D
Y=-—-C (3)
g c
Putting (3) into (2) the final correction equations result:
deine n ai
—'C, t€
3 y
dx' (ex) = a ex (4)
D
In analogy the correction of image coordinates y' yield to
y 7 y'&dy (ey) * dy (ez)
y^ey
and hence dy'(ey)=- D and
a ‘Cr +ey
dy'(ez) = d ez (5)
D
In order to compute the exterior orientations of the images
within the panorama the mathematical model of spatial
similarity transformation is used. By default it is based on
object coordinates:
x X
e, [7 A-R-|Y (6)
y Z
The presented image corrections based on excentricity and the
formulation of object coordinates as angular directions the
following equation results:
x'+dx' (ey) + dx' (ex) sin(@)
Cr = À-R-| cos(a) (7)
y'+dy' (ey) + dy' (ez) tan()
and the modified collinearity equation can be derived as
xX +dx' (ey) + dx' (ex)y =
x Ail -sin(œ, ) + 51, : cos(a) * 03, -tan( B)
m.
: 3, - sin(@,) + 753, * COS(O; ) + 733, -tan(B;)
1 1 : 1 : (8)
V'u +dy' (ey);, + dy' (ez)x =
Fy, -Sin(@y) + ry, * COS(@ ) + 732, -tan(B,)
hs, -Sin(at) * 755, : cos(a ) * 73, -tan(B,)
=C;
These equations consists of all parameters that have to be
estimated:
= excentricity parameters ex, ey and ez
= direction, tilt angle and roll of image (7) of the rotation
matrix R(e,œ,K)
= direction and slope to object poin (k) (a and p)
x target point
image plane A er diei)
image plane
/
perspective centre 4 /
ei)
© tripod axis
Fig. 8: Excentricity ex and ey
3.3 Practical process for panorama images generation
For the practical realisation of geometric exact panoramas the
determination of the excentricity parameters are of major
importance. Only for the case that the excentricity vanishes to
zero, exact panoramas can be calculated from single frame
images. Therefore the "imaging system" camera/tripod is
calibrated in advance and justified according to the following
steps.
Firstly, circular targets with known diameter are distributed in
object space that is imaged as a panorama by the camera to be
calibrated. A least-squares adjustment based on the observation
equations (8) yields the excentricity quantities. Subsequently
the camera positioin on the tripod is corrected. In the following
an arbitrary number of panoramas can be aquired by this
system.
The panoramas themselves can be computed from the single
images by resampling based on (8) in addition with a
radiometric matching of the overlapping zone (Fig. 9). Tie
points are measured in order to calculate the local orientation
parameters of the images (similar to relative orientation). The
resulting quality can be assessed by the remaining residuals of
tie points. A number of experiments has shown that the stability
of the mechanical adjustment of the tripod/camera system is
better than 1mm, even if the camera has been dismounted
several times.
Fig. 9: Panorama, calculated using orientation data
-184—
en a > oO € t Oo
me Aa = A de pun