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An alternate approach to the plane-based calibrationmethod in Section 6.3.2 is to
take a single picture of a calibration device containing multiple squares in different
orientations. Using the corners of each square to estimate a different projective trans-
formation is mathematically equivalent to imaging a single square in different orien-
tations indifferent images. However, using a calibrationdevicemaybemore awkward
than simply imaging a single plane in different positions. The camera calibration
toolbox provided by Bouguet ( http://www.vision.caltech.edu/bouguetj/calib doc/ )
is often used to automate the process. Other possibilities for internal parameter esti-
mation include using information about known angles and length ratios in a real
scene (e.g., [ 282 ]), or reasoning about the vanishing points of parallel lines such as
the edges of buildings (e.g., [ 81 , 283 ]).
As mentioned in Section 6.3.2 , the matrix
KK ) 1 is also called the image
ω = (
of the absolute conic . While our derivation of
was entirely algebraic, it also has
an important geometric interpretation. That is, the set of points in the image plane
defined by the second-degree equation
ω
] =
0 is the projection
of a special set of points in the environment called the absolute conic, defined in
homogeneous world coordinates by the
[
x , y ,1
] ω [
x , y ,1
] that satisfy X 2
Y 2
Z 2
[
X , Y , Z , W
+
+
=
0
and W
0. This conic lies on the plane at infinity, and is invariant under any similarity
transform of the world coordinates. Unfortunately, while they are extremely useful
for camera calibration, both the absolute conic and its image are purely imaginary
constructs and cannot be visualized. An alternate classical method for camera self-
calibration, not discussed here, is based on the Kruppa equations (e.g., [ 308 ]), which
are especially suitable when only two views are available.
In addition to the absolute conic and similar constructs, there are many other
important issues inprojective geometry that we have omitted in this chapter for space
and readability. For example, we only alluded briefly to the trifocal tensor [ 439 ], an
analogue to the fundamental matrix that relates point and line correspondences in
triples of views. The concept of an affine reconstruction is also frequently used as
an intermediate stage when upgrading a projective reconstruction to a Euclidean
one. To get a better intuition for these projective geometry concepts, see the topics
mentioned previously [ 137 , 188 ].
Unfortunately, the problem of choosing a good gauge for bundle adjustment, as
mentioned in Section 6.5.3.1 , is tricky. Setting the first camera to the canonical form
has the disadvantage of biasing the result; that is, we would get a different solu-
tion to the minimization problem if we renumbered the cameras. Another troubling
problem is that assessing the quality of a solution to bundle adjustment (for exam-
ple, understanding how uncertainty in the estimated feature locations propagates
to uncertainty in the estimated locations of the 3D points) seems to depend on the
gauge. McLauchlan [ 321 ] and Morris et al. [ 337 , 338 ] thoroughly discuss these issues.
Solving the structure from motion problem lies at the core of the matchmov-
ing software used for visual effects production in movies. However, as discussed in
Section 6.7 , obtaining a good matchmove for difficult sequences is a “black art” and
requires an expert's touch. For example, the input video may contain significant
motion blur, soft focus, or interlacing, which require preprocessing in the best case
and manual feature tracking in the worst case. The matchmover (aided by software)
must detect segments with low motion and little parallax that correspond to pure
rotations (called nodal pans ); in this case, it may be necessary to add “helper frames”
=
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