is much worse than with a wide angle objective. Because
of the triangular geometry of photogrammetry, the results
remain valid also for other principal distances, if one ap-
plies the respective factors.
About 5.000 points were manually plotted at an analytical
plotter by static profiling in both data sets. Especially the
large scale data set was measured with great care. In a
second step the small scale images were scanned on a
Zeiss Photo Scanner PS-1, and the software Match-T was
used to automatically derive a DEM at an Intergraph
Image-Station 1.
4.2 The Correlations Between the Height Errors
4.2.1 The Strategy: The test data set was used to create
a series of DEMs by different interpolation techniques. For
all the resulting data sets the correlation coefficients were
computed for a submatrix of 9x9 points (i. e. up to 4 grid
points distance from the center point) according to equa-
tion (6). The next step was to examine the relation be-
tween the correlation coefficients and the local terrain
slope in the grid point. To do so, the terrain was sub-
divided in several zones of hill-slope. For all zones the
calculation of the correlation was repeated. The result were
the same 9x9-correlation matrices as mentioned above, but
now one matrix for each slope-zone. At last the relation
between correlation and slope was tested by statistical
analyses (regression).
4.2.2 The Analysis: The test data set h was used to test
the correlation between a grid point and its neighbours.
First of all the differences in all grid points between the test
data set h and the reference data set H were calculated,
resulting in the matrix of the "true errors’, d, :
d,-h-H. (24)
The "true errors" are reduced by their mean value in order
to eliminate the region wide constant error which does not
have any influence on the slope vector, resulting in the
matrix of the height errors, d,:
nr nc
YS (25)
d = d; _ rl el
ha nc-nr
For d, the covariances are computed according to equa-
tion (4) and the correlation coefficients according to equa-
tion (6) for the 9x9-neighbourhood of any grid point. The
test was done with three different computation levels: a)
The original data as plotted at a Zeiss P3 Analytical Plot-
ter; b) interpolation of the grid through a convolution oper-
ation; c) interpolation of the grid via a weighted average
function. Tab. 3 to Tab. 5 show the resulting submatrices
of the correlation coefficients for the three stages (the
original data set was smoothed by a convolution, too,
though the influence of the smoothing operation was small
on the results of the correlation coefficients of the test data
set):
694
0.06 | 0.10 | 0.15 | 0.21 | 0.27 | 0.27 | 0.24 | 0.19 | 0.14
0.10 | 0.15 | 0.23 | 0.32 | 0.40 | 0.37 | 0.30 | 0.21 | 0.14
0.14 | 0.21 | 0.32 | 0.46 | 0.56 | 0.49 | 0.35 | 0.22 | 0.14.
0.14 | 0.22 | 0.39 | 0.59 | 0.74 | 0.61 | 0.40 | 0.22 | 0.14
0.15 | 0.26 | 0.46 | 0.74 | 1.00 | 0.74 | 0.45 | 0.24 | 0.15
0.13 | 0.23 | 0.40 | 0.61 | 0.74 | 0.58 | 0.36 | 0.19 | 0.13
0.11 | 0.20 | 0.34 | 0.49 | 0.56 | 0.46 | 0.30 | 0.18 | 0.13
0.11 | 0.19 | 0.28 | 0.37 | 0.40 | 0.32 | 0.22 | 0.15 | 0.11
0.10 | 0.18 | 0.24 | 0.28 | 0.28 | 0.22 | 0.15 | 0.11 | 0.09
Tab. 3 Correlation coefficients for the original data
0.49 | 0.51 | 0.58 | 0.55 | 0.64 | 0.55 | 0.59 | 0.52 | 0.52 |
0.53 | 0.56 | 0.65 | 0.62 | 0.73 | 0.62 | 0.65 | 0.55 | 0.54
0.57 | 0.61 | 0.73 | 0.71 | 0.85 | 0.70 | 0.73 | 0.60 | 0.58
0.60 | 0.65 | 0.80 | 0.78 | 0.94 | 0.78 | 0.79 | 0.65 | 0.61
0.61 | 0.67 | 0.83 | 0.82 | 1.00 | 0.83 | 0.83 | 0.69 | 0.64
0.59 | 0.65 | 0.79 | 0.78 | 0.94 | 0.80 | 0.80 | 0.68 | 0.64
0.57 | 0.61 | 0.73 | 0.71 | 0.85 | 0.73 | 0.75 | 0.65 | 0.62
0.54 | 0.58 | 0.67 | 0.65 | 0.75 | 0.65 | 0.67 | 0.61 | 0.58
0.52 | 0.55 | 0.62 | 0.59 | 0.66 | 0.59 | 0.60 | 0.56 | 0.53
Tab. 4 Correlation coefficients for the convolution
0.15 | 0.18 | 0.23 | 0.30 | 0.29 | 0.28 | 0.35 | 0.34 | 0.28
0.22 | 0.18 | 0.26 | 0.37 | 0.40 | 0.41 | 0.37 | 0.29 | 0.30
0.27 | 0.31 | 0.40 | 0.43 | 0.49 | 0.56 | 0.46 | 0.34 | 0.32
0.24 | 0.36 | 0.52 | 0.58 | 0.67 | 0.58 | 0.43 | 0.42 | 0.37
0.31 | 0.40 | 0.48 | 0.71 | 1.00 | 0.70 | 0.46 | 0.39 | 0.32
0.34 | 0.42 | 0.44 | 0.58 | 0.67 | 0.56 | 0.50 | 0.34 | 0.24
0.28 | 0.33 | 0.46 | 0.56 | 0.49 | 0.42 | 0.39 | 0.29 | 0.28
0.27 | 0.28 | 0.38 | 0.41 | 0.39 | 0.37 | 0.24 | 0.17 | 0.24
0.23 | 0.32 | 0.35 | 0.27 | 0.27 | 0.28 | 0.20 | 0.17 | 0.17
Tab. 5 Correlation coefficients for the interpolation through
weighte average.
The original data have been measured in West-East direc-
tion. The correlation is slightly higher in that direction, but
the difference is small enough compared to the value in
order to assume rotational symmetry. The correlation in all
diagonal directions is constant, so equations (19) respec-
tively (23) may generally be used instead of the much
more complex forms for the non-symmetric cases. Gener-
ally, the following conclusions can be drawn from the tests:
o Rotational symmetry is always fulfilled for the diago-
nals and the second neighours along the axes. That
is one more advantage when the slope vector is cal-
culated from the differences of the neighboured points
to a grid point rather than involving the grid point
itself, as shown in equation (8).
o The elevation errors of photogrammetric data ob-
tained through static profiling are nearly uncorrelated
from one point to its second neighbours. Furthermore
the correlation is independent from the axis-direction
(scanning direction).
o The algorithms implemented in the software MSM of
the Image Station do not show any dependence from
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
ah des ga m — a a uade equ ade Zu. SES gauiuih cuiu cut -- AD .- ade M | FIN A ES AN Ar MR ER MA
as Peut ap Var n
