Dealing with sets of mixed pixels

Figure 10 shows 4 sets of pixels: Three of them represent three different pure classes and the one in the middle represents a mixed region.

Figure 10

One can view a mixed pixel as an instantiation of the outcome of a random process. All pixels that come from the same mixed region then constitute an ensemble of outcomes of the same random process. In a similar way, the pixels that represent a single pure class are instantiations of the outcomes of the random process that creates the intra-class variability. In figure 10 then, the sets that represent the pure classes represent the distributions of three independent random variables, and the set that represents the mixed region represents the distribution of a linear combination of these variables. Bosdogianni, Petrou and Kittler used this idea to work out the proportions of the mixed region by trying to match the shapes of the distributions of the pure and the mixed pixels (Mixture models with higher order moments, P Bosdogianni, M Petrou and J Kittler, IEEE Trans. Geoscience and Remote Sensing,vol 35, pp 341-353, 1997). The shape of a distribution is expressed by the moments of the distribution: Its 1st moment (i.e. the mean), its 2nd moments (i.e. the elements of its covariance matrix), its 3rd moments etc. From the theory of random variables, we know the relationships between the moments of the independent variables and the corresponding moments of any linear combination of them. This way an augmented set of equations is created and it can be solved in terms of the unknown mixing proportions.

Practical details  

Advantages of the approach  

Disadvantages of the approach 

Relationship with the simultaneous unmixing of several mixed pixels