In this paper, we aim to propose a consistent non-Gaussian Bayesian filter of which the system state is a continuous function. The distributions of the true system states, and those of the system and observation noises, are only assumed Lebesgue integrable with no prior constraints on what function classes they fall within. This type of filter has significant merits in both theory and practice, which is able to ameliorate the curse of dimensionality for the particle filter, a popular non-Gaussian Bayesian filter of which the system state is parameterized by discrete particles and the corresponding weights. We first propose a new type of statistics, called the generalized logarithmic moments. Together with the power moments, they are used to form a density surrogate, parameterized as an analytic function, to approximate the true system state. The map from the parameters of the proposed density surrogate to both the power moments and the generalized logarithmic moments is proved to be a diffeomorphism, establishing the fact that there exists a unique density surrogate which satisfies both moment conditions. This diffeomorphism also allows us to use gradient methods to treat the convex optimization problem in determining the parameters. Last but not least, simulation results reveal the advantage of using both sets of moments for estimating mixtures of complicated types of functions. A robot localization simulation is also given, as an engineering application to validate the proposed filtering scheme.

翻译：本文旨在提出一种系统状态为连续函数的一致非高斯贝叶斯滤波器。真实系统状态、系统噪声与观测噪声的分布仅假设为勒贝格可积，且不预设其所属的函数类别。此类滤波器在理论与实践中均具有显著优势，能够改善粒子滤波（一种通过离散粒子及其对应权重参数化系统状态的流行非高斯贝叶斯滤波器）的维数灾难问题。我们首先提出一种新型统计量——广义对数矩。该统计量与幂矩共同用于构建参数化为解析函数的密度替代模型，以逼近真实系统状态。本文证明从所提密度替代模型参数到幂矩与广义对数矩的映射为微分同胚，从而确立了存在唯一满足两类矩条件的密度替代模型。该微分同胚特性还允许我们使用梯度法处理参数确定过程中的凸优化问题。最后，仿真结果揭示了联合使用两类矩对复杂函数类型混合进行估计的优势。本文还给出机器人定位仿真作为工程应用，以验证所提滤波方案的可行性。