The neural panorama is extremely large and neural algorithms
have been developed to resolve very different kinds of
applications: it is the choice of application that determines the
choice of algorithm.
Attention has been paid to the MLP algorithm (Multi Layer
Perception) to obtain a geometric correction of satellite images.
Function approximation and estimation properties of this
algorithm (non-linear) have already been widely described in
literature. The basic idea is that of substituting the upward
projection model that relates the image coordinates ( 5,77) to
the coordinates of the object (X, Y and Z) with an MLP neural
network opportunely trained on the basis of the GCPs.
The reasons for this choice arose out of an analysis of the
problems connected to the previously described RFM approach.
Neural networks preserve from a forced linearisation of the
equations around an approximated solution. They give a non-
linear response to a non-linear problem, whose efficiency
increases, just similarly to RFMs, with the growing of the
number of GCPs and with the decrease of the original image
deformations.
In the MLP network each neuron performs a very simple
operation that consists in generating, through an opportune
function, which is known as the transfer function, a response to
the signals that converge on it through communication
channels. These channels simulate the biological synapses and
their duty consists in “weighting” the intensity of the
transmitted signals: for this reason they are known as “synaptic
weights” or simply “weights”.
Formally the response signal (u;) that is restituted by the generic
neuron i" is equal to:
N
H, JO wp, t b) (8)
j=l
where / is the transfer function which normally takes the shape
of a hyperbolic tangent (9) or of a logical sigmoid (10), w; are
the weights of the i" neuron, pi are the input at the i" neuron
(N) and 5; are scalar additives, called bias, that are considered
as weights of unitary additional input (Figure 2).
i
f(x) = ET Hyperbolic tangent (9)
; ]-e"
f(x) x Logical sigmoid (10)
+e“
i h
ROT
x) ^b
IARE AER d:
Dur n y
ip Zn
“Z} b
p Vu
?
Figure 2 — Mathematical model of a two-layer computational
MLP neural network (hidden and output).
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
This type of algorithm belongs to the feed-forward family of
neural networks, that is, networks in which the information
travels in parallel and in a single direction.
The MLP network therefore constitutes a mathematical model
in which the parameters are the weights and the biases of the
hidden and of the output layers. The estimation of the values of
these parameters on the basis of opportune samples (patterns),
represents the training step of the network. In this application
the training algorithm is the optimised (for a greater
convergence speed) Error Backpropogation (EBP), known as
Levenberg-Marquardt (LM).
In the EBP algorithm the network weights assume values that
minimise (local minimums) the Performance Function (PF).
This is defined, for a batch training, as:
2 K pr
PEW (1) = > i nda EE aed.)
=
where W(t) = [wy wa.....wy]' is the weight vector of the
network at epoch /, ¢ counts the epochs of the training process
and it is fixed by the operator, d;, is the expected value (rarger)
of the A output relative to the p" training pattern, Hp isthe
value of the 4" output calculated by the network,
E(t)=[e;1.621..-€41.€12--€12€;p-exp]" , in which ej (dif) .
keep Ko p-e1- , P, is the cumulative error of a batch
training.
MATLAB 5.3 Neural Network Toolbox routines have been
used. An upward "orthoprojection" approach has been adopted
so that the coordinates of the object (X, Y, Z) constitute the
network input and the coordinates of the image (4,7)
e
constitute the output. Only one hidden layer and one output
layer have been foreseen. Two network configurations have
been verified and implemented with respects to the possible
transfer functions that can be used. In the first case, the transfer
function adopted for the hidden layer, which results more
appropriate for the treatment of pushbroom images (to which
the here presented results refer) is a hyperbolic tangent (9)
while, for the output layer, it is a simple linear function (purely
weighted sum).
[In the second case, which is considered more suitable for the
treatment of whiskbroom images, a logical sigmoid transfer
function (10) has been adopted for the hidden layer while, also
in this case, a simple linear function has been considered for the
output layer.
The number of neurons (of the hidden layer) that drive to best
performances has to be determined each time on the basis of
repeated tests: in the MATLAB developed routine it is the
calculator itself that does this automatically.
It should be recalled that an approximate estimation (as we are
working in a non-linear ambit, these considerations are purely
indicative) of the maximum number of admissible neurons
could be obtained by comparing the training pattern number
(the GCPs) with that of the parameters to be estimated (weights
and bias). This results to be equal in number to:
7
= ; 2
param = (3 viz) M prsimicdten) * M nistiadon T £u M +2 (12)
where M is the number of neurons of the hidden layer.
More precise indications could be derived from a careful
analysis of the results (residuals), verifying the possible
appearance of overfitting phenomena, which, as a first
approximation, can be identified in the progressive spread of
Internatior
the differe
the Check
The accur:
of residua
extent witl
weights an
of the trair
quite negli
The archit
is able to
has to sup}
e the ra
maxim
to be d
GCPs
numbe
capaci
approx
e the nm
archite
repeat
results
neuron
how tl
initiali
minim
the tra
times f
e level o
on the
basis 0
where
The thresl
orthoimage
helps, in a
network (t
approxima
RMSE.
The RFM
images acc
of the pl
evaluation
the estima
(which wei
of the resi
error. A g
on the enti
validity o
increase in
