几何本征非线性递归滤波器II：基础
Geometrically Intrinsic Nonlinear Recursive Filters II: Foundations
摘要：
本文从随机微分几何的角度出发，给出了构造几何内禀非线性递归滤波器的技术基础。流形N上的扩散X在时间间隔T内运行，具有随机初始条件。存在由X（T）的非线性函数组成的单个观测，被噪声破坏，并且具有另一个流形M中的值。X的噪声协方差和观测协方差本身分别在M和N上诱导几何。使用这些几何形状，我们计算近似但无坐标公式的“最佳估计”的X（T），给定的观察，其条件方差。计算的基础上使用的雅可比字段和“内在的位置参数“，一个概念来自调和映射的热流。当存在任何非线性时，所得公式与连续-离散扩展卡尔曼滤波器的公式不同。一个辅助结果是一个公式，用于近似计算流形上的随机变量S的“指数重心”，即点z，使得S在z处的指数映射下的逆像在z处的切空间中具有均值零。
英文摘要：
This paper contains the technical foundations from stochastic differential geometry for the construction of geometrically intrinsic nonlinear recursive filters. A diffusion X on a manifold N is run for a time interval T, with a random initial condition. There is a single observation consisting of a nonlinear function of X(T), corrupted by noise, and with values in another manifold M. The noise covariance of X and the observation covariance themselves induce geometries on M and N, respectively. Using these geometries we compute approximate but coordinate-free formulas for the "best estimate" of X(T), given the observation, and its conditional variance. Calculations are based on use of Jacobi fields and of "intrinsic location parameters", a notion derived from the heat flow of harmonic mappings. When any nonlinearity is present, the resulting formulas are not the same as those for the continuous-discrete Extended Kalman Filter. A subsidiary result is a formula for computing approximately the "exponential barycenter" of a random variable S on a manifold, i.e. a point z such that the inverse image of S under the exponential map at z has mean zero in the tangent space at z.
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