Page 146 - Introduction to Autonomous Mobile Robots

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Perception
•A point in the scene visible to both cameras produces a pair of image points (one via
each lens) known as a conjugate pair. Given one member of the conjugate pair, we know
that the other member of the pair lies somewhere along a line known as an epipolar line.
In the case depicted by figure 4.22, because the cameras are perfectly aligned with one
another, the epipolar lines are horizontal lines (i.e., along the direction).
x
However, the assumption of perfectly aligned cameras is normally violated in practice.
In order to optimize the range of distances that can be recovered, it is often useful to turn
the cameras inward toward one another, for example. Figure 4.22 shows the orientation
vectors that are necessary to solve this more general problem. We will express the position
P
of a scene point in terms of the reference frame of each camera separately. The reference
frames of the cameras need not be aligned, and can indeed be at any arbitrary orientation
relative to one another.
P
For example the position of point will be described in terms of the left camera frame
,
,
as r' = ( x' y' z' ) . Note that these are the coordinates of point , not the position of its
P
l l l l
counterpart in the left camera image. can also be described in terms of the right camera
P
,
,
frame as r' = ( x' y' z' ) . If we have a rotation matrix and translation matrix r relat-
R
r r r r 0
ing the relative positions of cameras l and r, then we can define r' r in terms of r' l :
⋅
r' = Rr' + r (4.29)
r l 0
R
where is a 3 x 3 rotation matrix and r 0 is an offset translation matrix between the two
cameras.
Expanding equation (4.29) yields
x' r r r x' r
r 11 12 13 l 01
y' r = r 21 r 22 r 21 y' l + r 02 (4.30)
z' r r r z' r
r 31 32 33 l 03
The above equations have two uses:
1. We could find r' if we knew R, r' and r . Of course, if we knew r' then we would
r l 0 l
have complete information regarding the position of relative to the left camera, and
P
so the depth recovery problem would be solved. Note that, for perfectly aligned cameras
as in figure 4.22, R = I (the identify matrix).
2. We could calibrate the system and find r , r … given a set of conjugate pairs
12
11
,
( { x' y' z', , ) x' y' z' )} .
,
(
,
l l l r r r