  Angular path integration is the ability of a system to estimate its own
heading direction from potentially noisy angular velocity (or increment)
observations. Non-probabilistic algorithms for angular path integration, which
rely on a summation of these noisy increments, do not appropriately take into
account the reliability of such observations, which is essential for
appropriately weighing one's current heading direction estimate against
incoming information. In a probabilistic setting, angular path integration can
be formulated as a continuous-time nonlinear filtering problem (circular
filtering) with observed state increments. The circular symmetry of heading
direction makes this inference task inherently nonlinear, thereby precluding
the use of popular inference algorithms such as Kalman filters, rendering the
problem analytically inaccessible. Here, we derive an approximate solution to
circular continuous-time filtering, which integrates state increment
observations while maintaining a fixed representation through both state
propagation and observational updates. Specifically, we extend the established
projection-filtering method to account for observed state increments and apply
this framework to the circular filtering problem. We further propose a
generative model for continuous-time angular-valued direct observations of the
hidden state, which we integrate seamlessly into the projection filter.
Applying the resulting scheme to a model of probabilistic angular path
integration, we derive an algorithm for circular filtering, which we term the
circular Kalman filter. Importantly, this algorithm is analytically accessible,
interpretable, and outperforms an alternative filter based on a Gaussian
approximation.
