Nonlinear stochastic motion presents significant challenges for Bayesian particle tracking. To address this challenge, this paper proposes a framework to construct an invertible transformation that maps the nonlinear state-space model (SSM) into a higher-dimensional linear Gaussian SSM. This approach allows the application of standard linear-Gaussian inference techniques while maintaining a connection to the dynamics of the original system. The paper derives the necessary conditions for such transformations using Ito's lemma and variational calculus, and illustrates the method on a bistable cubic motion model, radial Brownian process model, and a logistic model with multiplicative noise. Simulations confirm that the transformed linear systems, when projected back, accurately reconstruct the nonlinear dynamics and, in distinct regimes of stiffness and singularity, yield tracking accuracy competitive with conventional filters, while avoiding their structural instabilities.

翻译：非线性随机运动为贝叶斯粒子跟踪带来了重大挑战。为应对这一挑战，本文提出一种框架，用于构建可逆变换，将非线性状态空间模型映射至高维线性高斯状态空间模型。该方法使得标准线性高斯推断技术得以应用，同时保持与原系统动力学的关联。本文利用伊藤引理和变分法推导了此类变换的必要条件，并通过双稳态立方运动模型、径向布朗过程模型以及具有乘性噪声的逻辑模型对该方法进行了验证。仿真结果表明，经变换的线性系统在投影回原空间后，能够准确重构非线性动力学；在刚度与奇异性不同的区域中，其跟踪精度可与传统滤波器相媲美，同时避免了传统方法的结构不稳定性。