AQ ALI eere for Ag) = Ag)
BO CB Ce for Aa — An)
ADA = AD AD ren for Ar) = Arc)
b® = b®):
i k
pO = p
6) nd
by = by
i k
pO = p
Before these six equations can be added as constraints
of the bundle adjustment, the formulas must be linarized
with respect to the original exterior orientation parameters
of the images (the perspective center and the three rotation
angles of each image). This is rather complicated as the
relative rotation matrices depend on parameters of both
photos.
The distance between the two perspective centers of any
image-pair is constrained, which is the base-distance do that
was measured by theodolites (7). Its accuracy is about 0.5
mm. This constraint defines the scale of the local
coordinate system in which 3-dimensional positioning is
possible.
d, 2 A (Xi — Xp)? + (YL — YRŸ + (ZL — Zr)?
= Wb2 + b3 + b2 (7)
with: XL, YL #1. coordinates of the left perspective
center,
XmYm ZR... coordinates of the right perspective
center,
NE distance measured between
perspective centers,
bx, by, b;......... components of the base vector.
This constraint is only specified for one stereo-pair; all
other base distances are automatically equaled by condition
(6). Therefore, there are (m-1) relative orientation
constraints for m stereo-pairs, and one base-distance
observation.
By collecting all the formulas mentioned before (1), (6),
(7), and writing a system of observation equations and
constraints, we get (8). The least squares solution is
computed by (9).
(8)
b
observation equations: v = A X —
Iz
px
+
le
constraints: EG
with: A... linearized collinearity equations (1) and
distance equation (7),
dee unknown orientation parameters:
Xp: Jp. €, A], A2, A3, A4, A5, AG for each camera,
Xo, Yo, Zo, 0, 9, Kfor each image.
I se discrepancies (observation minus
approximation),
D... linearized constraints of the relative
orientation (6),
PA absolute terms of constraints,
a ee residuals of the observations,
DE zero vector.
jen Bt I= | AT]
_— = -—- 9
B 9 -t 2
: Once these orientation parameters (x) are available, 3-
dimensional coordinates can be computed for any point
identified in both images. This is useful for measuring
spatial distances or for creating local 3-dimensional models
of small objects in the field of view of the stereo-vision
system. However, the positions are only defined in a local
coordinate system relative to the two cameras. These
coordinates are referred to as local camera coordinates (X...
Y., Zc) and will be transformed into a global system in the
next chapter.
4. ABSOLUTE POSITIONING
The local coordinate system derived above corresponds
to a stereo-model. To obtain absolute positions of points
and features the absolute orientation of this model must be
established. It consists of six parameters (3 translations and
3 rotations) to convert points from the camera system (Xc,
Yc, Zc) into a topocentric system (Xt, YT, ZT) with its
origin at the GPS antenna, the XT axis pointing east, the YT
axis pointing north, and the Zr axis identical to the vertical
at the ellipsoid. The scales of the two coordinate systems
are equivalent. The six transformation parameters are
derived from the GPS position of the van (3 coordinates,
which correspond to 3 translations) and the inertial
measurements (gyros), which define 3 rotations (direction,
pitch, roll). To visualize this transformation in a better
way, it is separated into two steps:
4.4 Transformation to a Vehicle System
The vehicle coordinate system (Xy, Y y, Z,) is directly
connected to the van; it is defined by the Yy-axis pointing
in the driving direction and the Xy-axis parallel to the left
image plane, as well as to the rear axle of the vehicle. The
Zy-axis is vertical if the van is positioned on an horizontal
surface. The vehicle coordinate system is assumed to be
parallel to the gyro-axes of the inertial system. This
coordinate system can be found by repeated measurements
of the GPS antenna and the cameras by theodolite
