0 0 0
ve ams! (2) a + & dk -
am, ot ok (5)
T (m; T (Morr 0 K°)
where dm,,, di and dk are the unknowns of the
observation equation (They are corrections to approxi-
mate values of M, ( and k. The superscript ° indicates
that for function f the approximate values have to be
applied. )
«secondly, provide approximate values for the unknowns
as initialisation. It is rather simple to estimate those
values as the expected ideal values are known:
m,,~128, (=0 and k=1 are appropriate choices (i.e. the
values for an ideal Lambertian reflector with an average
greyvalue of 128)
The adjustment process is iterative. The approximate
values of the current step (n) are corrected by the solutions
for dm_,,, di and dk, thus leading to the new approximate
values for the following step (n+1). The iteration process
terminates when the correction values fall below a given
threshold [e.g.: Albertz et al., 1989].
Independent of the size of the image the solution will always
be very fast due to the small equation system and to the
rather good convergence behaviour as practical tests
demonstrated. In most cases not more than three to five
iterations are necessary. The number of observation
equations depends on the iclasses for which the greyvalue
means are to be calculated. For 10? intervals of ij, for
instance, ten observation equations may be obtained (1*' for
the interval from 0° to 10°, 2™ for 10° to 20°, ..., 9" for 80° to
90° and 10" for above 90°). It turned out advantageous to
eliminate the flat as well as the very steep terrain regions
from the adjustment. This can easily be done by calculating
a slope image, which usually exists anyway in the course of
the calculation of j, and eliminate all areas with slopes below
and above a certain threshold. Very often flat areas contain
much more and some rather different object types than
slopes (see lakes in our example for instance) and thus
may confuse the algorithm while it does not do any harm if
observations are missing for a small interval of i as the
correction function will bridge this gap anyway. By excluding
steep slopes also all those virtual slopes will be excluded
that may occur on the edges of the DTM image or that may
occur even in rather flat terrain due to a possibly coarse
discretisation of the elevation values. Also those ;classes
should not be taken into consideration that do not contain a
given minimum number of pixels.
The advantage of the adjustment procedure is the con-
current delivery of accuracy values of the fitting process.
Beside the mean error (0, = VEVA/(n-u), where n is the
number of observation equations and u=3 is the number of
unknows) one also can easliy obtain the mean errors for the
unknowns (0, 9, and o,). If necessary the actual residuals
v, for each individual observation may be calculated, too.
5. THE NORMALISATION PROCESS
For the topographic normalisation one just needs to apply
the correction function (4a) to the original image:
1
cor 7 9i x - gi
g g (e+ (1-0)-coski)
(7)
Each pixel of the image will be corrected according to the
incidence angle on its location. The mean values within
each incidence class of the corrected image will then be
approximately the same. One should be aware that together
with the correction of the mean values also the standard
deviations are affected. We'll come back to this later.
In order to estimate the influence of the inaccuracy of the
parameters, the correction term K (see equ. 4a) has to be
partially derived with respect to its parameters ¢, k and |
(equation 8):
eK. K?- (t - 1): cos*i-In(cos)
ok
Sé - K?-(cos*i - 1) (8)
= = K?-(1 -0): k:cos* ' i: sini
/
From this derivatives one can easily obtain the root mean
square error of the topographically corrected greyvalue by
applying:
retire ie
ok ol 9i (9)
I
where o, and o, can be obtained by the adjustment proced-
ure and o, can be estimated from the DTM accuracy. By
analysing the above formula it becomes obvious that the
accuracy of normalisation deteriorates rapidly towards
i=90°.
6. EXAMPLE
A practical example should prove the suitability of the
presented correction model. The terrain heights of the
“Salzkammergut” test area range from some 400 m to
1800 m a.s.l. The image is a Landsat TM scene from
September 1991 (see Fig.2). The digital terrain model was
available as 16bit-integer intensity image with directly coded
terrain heights. From the DTM image have been derived
- a slope image (used for the calculation of i and for the
exclusion of flat and very steep areas)
- and - together with the given sun position - the cos-
image.
The image has been classified into seven incidence classes
(0-15°, 15-30°, 30-45°, 45-60°, 60-75°, 75-90°, >90°), the
mean values and standard deviations were calculated for
each spectral band (except band 6), obtaining e.g. for band
1 and 4:
12 International Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 7, Budapest, 1998
Tab.1:
This le
band
adjustr
values
54.6
22.8
20.1
75.4
64.9
22.3
Tal
80
60
40
20
One ca
very ck
factor (
than 0
the blue
importe
correct
accura
