Page 45 - Dynamic Vision for Perception and Control of Motion

2.1 Three-dimensional (3-D) Space and Time 29

tive downward. The corresponding HTM is T Rc with entries y gc and H c in the last
column of the second and third rows.
5. The camera-oriented CS (indexed “c”): The gaze direction of the camera is as-
sumed to be fixed on the center of the road at the look-ahead distance of the ob-
ject center. The elevation H c of the camera above the ground and its lateral off-
set y gc yield the camera pan (yaw) and tilt (pitch) angles \ c and 4 c. The camera
CS is obtained from CS 4 by two rotations: First, rotating around the Z gc -axis
(vertical line through camera projection center, not directly shown but indicated
by the vector H c in the opposite direction) by the amount \ c yields the horizon-
tal direction of sight. The corresponding HCT–matrix is R \c. Now the X-Z-plane
cuts the axis X gR at distance x co. Within this X-Z-plane now the intermediate x-
axis has to be rotated until it also cuts the axis X gR at distance x co (pitch angle -
4 c) and becomes X c. The corresponding HTM is R 4c [with a 1 in position (2, 2)
for rotation around the intermediate y-axis].
6. The image CS into which the scene is now mapped by perspective projection.
7. The CS for the image matrix of pixel points in computer memory. During data
acquisition and transfer, shifts may occur: By misalignments of a frame grabber,
unintentional shifts of the image center may occur. Intentionally, the origin of
the image CS may be shifted to the upper left corner of the image (see coordi-
nates u, v in Figure 2.5).
Since all these transformations can be applied to only one point at a time, ob-
jects consisting of sequences of straight lines (so-called “polygons”) have to be de-
scribed by the ensemble of their corner points. This will be treated in Section 2.2.
Note here that each object described in a certain CS has to be given by the ensem-
ble of its corner points. For example, the rod is given by the two endpoints E and T
on the X o -axis at –L/2 and +L/2. The straight road is given by its left and right
boundary lines Bl and Br, and by the centerline Cl R in the road-CS. All three lines
are realized by a straight line connection between two end points with indices 1
(left side of Figure 2.6) and 2 (right); all end points of lines defining the road lie at
z = 0. The points on the left-hand side of the road are at y = íB/2, and those on the
right-hand side are at + B/2. The centerline Cl R is at y = 0.
Let us consider the transformation of the endpoint T of the rod into the image
taken by the camera according to Figure 2.6. In the 3-D homogeneous object CS of
T
the rod, this point has the coordinate description x o = ( L/2, 0, 0, 1). After transi-
tion to the geodetic CS X go , the state vector according to point 2 in the list above
changes to
x go R ˜ \ o x . (2.6)
o
To describe the point T in the road-oriented CS, the second HTM T Ro for trans-
lation from C to 0 Rc according to point 3 above has to be applied:
x gR T ˜ Ro R ˜ \ o x . (2.7)
o
For the transition to the geodetic CS of the camera, multiplication by the HTM
T Rc with entries y gc and H c has to be performed according to point 4 above:
x gc T ˜ Rc T ˜ Ro R ˜ \ o x . (2.8)
o
The two translations may be combined into a single matrix containing in the up-
per three rows of the fourth column the sum of the elements in the corresponding