Aa. Nielsen, Spectral mixture analysis: Linear and semi-parametric full and iterated partial unmixing in multi- and hyperspectral image data, J MATH IM V, 15(1-2), 2001, pp. 17-37
As a supplement or an alternative to classification of hyperspectral image
data linear and semi-parametric mixture models are considered in order to o
btain estimates of abundance of each class or end-member in pixels with mix
ed membership. Full unmixing based on both ordinary least squares (OLS) and
non-negative least squares (NNLS), and the partial unmixing methods orthog
onal subspace projection (OSP), constrained energy minimization (CEM) and a
n eigenvalue formulation alternative are dealt with. The solution to the ei
genvalue formulation alternative proves to be identical to the CEM solution
. The matrix inversion involved in CEM can be avoided by working on (a subs
et of) orthogonally transformed data such as signal maximum autocorrelation
factors, MAFs, or signal minimum noise fractions, MNFs. This will also cau
se the partial unmixing result to be independent of the noise isolated in t
he MAF/MNFs not included in the analysis. CEM and the eigenvalue formulatio
n alternative enable us to perform partial unmixing when we know one desire
d 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 constrai
ned interference minimized filter (TCIMF) is described. Spectral angle mapp
ing (SAM) is briefly described. Finally, semi-parametric unmixing (SPU) bas
ed on a combined linear and additive model with a non-linear, smooth functi
on to represent end-member spectra unaccounted for is introduced. An exampl
e with two generated bands shows that both full unmixing, the CEM, the iter
ated CEM and TCIMF methods perform well. A case study with a 30 bands subse
t 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 tha
n CEM. A study with AVIRIS spectra generated from real spectra shows (1) th
at ordinary least squares in this case with one unknown spectrum performs b
etter than non-negative least squares, and (2) that although not fully sati
sfactory the semi-parametric model gives better estimates of end-member abu
ndances than the linear model.