Unmixing methods based on nonnegativity and weakly mixed pixels for
  astronomical hyperspectral datasets
  [Abridged] An increasing number of astronomical instruments (on Earth and
space-based) provide hyperspectral images, that is three-dimensional data cubes
with two spatial dimensions and one spectral dimension. The intrinsic
limitation in spatial resolution of these instruments implies that the spectra
associated with pixels of such images are most often mixtures of the spectra of
the "pure" components that exist in the considered region. In order to estimate
the spectra and spatial abundances of these pure components, we here propose an
original blind signal separation (BSS), that is to say an unsupervised unmixing
method. Our approach is based on extensions and combinations of linear BSS
methods that belong to two major classes of methods, namely nonnegative matrix
factorization (NMF) and Sparse Component Analysis (SCA). The former performs
the decomposition of hyperspectral images, as a set of pure spectra and
abundance maps, by using nonnegativity constraints, but the estimated solution
is not unique: It highly depends on the initialization of the algorithm. The
considered SCA methods are based on the assumption of the existence of points
or tiny spatial zones where only one source is active (i.e., one pure component
is present). In real conditions, the assumption of perfect single-source points
or zones is not always realistic. In such conditions, SCA yields approximate
versions of the unknown sources and mixing coefficients. We propose to use part
of these preliminary estimates from the SCA to initialize several runs of the
NMF to constrain the convergence of the NMF algorithm. Detailed tests with
synthetic data show that the decomposition achieved with such hybrid methods is
nearly unique and provides good performance, illustrating the potential of
applications to real data.
