  We develop a general framework for state estimation in systems modeled with
noise-polluted continuous time dynamics and discrete time noisy measurements.
Our approach is based on maximum likelihood estimation and employs the calculus
of variations to derive optimality conditions for continuous time functions. We
make no prior assumptions on the form of the mapping from measurements to
state-estimate or on the distributions of the noise terms, making the framework
more general than Kalman filtering/smoothing where this mapping is assumed to
be linear and the noises Gaussian. The optimal solution that arises is
interpreted as a continuous time spline, the structure and temporal dependency
of which is determined by the system dynamics and the distributions of the
process and measurement noise. Similar to Kalman smoothing, the optimal spline
yields increased data accuracy at instants when measurements are taken, in
addition to providing continuous time estimates outside the measurement
instances. We demonstrate the utility and generality of our approach via
illustrative examples that render both linear and nonlinear data filters
depending on the particular system.
