Graphics Reference
In-Depth Information
from the image plane to the camera center is called the focal length , denoted by f .
From Figure 6.3 a we can see that the film/CCD image is “upside down” with respect
to the scene because of the way light rays hit the image plane. In computer vision,
we usually make the mathematically convenient assumption that the image plane
actually lies between the camera center and the scene. As we can see fromFigure 6.3 b,
this results in the same image, but one that is already “right-side up.”
6.2.1
Internal Parameters
To create an image, we consider 3D points in the world that are specified in the
camera's coordinate system , as depicted in Figure 6.4 a. The origin of the coordinate
system is the camera center C . The image plane is located at a distance f (the focal
length) along the positive Z axis and is parallel to the XY plane. The x and y axes of
the image plane are parallel to the X and Y axes of the camera coordinate system,
and the Z axis of the camera coordinate system (also called the principal axis ) points
outward into the scene. Therefore, any point the camera can “see” has a positive
Z -coordinate. Note that our usual Cartesian conventions of x and y mean that the
camera coordinate system is left-handed rather than right-handed.
Let's consider a point
in the camera coordinate system. Figure 6.4 b
shows that by reasoning about similar triangles, this point's projectionon thephysical
image plane is given by:
(
X c , Y c , Z c
)
f X c
Z c
f Y c
Z c
x
˜
=
y
˜
=
(6.1)
˜
˜
y in Equation ( 6.1 ) aren't the same as the pixel coordinates in a
digital image, since they're measured in the same physical units as the environment,
and thus are likely to be extremely small numbers. For example, a digital camera's
CCD is only a few mm wide, so each pixel is physically on the order of 1
However, the
x and
µ
m wide.
( X c , Y c , Z c )
Y
Z
Y
image
coordinates
~~
( x , y )
y
Y c
x
image
plane
~
y
f
Z
f
Z c
X
camera coordinates
(a)
(b)
Figure 6.4. (a) The camera coordinate system and the image coordinate system. (b) Side view
of (a), showing that the projections of scene points onto the image plane can be computed by
considering similar triangles.
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