Optimizing the Endmembers Using Volume Invariant Constrained Model

The linear mixture model (LMM) plays a crucial role in the spectral unmixing of hyperspectral data. Under the assumption of LMM, the solution with the minimum reconstruction error is considered to be the ideal endmember. However, for practical hyperspectral data sets, endmembers that enclose all the pixels are physically meaningless due to the effect of noise. Therefore, in many cases, it is not sufficient to consider only the reconstruction error, some constraints (for instance, volume constraint) need to be added to the endmembers. The two terms can be considered as serving two forces: minimizing the reconstruction error forces the endmembers to move outward and thus enlarges the volume of the simplex while the endmember constraint acts in the opposite direction by driving the endmembers to move inward so as to constrain the volume to be smaller. Many existing methods obtain their solution just by balancing the two contradictory forces. The solution acquired in this way can not only minimize the reconstruction error but also be physically meaningful. Interestingly, we find, in this paper, that the two forces are not completely contradictory with each other, and the reconstruction error can be further reduced without changing the volume of the simplex. And more interestingly, our method can further optimize the solution provided by all the endmember extraction methods (both endmember selection methods and endmember generation methods). After optimization, the final endmembers outperform the initial solution in terms of reconstruction error as well as accuracy. The experiments on simulated and real hyperspectral data verify the validation of our method.