A straightforward method to solve for the vector of proportions, 
would be
This is the constrained least-squares approach, and has the following closed form solution []
where :
In our experiments, we find that this estimator satisfies
the sum to one constraint, but does not result in accurate
estimates of 
. Therefore, we have developed an
iterative algorithm that converges towards the least
squares solution. This section discusses details of this
algorithm.
The estimation error is given by
We attempt to minimize the squared error subject to the constraint that the
proportions must sum to one. A Lagrangian formulation follows directly and the
unconstrained minimization problem may be cast as
Minimize
We use the Newton-Raphson iterative approach to solve this
minimization problem . Defining the variable z
to be
, the iterative
step in this approach may be expressed as
where 
is the step size, 
is the Hessian
matrix and 
is the gradient vector.
This algorithm yielded good estimates of f in experiments with synthetic as well as actual AVHRR imagery of a portion of the African continent. These results are discussed in section IV.