A regularization based method for spectral unmixing of imaging spectrometer data

In spectral unmixing, the imaging spectrometer data is unmixed to yield underlying proportions (abundance maps) of the constituent materials after extracting (estimating) their spectral signatures (endmembers). Under linear mixing model, we consider an unmixing problem wherein given the extracted endmembers, the task is to estimate the abundances. This is a severely ill-posed problem, as the hyperspectral signatures are strongly correlated resulting in the ill-conditioned signature matrix which makes the estimation highly sensitive to the noise. Further, the acquired data often do not fully satisfy the simplex requirement imposed by the linearity, resulting in inaccurate extraction of endmembers. This in turn could lead to unstable solution in the subsequent estimation of abundances. In this paper, we adopt the regularization-based alternative to achieve stable solution by improving the conditioning of the problem. For this purpose, we propose to use Tikhonov regularization within the total least squares (TLS) estimation framework. The problem is formulated with a sum of the TLS as its data-term which takes care of the possible modeling errors in both abundances and endmembers, and the Tikhonov prior which imposes the smoothness constraints. The resultant energy function being convex is minimized by gradient based optimization technique wherein the solution space is restricted to yield nonnegative abundances. We show the analysis of the regularized solution and compare it with a TLS-based direct inversion. The experiments are conducted with different noise levels on the simulated data. The results are compared with the state-of-art approaches using different quantitative measures and observing the consistency within spatial patterns of the estimated abundances. Finally the proposed approach is applied on the AVIRIS data to obtain abundance maps of the constituent materials.