Kutschireiter Anna, Rast Luke, Drugowitsch Jan
Department of Neurobiology, Harvard Medical School, Boston MA, United States.
IEEE Trans Signal Process. 2022;70:686-700. doi: 10.1109/tsp.2022.3143471. Epub 2022 Jan 14.
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.
角路径积分是一个系统根据潜在有噪声的角速度（或增量）观测值来估计自身航向方向的能力。用于角路径积分的非概率算法依赖于这些有噪声增量的求和，没有适当地考虑此类观测的可靠性，而这种可靠性对于根据传入信息合理权衡当前航向方向估计至关重要。在概率设定中，角路径积分可被表述为一个具有观测状态增量的连续时间非线性滤波问题（循环滤波）。航向方向的圆对称性使得这个推理任务本质上是非线性的，从而排除了使用诸如卡尔曼滤波器等流行推理算法的可能性，使得该问题在分析上难以处理。在此，我们推导出循环连续时间滤波的近似解，它在通过状态传播和观测更新保持固定表示的同时，整合状态增量观测值。具体而言，我们扩展已有的投影滤波方法以考虑观测到的状态增量，并将此框架应用于循环滤波问题。我们还为隐藏状态的连续时间角值直接观测提出了一个生成模型，并将其无缝集成到投影滤波器中。将所得方案应用于概率角路径积分模型，我们推导出一种循环滤波算法，我们将其称为循环卡尔曼滤波器。重要的是，该算法在分析上是可处理的、可解释的，并且优于基于高斯近似的替代滤波器。
