  This paper is concerned with the problem of continuous-time nonlinear
filtering for stochastic processes on a compact and connected matrix Lie group
without boundary, e.g. SO(n) and SE(n), in the presence of real-valued
observations. This problem is important to numerous applications in attitude
estimation, visual tracking and robotic localization. The main contribution of
this paper is to derive the feedback particle filter (FPF) algorithm for this
problem. In its general form, the FPF provides a coordinate-free description of
the filter that furthermore satisfies the geometric constraints of the
manifold. The particle dynamics are encapsulated in a Stratonovich stochastic
differential equation that preserves the feedback structure of the original
Euclidean FPF. Specific examples for SO(2) and SO(3) are provided to help
illustrate the filter using the phase and the quaternion coordinates,
respectively.
