  A novel family of geometric signal detectors are proposed through medians of
the total Bregman divergence (TBD), which are shown advantageous over the
conventional methods and their mean counterparts. By interpreting the
observation data as Hermitian positive-definite matrices, their mean or median
play an essential role in signal detection. As is difficult to be solved
analytically, we propose numerical solutions through Riemannian gradient
descent algorithms or fixed-point algorithms.
  Beside detection performance, robustness of a detector to outliers is also of
vital importance, which can often be analyzed via the influence functions.
Introducing an orthogonal basis for Hermitian matrices, we are able to compute
the corresponding influence functions analytically and exactly by solving a
linear system, which is transformed from the governing matrix equation.
Numerical simulations show that the TBD medians are more robust than their mean
counterparts.
