Game Development Reference
In-Depth Information
where
A
is a matrix with dimension (
m
⋅
n
)
×
(
m
+
n
),
x
and
b
are vectors with
dimensions (
m
+
n
) and (
m
⋅
n
), respectively.
0
is a row vector with all (
m
− 2)
elements being 0, and
I
n
×
n
is a unit matrix with
n
rows and
n
columns.
The above equation is over-determined, and can be solved as follows:
T
m
−
2
(
) (
)
−
1
,
T
T
x
=
A
A
A
b
where the sparsity of the matrix
A
T
A
can be explored to improve the efficiency
of the computation.
However, the camera and frame set is not always complete, which means that
some frames are not visible in certain cameras. For this situation there are two
solutions: 1) The whole set can be decomposed into several subsets that are
complete by themselves. Then, for each subset the above calculations can be
done independently and subsequently combined into one WCS; and 2) The
problem is treated as a
missing data
system, which can be solved by the
interpolation method proposed in Sturm (2000).
Special Camera Calibration Techniques
In addition to the aforementioned approaches for passive camera calibration,
some other special techniques, such as those utilizing projective geometry
invariants and special calibration objects, have also been developed. For
instance, in Liebowitz & Zisserman (1998), instead of using calibration control
points with known metric, other types of constraints, such as a known angle, two
equal-but-unknown angles, and a known length ratio are utilized. The most
important ones are summarized in the following sections.
Vanishing points
By exploring the geometry property of vanishing points, the camera geometry
can be obtained to a certain degree. A vanishing point is the common intersection
of all image lines whose 3-D-space correspondences are parallel to each other
in the same direction before perspective projection. It can be located reliably in
an image.
Vanishing points have several interesting properties (Caprile & Torre, 1990): 1)
All vanishing points associated with the sets of lines that are parallel to a given
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