σ
(3
,
3) is now replaced by 0.
