International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
n 
Mm S CS; ( ! ) 
i=l 
where N (&,, o,) is the input membership function with the 
Gaussian distribution of mean Z,; and stand deviation o,; c,; and 
c, are tunable coefficients. The weight of rule r for a data point 
s is determined according to the distance between the vector s 
and the center of an n-dimensional Gaussian sphere with a 
mean (ó6,,.., &,) and a standard deviation o,, which is the 
product of input membership values: 
  
| « > = ] ) 
w, — exp| — ; s 6, - v = [Teo] 5 nié, = | 
i=] i=1 20, 
20 
(2) 
r 
Normalizing the weights of all rules: 
where A is the number of rules. Finally, the output of the 
rulebase is given by 
Q = S Pryr e 
rz] 
It has been shown that such a configuration can approximate 
any nonlinear function to any degree of accuracy if the number, 
the locations, and the variances of Gaussian spheres are 
allowed to change. 
The above approach to identifying an RBF rulebase from data 
is described in Procedure 1. It begins by creating an RBF rule 
with the Gaussian center coinciding with the first data point. 
When the next data point is encountered, the parameters of the 
first rule are adapted to account for both data points. If the error 
on the second data point is still too large, then a second rule is 
created centered at the second data point. The process 
continues until all data points have been considered. After all 
the data have been processed, the neurons are pruned to get rid 
of redundant rules and make knowledge more compact. This 
may lead to slightly higher error with a reduction in the 
rulebase. It is a very fast, one-pass algorithm, which also gives 
very good results as our experiments indicate. 
Let s and d be the input and output parts of each data point. For 
each sample (s, d) do | 
INFERENCE: 
O = Rulebase output when input s is presented; 
Let the index of the nearest Gaussian rule be 
i" = arg min Als - Eb 
Let the distance to the nearest rule be 4" — Is -é. li 
Let the nearest rule be 
w' z exp( -|s sé, [20° ) 
applicability of the 
Let the error of that rule be error = lo — dl: 
ADAPT PARAMETERS: 
Modify parameters by learning rules: 
^Ac,; = (d - O)p,s; 
AC, =n(O-d)y, 
  
  
  
p, rm p. XE, = $;) 
= 
r 
> 
Je 
Es 
  
] 
Ac, » (0 - d)y, — p, (1 - p,) 
o. 
ADD RULE: 
If (w' « à)( 
add neuron at s with 
d'[421n2 - 6. 
INFER and ADAP PARAMETERS; 
} else if (error > €){ 
add neuron at s with spread o,,;, 
INFER and ADAP PARAMETERS; 
i 
J 
spread 
1 
j 
PRUNE RULES: 
For each pair of remaining neurons 7, and n; (a « b) 
do { 
Let 0,,, = angle (hyperlane,;, hyperplane,;); 
Let d y = I. = €, 
If (max; 0, € w){ 
— if (0,7 d, ) (winner: = a; loser: = 
, 
  
  
bi 
else if (o, < d,; ){winner: = b; 
loser: = a} 
else consider next pair; 
move winner towards loser in 
proportion g” 
Hu Oo ? 
expand winner's radius to include 
loser's radius; 
delete loser neuron; 
1 
f 
\ 
J 
EXTRACT RULES from neurons; 
Procedure 1. Algorithm for extracting radial basis function 
rules from data 
To verify the effectiveness of the RBFC algorithm in the 
Landsat image classification, in the following section, we will 
compare it with the widely used multilayer neural network, 
Back Propagation Neural Network (BPNN) (Duda, 2001; 
Heermann, 1992). 
3. ASTUDY OF LAND COVER CLASSIFICATION 
3.1 Landsat 7 ETM+ Data Set Over Rio Rancho 
Landsat 7 carries the Enhanced Thematic Mapper Plus (ETM+) 
instrument—a nadir-viewing, multispectral scanning 
radiometer, and provides image data for the Earth’s surface via 
cight spectral bands (NASA, 2000; USGS Landsat 7, 2000). 
The bands are for the visible and near infrared (VNIR), the 
mid-infrared (Mid IR), and the thermal infrared (TIR) regions 
of the electromagnetic spectrum, as well as the panchromatic 
region. Table 1 lists the ETM+ Bands, spectral ranges, and 
nominal ground resolution. Data are quantized at 8 bits. The 
size of the image for one band is 744 lines x 1014 pixels, which 
is shown in Figure 1 over the Rio Rancho area with 3 bands 
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