corresponded to a thematic category. Therefore the
density values are the inputs and the class belongings the
outputs. If the image has n spectral bands (for Landsat
TM 6 without the thermal infra) the input will contain 6
values. Having m thematic classes we do have m outputs
between 0 and 1. These class belongings can be
interpreted as probabilities distinguishing from the
mathematical nomenclature they are named as "neural
probabilities".
Increasing the number of thematic classes the amount of
necessary neurons also increasing while the network
parameters grows drastic. This growth has radical effect
on the Jacobian so we must find more and more powerful
computers to be able to train these networks. We can use
an another solution: let's cut the problem into smaller
tasks that means let's design several independent
networks! The discipline what is followed is "one
thematic class — one neural network". From the whole set
of ground truth pixels we must define smaller but
representative training sets for all neural nets. It's desired
that all the class-own pixels should be in the training set
while the rest should be presented by sampling. With this
design we'll have "class responsible networks".
As preliminary tests have shown 3-layer neural networks
are satisfactory and usable in all classification cases. The
only little difficulty is the simulation of these networks:
having ten classes, must be evaluated ten neural networks
to get a single output for a pixel. Using feed-forward
networks with the same structure (3 layers — of course
with different number of processing elements) we can
transform the networks into an equivalent one. As
Figure | shows we can put together the networks by
defining fictive connections, which are easy to be defined
in weights and biases.
1st neural network
—— — MÀ —— —— mn A ee eu
2nd neural network
Figure 1
Transformation of two same structured neural networks
The bias vectors are the accumulation of the network
biases:
bz. (3)
where b' is the bias vector of the transformed network,
b;,b;..b, the bias vectors of the 1“ 2'4 and n®
independent nets. The weight matrix after transformation
will be a hyperdiagonal matrix
W000
edie Wy
M - : : ee : (4)
0 0
where Wi, W;...W, are the weight matrices of the current
layer for the input networks. There's only an exception:
the first layer, where the derived matrix is defined as
W; =[W, W, … W,] (5)
because all the input intensities must be feed into all
networks.
The described transformation has three important
advantages:
+ the maintenance of the classification networks is
getting much simple; there's only a single
network to store and simulate
+ after the training of the independent networks
it’s totally flexible which “subnets” is to collect
for the simulation
+ simulating just one network instead of several
ones there's a sensible speed acceleration of the
network simulation.
2.2. Elements of fuzzy logic
Fuzzy logic operates with extended set memberships.
There aren’t only two belongings, 0 and 1 but infinitely
lot in this range. The logic basing on the new set theory
has own basic functions for OR, AND, NOT and for all
further derivable functions. L. A. Zadeh gives the mostly
used definition:
AND(x,,, x) = min( 44, (x), 4, (x)
OR(X 4, Xp) = max(u (x), ug (x))
NOT (x) =1- (x)
(6)
Fuzzy decision making is a very efficient use of fuzzy
logic. In the decision making there’re further AND-type
functions defined, the most interesting one is the
Zimmermann-Zysno which is a kind of combination of
AND and OR:
[4,09 us CO] = {0 = p00) - 65 CON
(7)
324 Intemational Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 7, Budapest, 1998
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