95.

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Page 183

mixture model will not necessarily provide a valid solution. Bosdogianni et al. (1997) acknowledge such a problem and propose a method that is less constrained by the number of bands.

The approach developed by Bosdogianni et al. (1997) is concerned with the estimation of proportions present in sets of mixed pixels, rather than in single pixels considered separately. The system of linear equations is generated using the mean and covariance of a set of pixels. If the assumption of statistical independence between the end members is made, Equation (4.43) can be generalised into:

(4.47)

where _i, ā_i, and denote the mean values of corresponding variable in band i. The relationship between these variables can also be expressed in terms of covariance matrix as:

(4.48)

In the case of a three-band image, Equations (4.47) and (4.48) generate six equations (i.e. three equations defining the means and three equations defining the covariances) while in the case of a six-band image it can form twenty-one equations (i.e. six mean equations and fifteen covariance equations). The total number of equation can be calculated as:

(4.49)

where n is the number of spectral bands. It is clear that, as the number of bands increases, the number of available equations increases disproportionately.

Once the set of equations has been constructed, the least squares method is used to perform end member proportion estimation. The squared errors for Equations (4.47) and (4.48) are given by:

(4.50)

Both squared errors are then weighted according to the inverse of the standard errors. The weighting parameters are set so that the equations of smaller error should contribute more to the sum. The standard error for

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