  In this paper, we propose a class of median-based matrix information geometry
(MIG) detectors with a manifold filter and apply them to signal detection in
nonhomogeneous environments. As customary, the sample data is assumed to be
modeled as Hermitian positive-definite (HPD) matrices, and the geometric median
of a set of HPD matrices is interpreted as an estimate of the clutter
covariance matrix (CCM). Then, the problem of signal detection can be
reformulated as discriminating two points on the manifold of HPD matrices, one
of which is the HPD matrix in the cell under test while the other represents
the CCM. By manifold filter, we map a set of HPD matrices to another set of HPD
matrices by weighting them, that consequently improves the discriminative power
by reducing the intra-class distances while increasing the inter-class
distances. Three MIG median detectors are designed by resorting to three
geometric measures on the matrix manifold, and the corresponding geometric
medians are shown to be robust to outliers. Numerical simulations show the
advantage of the proposed MIG median detectors in comparison with their
state-of-the-art counterparts as well as the conventional detectors in
nonhomogeneous environments.
