A second mixture model, implementing Fuzzy 
mathematical concepts, was proposed by Wang 
(1990). The concept of class membership is 
introduced to account for multiple member- 
ship, i.e., for the mixture pixel, Any 
Pixel may belong to more than one class. A 
membership function is introduced to 
estimate the degree to which a pixel belongs 
to each class. The membership degree is 
then associated to proportion of an 
information class in a pixel. 
2. THE LINEAR MIXTURE MODEL 
The linear mixture model assumes that in 
any spectral band 'i', the reflectance Rp 
associated with a mixture pixel can be 
equated to a linear function of the re- 
flectances R4 associated with the component 
("pure") classes. Each component class 
reflectance is weighted according to its 
proportion in a pixel: 
n 
Rai > R. i x. +. (1) 
, j=l Jo» j i 
where: 
m.i 7 Bean spectral reflectance associated 
2 with a mixture pixel for spectral 
band i. 
R. . 7 spectral reflectance associatedwith 
Jak the component j for spectral band i. 
x, = proportion of component j in a pixel 
j 1,2,...,n(n=number of components). 
i = 1,2,...,k(k=number of spectral bands) 
£i 7 error term associated with spectral 
band 'i',. 
To implement this model, the digital image 
must be converted from digital numbers 
available on CCTs into reflectances. This 
procedure is reported by some authors(eg.: 
Robinore, 1982; Markham and Barker,1986). 
The reflectances Rj i of the component 
classes can then be estimated for the 
training sets available. 
Usually the number of spectral bands 
utilized is larger than the number of 
component classes (K >n). In this case,the 
system (1) becomes overdetermined and a 
least squares procedure is then applied. 
Then, the numerical values for X3€¢3=1,..., 
n) should be such that they minimize the 
suns of the squares of errors £i 
K 
2 £. = minimum (2) 
Also, two additional condition equations 
should be added in order to allow for a 
phisically meaningful solution to the pro- 
portions xs 
X,.5. 104,11 % j 
J 
: (3) 
y X, = | 
jz1,.3 
The Linear Mixture problem can thus be 
written as: 
n 
fe RR = aj OUR. 0X. 
i m,i =}; 1» j 
908 
K 
Minimize X £2 as a function of the pro- 
i=1 1 portions x 
Subject to: 
«x xj (4) 
Numerical methods to solve this constrained 
least squares problem are presented in Shi- 
mabukuro (1987). 
The Linear Mixture model, applied to each 
Pixel individually, estimates the proportions 
of every component in it. 
Another approach (Haertel, 1991)consists in 
using equation (1) to estimate the mean 
vector and covariance matrix for given 
mixture proportions (X.). Mixture classes 
can then be selected and the entire image 
classified into the existing"pure" classes 
and the selected "mixture" classes, using 
a maximum likelihood classifier. This approach 
proved to be very useful when the mixture 
classes of interest are previously defined. 
3. THE FUZZY MATHEMATICAL MODEL 
Mathematical fuzzy techniques for Remote 
Sensing image classification were proposed 
by Wang (1990a) and Wang (1990b), by. im- 
plementing the concept of partial and 
multiple membership, instead of the one- 
pixel-one-class conventional methods. The 
multiple membership concept can be understood 
as a result of multiple classes in a pixel 
i,e., the class membership grades measure 
the proportion of the component ("pure") 
classes in a pixel. 
Wang (1990a) comments on the information 
loss that occurs in image classification 
methods such as the Gaussian Maximum Like- 
lihood. Pixel probabilities of belonging 
to each of the information classes are 
estimated. The pixel is then assigned to 
the class associated with the largest pro- 
bability. A11 the remaining probabilities 
are discarded, regardless of their  magni- 
tude, i.e., the possibility that a pixel 
may partially belong to more than one class 
is excluded. 
The fuzzy classification method attempts to 
make use of the information contained in 
these discarded probabilities. This attempt 
is implemented via the concept of multiple 
partial membership. For each class a member- 
ship function is defined. These functions 
segment the multispectral space into fuzzy 
partitions rather than "hard" ones as in 
conventional classification, which me ans 
that a point (pixel) may partially belong 
to more than one class. 
Wang (1990a) defines the membership function 
associated to class 'i' by: 
fO d C (5) 
i 
nN NA HH 
Heh tH BY BA oH