Graphics Reference
In-Depth Information
Here, the
notation denotes the horizontal concatenation of matrices that have the
same number of rows. The camera matrix
P
completely determines how a cam-
era obtains its image of a point in the world coordinate system. It operates on the
homogeneous coordinate of a 3D point in the world coordinate system by simple
multiplication:
|
x
y
1
X
c
Y
c
Z
c
∼
K
%
+
&
X
Y
Z
=
K
R
t
(6.10)
X
Y
Z
1
=
P
FromEquation (
6.10
), we can see that the cameramatrix
P
is a homogenous quan-
tity, since anymultiple of
P
will produce the same projection relationship. Thus, there
are at most eleven degrees of freedom in the twelve entries of
P
(in fact, using our
standard assumptions there are ten degrees of freedom: four for the internal param-
eters and six for the external parameters). We commonly abbreviate the relationship
in Equation (
6.10
)as
x
∼
P
X
(6.11)
×
×
where the 3
1 vector
X
are the homogeneous coordinates of
corresponding image and world points, respectively.
Therefore, matchmoving is equivalent to determining the camera matrix
P
corre-
sponding to each given image. The following sections discuss different circumstances
of this general estimation problem.
1 vector
x
and the 4
6.3
SINGLE-CAMERA CALIBRATION
In this section, we discuss the estimation of the parameters of a single camera that is
fixed in position. We'll consider two scenarios. In the first, the camera takes a single
image of an environment that contains 3D landmarks with known locations in world
coordinates. For example, these landmark points may result from an accurate survey
of amovie set acquired using a range sensor (see Section
8.1
). In this case, we're given
both the known 3Dworld coordinates and corresponding 2D image coordinates, and
it's straightforward to estimate the cameramatrix using a process called
resectioning
.
In Section
6.3.1
we describe resectioning and show how the internal and external
parameters of the camera can be recovered from the camera matrix.
The secondproblemwe consider is the estimationof the unknown internal param-
eters of a camera from several images of a plane — for example, a checkerboard
pattern with known dimensions that is shown to the stationary camera in several
