Spectral Mixture Analysis: Linear and Semi-parametric Full and Iterated Partial Unmixing in Multi- and Hyperspectral Image Data

As a supplement or an alternative to classification of hyperspectral image data linear and semi-parametric mixture models are considered in order to obtain estimates of abundance of each class or end-member in pixels with mixed membership. Full unmixing based on both ordinary least squares (OLS) and non-negative least squares (NNLS), and the partial unmixing methods orthogonal subspace projection (OSP), constrained energy minimization (CEM) and an eigenvalue formulation alternative are dealt with.

The solution to the eigenvalue formulation alternative proves to be identical to the CEM solution. The matrix inversion involved in CEM can be avoided by working on (a subset of) orthogonally transformed data such as signal maximum autocorrelation factors, MAFs, or signal minimum noise fractions, MNFs.

This will also cause the partial unmixing result to be independent of the noise isolated in the MAF/MNFs not included in the analysis. CEM and the eigenvalue formulation alternative enable us to perform partial unmixing when we know one desired end-member spectrum only and not the full set of end-member spectra.

This is an advantage over full unmixing and OSP. The eigenvalue formulation of CEM inspires us to suggest an iterated CEM scheme. Also the target constrained interference minimized filter (TCIMF) is described. Spectral angle mapping (SAM) is briefly described. Finally, semi-parametric unmixing (SPU) based on a combined linear and additive model with a non-linear, smooth function to represent end-member spectra unaccounted for is introduced.

An example with two generated bands shows that both full unmixing, the CEM, the iterated CEM and TCIMF methods perform well. A case study with a 30 bands subset of AVIRIS data shows the utility of full unmixing, SAM, CEM and iterated CEM to more realistic data. Iterated CEM seems to suppress noise better than CEM.

A study with AVIRIS spectra generated from real spectra shows (1) that ordinary least squares in this case with one unknown spectrum performs better than non-negative least squares, and (2) that although not fully satisfactory the semi-parametric model gives better estimates of end-member abundances than the linear model.