Linear spectral unmixing for single pixels

Let us repeat here equation (1):

 w_i = a_i x + b_i y + c_i z                         (1)

where

w_i : vector of values of mixed pixel i in K spectral bands.

x : vector of values of the 1st pure class in the K spectral bands

y : vector of values of the 2nd pure class in the K spectral bands.

z : vector of values of the 3rd pure class in the K spectral bands.

a_i,b_i,c_i : proportions of the three pure classes in mixed pixel i.

w_i, x, y, z  : known

a_i,b_i,c_i : unknown.

It is possible to supplement equation (1) with the requirement that the mixing proportions have to add to 1:

a_i + b_i + c_i = 1                                       (2)

Then equation (1) represents a system of K equations in 2 unknowns. In general, we are going to have a system of K equations in N-1 unknowns, where N is the number of pure classes we have. 

If K=N-1 we have as many linear equations as unknowns, and we can solve the system.

If K>N-1 we have more equations than unknowns and we can solve the system in the least square error sense.

If K<N-1 we have fewer equations than unknowns and the system cannot be solved. That is why it is said that one of the limitations of this method is that we cannot recover more classes than the number of available spectral bands. 

In practice this method is never applied because it gives very unreliable results. To solve this problem, the simultaneous unmixing of several pixels is usually attempted.

Click here if you want to learn more about the simultaneous unmixing of many pixels