  The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is
designed to estimate an arbitrary continuous-time Markov diffusion process X
subject to nonlinear discrete-time observations. The GI Filter is fundamentally
different from the much-used Extended Kalman Filter (EKF), and its second-order
variants, even in the simplest nonlinear case, in that: (i) It uses a quadratic
function of a vector observation to update the state, instead of the linear
function used by the EKF. (ii) It is based on deeper geometric principles,
which make the GI Filter coordinate-invariant. This implies, for example, that
if a linear system were subjected to a nonlinear transformation f of the
state-space and analyzed using the GI Filter, the resulting state estimates and
conditional variances would be the push-forward under f of the Kalman Filter
estimates for the untransformed system - a property which is not shared by the
EKF or its second-order variants.
  The noise covariance of X and the observation covariance themselves induce
geometries on state space and observation space, respectively, and associated
canonical connections. A sequel to this paper develops stochastic differential
geometry results - based on "intrinsic location parameters", a notion derived
from the heat flow of harmonic mappings - from which we derive the
coordinate-free filter update formula. The present article presents the
algorithm with reference to a specific example - the problem of tracking and
intercepting a target, using sensors based on a moving missile. Computational
experiments show that, when the observation function is highly nonlinear, there
exist choices of the noise parameters at which the GI Filter significantly
outperforms the EKF.
