  In this paper, we revisit batch state estimation through the lens of Gaussian
process (GP) regression. We consider continuous-discrete estimation problems
wherein a trajectory is viewed as a one-dimensional GP, with time as the
independent variable. Our continuous-time prior can be defined by any
nonlinear, time-varying stochastic differential equation driven by white noise;
this allows the possibility of smoothing our trajectory estimates using a
variety of vehicle dynamics models (e.g., `constant-velocity'). We show that
this class of prior results in an inverse kernel matrix (i.e., covariance
matrix between all pairs of measurement times) that is exactly sparse
(block-tridiagonal) and that this can be exploited to carry out GP regression
(and interpolation) very efficiently. When the prior is based on a linear,
time-varying stochastic differential equation and the measurement model is also
linear, this GP approach is equivalent to classical, discrete-time smoothing
(at the measurement times); when a nonlinearity is present, we iterate over the
whole trajectory to maximize accuracy. We test the approach experimentally on a
simultaneous trajectory estimation and mapping problem using a mobile robot
dataset.
