Section 5 presents the final conclusions and some 
important considerations about the use of Mixing 
Model. 
2. ESTIMATION OF THE PROPORTIONS OF COMPONENTS 
WITHIN A PIXEL 
2.1 Linear Mixing Model 
By adopting the linear Mixing model, the pixel 
value in any spectral band is given by the linear 
combination of the spectral response of each 
component within the pixel. The model can be 
expressed as: 
r, = a x; + à Xo d i. + 
zz = H Rl + at? x + + 
934 M4 99: Ey T ines 2n *n 2 
m ml | m2 2 mn m 
that is: 
n 
r - X (a. X.) te,. 1 1, ..., m (number of 
Ek 4 48b 3 2 bands) (4.1) 
371, ...5 ‘nn (number 
of components) 
where, 
r, = spectral reflectance of the 25 spectral band 
of a pixel which contains one or more 
components; 
24 = know spectral reflectance of the im 
J component within the pixel on the i-— 
spectral band; 
x. - value of the 2 component proportion within 
J the pixel; and 
th 
e, = error for the i— spectral band. 
The estimates for x. are subject to the following 
restrictions: J 
n 
S x = 1 (4,2) 
Q«x «i1 (4.3) 
since they represent area proportions within a 
resolution element. 
2.2 Methods for proportions estimations 
  
The proposed methods in the literature to estimate 
classes proportions within a pixel select those 
components proportions such that their spectral 
signature be the best approximation of the observed 
pixel value (Ranson, 1975). 
The methods used in this work are based on the 
criteria of Least Squares, namely Constrained Least 
Squares (CLS) and Weighted Least Squares (WLS). The 
objective is to estimate the proportions x, by 
minimizing the sum of the squares of the errors e,, 
subject to the restrictions given by the 
expressions (4.2) and (4.3). The detailed 
description of these methods can be found in 
Shimabukuro et al. (1991) and Aguiar (1991). 
Once the proportions x,, .j ^ 1,...,n (number of 
components) are obtained, n synthetic bands that 
are linearly related to the estimated proportions 
x. are generated by multiplying the proportions in 
edch pixel by a scale factor 255 (maximum value of 
the pixel value). 
3. DIMENSIONALITY REDUCTION OF THE FEATURE SPACE 
This reduction can be obtained through the 
selection of an appropriate subset of features or 
through the transformation of the feature space 
into a smaller dimension space. 
The selection of an appropriate feature subset 
cannot be done in an indiscriminate manner. It is 
necessary to perform a comparison between the 
candidate subsets, based on the classification 
performance. Ideally, the criterion should be based 
on the error probability, but its computation is 
usually very difficult. Therefore, indirect 
criteria that express distances between 
distribution are used and they provide upper and/or 
lower bounds on the error probability. 
For multiple classes problems, frequently found in 
Remote Sensing, the Jeffries-Matusita (J-M) 
distance is one of the most widely used. For a 
detailed description of this distance, see Swain 
and Davis (1978) and Richards (1986). 
Another method to reduce the dimensionality of the 
feature space is transform the data into a new 
space in which the new features to be discarded 
become evident. Several transformations can be used 
for this objective. In Remote Sensing one of the 
most used transformations is the Karhunen-Loëve 
transformation, also known as the Principal 
Components transformation. 
The Principal Components transformation provides a 
mapping of the original pixel values into a new 
coordinate system, in which the new random 
variables are decorrelated. Also, it offers the 
maximum compression, under the mean square error 
criterion, for a given dimensionality reduction. It 
should be observed that the K-L transformation is 
optimum in the sense of representation of the 
mixture of classes and does not aim class 
separability. However, it has been frequently 
observed that, in Remote Sensing, the data is 
usually distributed in the direction of the first 
principal components and, in this case, the K-L 
transformation is also effective for class 
separability. 
A method that is specifically derived to optimize 
this last criterion is the so-called Canonical 
Analysis or Multiple Discriminant Analysis (Duda 
and Hart, 1973). This transformation rotates the 
data into a new space, with a maximum 
dimensionality equal to the number of classes minus 
one, by maximizing the ratio of the spread of the 
data between and within classes. 
The proposal of this work is to compare the use of 
these conventional feature reduction methods with 
the method derived from the Mixing Model, which can 
be regarded as a way to reduce the dimensionality 
of the data to the number of primary components 
within a pixel. In this case, the original bands of 
the multispectral images are transformed into the 
synthetic bands, as it was described earlier. In 
260 
  
  
  
  
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