Graphics Reference
In-Depth Information
These canbe easily extractedby consideringEquation ( 6.9 ). If wedenote M as the 3
×
3
matrix on the left side of P , we can see that M
KR ; that is, an upper triangular matrix
with positive diagonal elements multiplied by a rotation matrix. From linear algebra,
we know this factorization is unique and can be easily computed, for example, using
the Gram-Schmidt process [ 173 ]. 6 Once we have factored M
=
=
KR , we obtain t by
multiplying K 1 by the last column of P .
6.3.2
Plane-Based Internal Parameter Estimation
Amore common approach for obtaining the internal parameters of a stationary cam-
era, which requires no precise measurements of the environment, is to take several
pictures of a planar surface in different orientations. For example, we can simply
use a calibration pattern composed of a checkerboard of black and white squares
mounted on a flat board. Alternately, we can display the calibration pattern on a flat-
screen monitor, as illustrated in Figure 6.6 . The main consideration is that we must
be able to associate features (e.g., checkerboard corners) in different images of the
plane, so it should be clear where the top left corner of the planar pattern is in any
given image.
In practice, the camera is fixed in place and we move the planar surface around
in front of it. However, it's mathematically equivalent to view the configuration as
a fixed plane in world coordinates that we move the camera around. We define the
problem in the latter case. In particular, we assume that the fixed plane lies at Z
0
in world coordinates. Let's rewrite Equation ( 6.10 ) in this special case, for a point
(
=
X , Y ,0
)
on the planar surface and its projection
(
x , y
)
in the image:
X
Y
0
1
x
y
1
K
[
R
|
t
]
(6.15)
X
Y
0
1
=
K
[
r 1 r 2 r 3 t
]
(6.16)
X
Y
1
=
K
[
r 1 r 2 t
]
(6.17)
X
Y
1
H
(6.18)
Here, we've denoted the columns of the rotationmatrix as r 1 , r 2 , r 3 in Equation ( 6.16 ),
and denoted H in Equation ( 6.18 )tobethe3
×
3 matrix
H
K
[
r 1 r 2 t
]
(6.19)
6 The key linear algebra concept is the RQ decomposition , which is a little confusing in this context,
since theR stands for anupper (right) triangularmatrix and theQstands for anorthogonal (rotation)
matrix.
 
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