motion estimation and compensation for very low bitrate video coding

42 Chapter 2. Motion in the Framework of Video Coding
Y
X
f
y
r
x
2D image plane
R
3D object
Z
camera axis
Figure 2.2: Perspective projection geometry: from 3D to 2D
dinate systems have to cooperate: the camera and the image plane.
The rst is a 3D Cartesian coordinate system with its origin at the
camera lens and the Z-axis corresponding to the camera axis. The
second is the 2D (and space-limited) system of the image plane. Let
R =(XY Z) T be the position vector of a point in the 3D space and
r =(xy) T be the position vector of the projected point on the image
plane. Under a perspective projection, the image r of R is the intersection
of the plane with a ray linking the camera lens and R. If one
considers the similar triangles in the projection geometry, the following
relations are obtained:
f Z f
= ;
x X y
Z
= ; (2.1)
Y
where f is the distance between the camera and the image plane, referred
to as the focal length, and related to the focus of expansion [51]. The
perspective projection equation for every point isthus:
r =
x
y
!
= f
Z
X
Y
!
: (2.2)