  Hyperspectral images provide much more information than conventional imaging
techniques, allowing a precise identification of the materials in the observed
scene, but because of the limited spatial resolution, the observations are
usually mixtures of the contributions of several materials. The spectral
unmixing problem aims at recovering the spectra of the pure materials of the
scene (endmembers), along with their proportions (abundances) in each pixel. In
order to deal with the intra-class variability of the materials and the induced
spectral variability of the endmembers, several spectra per material,
constituting endmember bundles, can be considered. However, the usual abundance
estimation techniques do not take advantage of the particular structure of
these bundles, organized into groups of spectra. In this paper, we propose to
use group sparsity by introducing mixed norms in the abundance estimation
optimization problem. In particular, we propose a new penalty which
simultaneously enforces group and within group sparsity, to the cost of being
nonconvex. All the proposed penalties are compatible with the abundance
sum-to-one constraint, which is not the case with traditional sparse
regression. We show on simulated and real datasets that well chosen penalties
can significantly improve the unmixing performance compared to the naive bundle
approach.
