Level
0 land-cover
eer quA omes, MD
1 non-vegetated vegetated
2 water road blond sand wood herbaceous
3 deciduous hs1 thin grass/herb cover with blond sand
coniferous
sea buckthorn
privet/creeping willow
hs2 intermediate herb/moss cover with grey sand
hs3 high moss cover
hs4 high moss and low grass cover
hs5 high grass/herb cover with litter
Fig 3. Example land cover hierarchy.
3.3 Aggregation of fuzzy data
The image interpretation in the previous section yields
a 7-fold vector of membership values (MV... ,, MV,.,,
MV, MV, .,, MV, MV,.,, MV, ,,,) for each image pixel.
Although it is possible to detect temporal changes in
membership values on the pixel level, it is usually better
to aggregate the membership values to larger cell sizes
in order to minimize the effects of classification errors
on the temporal analysis.
The aggregation of fuzzy data is dealt with by
Klir and Folger (1988). First step is the calculation of a
pseudo-frequency N(c) for each state of the aggregate c
€ (sand, hs1, ..hs5, wood]. Since values of N(c) need not
be whole numbers, it is better not to use the therm
frequency. In order to accomplish that each cell
contributes equally to the pseudo-frequencies the
membership values of a single cell have to sum to one.
If not so, the membership values have to be normalized
in this sense. The pseudo-frequency for each state or
class is calculated as the sum taken over ail cells within
the segment defining the spatial extent of the
aggregate.
The pseudo-frequencies are now used to
estimate the value of a fuzzy measure, i.e. possibility or
probability, indicating the strength of the relationship
between the aggregate and a class. The (pseudo-
probability distribution is calculated by dividing a
pseudo-frequency by the sum of all pseudo-fre-
quencies:
p(c) = N(c)/ Y, N(z)
z c HS
The aggregation did not change the vector of attribute
names, unlike their measure for quantification, which
changed from membership values to pseudo-
probability values (p. Phe Phe Pus Psst Phssr Pood):
Note that the link between the concept of fuzzy sets
and fuzzy measures is effectuated by the pseudo-
frequency distribution. See table 2 for an example.
Tab. 2. Illustration of probability (p) distribution
estimates derived from a pseudo-frequency distribution
N calculated over 5 cells.
cicell..4- 2.3.44: 5 N(c) p(c)
sand 0:0: 0:10 15:0:5,1 1.0 0.20
herb1 02 03.05 .0,.:0 1.0... 0.20
herb2 - 0.8..0:2.-0:15-01: 50 3.6,,0.32
herb3 > +0:<::0:: 04:0 11.0 0.4 0.08
herb4 :: 0.0. -.0,+0-+<0 0.0 0.00
herb5 4:045 :0::50./0 0/540 0.0 0.00
wood Qs po: Oiosdos 1.0... 0.20
4 QUANTIFICATION OF ECOLOGICAL PROCESSES
By the proposed aggregation procedure the objects
and fields are converted in a field. In this field the
presence of each vegetation type is expressed by a
probability, resuiting in a vector of 7 probability values
(Psand Phstr Pur Phsar Pusar Press! Prod)” Obviously, this
vector specifies a point in a 7-dimensional vegetation
space, where each vegetation type defines an axis of
this space.
Changes in the vegetation composition. on a
specific site (x,y) from date t to t+At result in a move
through this space from point (P_, , Du, Du Phy Pres
Phes/ Progehsy to point (m Pu Puy Pus; Pu Pus
woody Tespectively. Each cell shows a specific
change in the vegetation space and from all these
movements general processes of vegetation change
can be calculated. Because a single vegetation
composition on different sites might develop into
different directions, each vegetation composition on
time t relates to many possible vegetation compositions
on time t+At. This set of possible future compositions
forms a cluster in the vegetation space. The cluster of
possible vegetation compositions can be conveniently
described by its point of gravity and standard deviation.
218
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
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