The International Archives oj the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008
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For each pixel of the reference patch an equation can be formed parameters are used. For this, additional observation equations
in every image patch. The discretisation leads to the following for bias and drift have to be introduced:
linear system:
v = ~g‘ x (x,y)da i - g‘ y (x,y)db‘ +dG(x,y)
~g‘(x,y)+G 0 (x,y)
where x, y = pixel coordinates
g‘(x, y) = grey value of image patch i (/= 1,..., n)
g°(x, y) = grey value of fixed image patch 0
g' x ~ grey value gradient with respect tox
g' = grey value gradient with respect to y
G 0 (x, y) = initial value of theoretical grey value
da' ,db‘ = unknown translations of image patch i
dG(x,y) - unknown change of theor. grey value
To avoid a datum defect the position of the first image patch 0
has to be fixed, i.e. the translations are set to zero:
v = dG(x,y)~ g°(x,y)+G 0 (x,y) (3)
With these two equations a Gauss-Markov model is set up and
solved iteratively. The resulting final image coordinates are as
already mentioned converted back to the level-2 coordinate
system and serve as input for the subsequent bundle adjustment.
3. BUNDLE ADJUSTMENT
The bundle adjustment approach estimates the parameters of the
exterior orientation only at few selected image lines, at the so-
called orientation points. To fit the points obtained from the
matching into the existing reference system on Mars the MOLA
DTM is used as control information. Furthermore, the
parameters of interior orientation have been improved by a
similar bundle adjustment (Spiegel 2007a) and are assumed to
be constant in the following approach.
v Xm =X H -X m +X B +JX D
= Y m ~Y m + Y B + jY D (4)
^Z m = Ym ~ Z m + Z B + jZ D
where v x ,.v Y ..v z ~ residuals at orientation points m
X Ÿ Z = un ^ nown ext - orientation parameters
m ’ m » m a t orientation points
X y Z - observed ext. orientation parameters
m ’ m ’ m ~ at orientation points
X B ,Y B ,Z B = unknown bias
X D , Y d , Z D - unknown drift
j = number of image line
To use the MOLA DTM as control information the least
squares adjustment has to be extended with an additional
condition describing a relation between the MOLA DTM and
the HRSC points. A bilinear surface defined by four
neighbouring MOLA DTM Mj, i = 1, ,4 points is created for
each HRSC point H (see Figure 3).
Figure 3. Fitting HRSC point in surface defined by MOLA
The mathematical model for photogrammetric point
determination with a three-line camera is a least squares
adjustment and is based on the well known collinearity
equations. In principle it is necessary to improve the exterior
orientation parameters for each image line in which observed
image coordinates are available. But, this is not possible due to
geometric reasons. In case of Mars Express (satellite orbit) the
trajectory is assumed to be stable and the parameters of the
exterior orientation are improved only at few selected positions
(orientation points). In order to obtain exterior orientation
parameters for every sensor line the values lying in-between the
orientation points are interpolated with Lagrange polynomials
of grade three.
In case of Mars Express there are observed parameters of the
exterior orientation for each image line. At the orientation
points these observations can be introduced into the adjustment
with additional observation equations. Because of Doppler shift
measurements there are systematic effects in the observed
exterior orientation. To model these effects in the bundle
adjustment biases (offset) and drifts for the three position
The condition that the HRSC points have to lie on these
surfaces can be formulated as a constraint on the vertical
distance from the HRSC point to the bilinear surface.
Furthermore, this constraint can be substituted by a fictitious
observation used as an additional observation in the bundle
adjustment:
^z„ - %H
X„-X t
V
1-
Y -Y
1 U A/
V
1-
V -Y
1 h 1 M,
1-
x H -x Mi
V
Y -Y
1 h 1 M,
X„-X Mi
v
Y -Y
1 H 1 M,
(5)
