Fully Constrained Linear Spectral Unmixing: Analytic Solution Using Fuzzy Sets

The linear mixture model is a convenient way to describe image pixels as a linear combination of pure spectra - termed end-members. The fractional contribution from each end-member is calculated through inversion of the linear model. Despite the simplicity of the model, a nonnegativity constraint that is imposed on the fractions leads to an unmixing problem for which it is hard to find a closed analytical solution. Current solutions to this problem involve iterative algorithms, which are computationally intensive and not appropriate for unmixing large number of pixels. This paper presents an algorithm to build fuzzy membership functions that are equivalent to the least square solution of the fully constrained linear spectral unmixing problem. The efficiency and effectiveness of the proposed solution is demonstrated using both simulated and real data.