Chapter 20 . Classification of Remotely Sensed Images
389
very different to the basic MLP with a single hidden layer that partitions the entire
feature space with hyperplanes along wh ich the activation level of the hidden
units is constant. Since the RBF units respond to local regions it is common for a
RBF network to require a more complex architecture, comprising more hidden
units, than a MLP network constructed for the same problem.
The RBF model provides a smooth interpolating function in which the number
of basis functions and so hidden units required is a function of the complexity of
the problem in-hand rather than the size of the data set (Bishop 1995). As there is
only a single hidden layer there are only two sets of weights in the network, one
connecting the hidden layer to the input layer and the other connecting the hidden
layer to the output layer. Those weights connecting to the input layer contain the
parameters of the basis functions. The weights connecting the hidden layer to the
output layer are used to form linear combinations of the activations of the basis
functions (hidden units) to generate the network outputs. Since the hidden units
are non-linear, the outputs of the hidden layer may be combined linearly and so
processing is rapid. The output of the network is derived from,
M
Yk(X) = LWk
J J
(20.4)
where M is the number of basis functions, x the input data vector, w kj represents
a weighted connection between the basis function and output layer and non-linear function of unit j, which is typically a Gaussian of the form
(20.5)
in which Il j and O"j are the parameters specifying the basis function' s centre and
width respecti vel y .
Training a RBF network involves two distinct stages. In the first stage, the
basis functions are determined using an unsupervised analysis. This effectively
defines the weights connecting the units in the input and hidden layers. The
determination of the basis function parameters in this first stage of training the
RBF network mayaiso make use of unlabelled cases (Bishop 1995). This can be
advantageous in remote sensing applications as ground data are often scarce or
costly to obtain but the remotely sensed data set is generally voluminous
(Shahshanhani and Landgrebe 1994; Fardanesh and Ersoy 1998). In the second
stage of training, the weights connecting the hidden and output layers are derived
using a linear supervised method (Bishop 1995).
389
very different to the basic MLP with a single hidden layer that partitions the entire
feature space with hyperplanes along wh ich the activation level of the hidden
units is constant. Since the RBF units respond to local regions it is common for a
RBF network to require a more complex architecture, comprising more hidden
units, than a MLP network constructed for the same problem.
The RBF model provides a smooth interpolating function in which the number
of basis functions and so hidden units required is a function of the complexity of
the problem in-hand rather than the size of the data set (Bishop 1995). As there is
only a single hidden layer there are only two sets of weights in the network, one
connecting the hidden layer to the input layer and the other connecting the hidden
layer to the output layer. Those weights connecting to the input layer contain the
parameters of the basis functions. The weights connecting the hidden layer to the
output layer are used to form linear combinations of the activations of the basis
functions (hidden units) to generate the network outputs. Since the hidden units
are non-linear, the outputs of the hidden layer may be combined linearly and so
processing is rapid. The output of the network is derived from,
M
Yk(X) = LWk
J J
(20.4)
where M is the number of basis functions, x the input data vector, w kj represents
a weighted connection between the basis function and output layer and non-linear function of unit j, which is typically a Gaussian of the form
(20.5)
in which Il j and O"j are the parameters specifying the basis function' s centre and
width respecti vel y .
Training a RBF network involves two distinct stages. In the first stage, the
basis functions are determined using an unsupervised analysis. This effectively
defines the weights connecting the units in the input and hidden layers. The
determination of the basis function parameters in this first stage of training the
RBF network mayaiso make use of unlabelled cases (Bishop 1995). This can be
advantageous in remote sensing applications as ground data are often scarce or
costly to obtain but the remotely sensed data set is generally voluminous
(Shahshanhani and Landgrebe 1994; Fardanesh and Ersoy 1998). In the second
stage of training, the weights connecting the hidden and output layers are derived
using a linear supervised method (Bishop 1995).