Computationally Efficient Adaptive Spline Filters for Nonlinear State Estimation

由于大多数实际系统中存在固有的非线性，非线性/非高斯滤波问题在文献中引起了极大的兴趣。状态估计问题中的非线性可能是由于其存在于状态-量测方程或状态本身的演化中而引起的。多个对象的存在将数据关联添加到混合中，从而使问题进一步复杂化。最优非线性状态估计器由多目标状态的条件(后验)pdf的计算组成，给定到当前时间的所有可用测量值。最优多目标非线性滤波通常是一个难以处理的问题，这不仅是因为计算的复杂性，而且还因为多目标pdf的多峰特性。 在这种情况下，人们需要一种能够通过识别时空非线性变化(在一个目标上或跨多个目标)来自动自适应的算法。我们的多目标状态传播将基于多维样条表示。在计算机科学、图形学、航空航天、汽车工业、统计学和数学中，使用有限节点集可以有效地使用样条线表示复杂的(和任意的)曲线和曲面。我们的创新方法是使用样条线来表示任意多目标pdf，然后基于标准的预测和更新步骤推导出随时间传播样条线的方程。样条线具有许多理想的性质：它们是连续的，可以处理多个模型，本质上能够测量非线性，不会退化或需要重采样，可以包含类似于路线图的约束，是传感器不可知的，并且可以通过在空间和时间上变化节点来自适应。本文对具有多目标、虚警、漏检和约束的更实际的状态估计问题进行了重要的理论扩展。

The problem of nonlinear/non-Gaussian filtering has generated significant interest in the literature due to the inherent nonlinearity in most practical systems. The nonlinearity in state estimation problems may arise due to its presence in the state-to-measurement equation or in the evolution of the state itself. The presence of multiple objects further complicates the problem by adding data association to the mix. The optimal nonlinear state estimator consists of the computation of the conditional (posterior) pdf of the multitarget state given all the measurements available up to the current time. Optimal multitarget nonlinear filtering is in general a non-tractable problem, not just because of computational complexity but also due to the multimodal nature of multitarget pdf. Under these circumstances, one needs an algorithm that is capable of automatically adapting itself by recognizing the spatio-temporal nonlinearity variations (over one target or across multiple ones). Our multitarget state propagation will be based on multidimensional spline representation. Splines have been used effectively to represent complex (and arbitrary) curves and surfaces in computer science, graphics, aerospace, automobile industry, statistics and mathematics using a finite set of knots. Our innovative approach is to use splines to represent any arbitrary multitarget pdf and then derive the equations for propagating the splines over time based on the standard prediction and update steps. Splines posses a number of desirable properties: they are continuous, can handle multiple models, inherently capable of measuring nonlinearity, do not suffer from degeneracy or need resampling, can incorporate road map-like constraints, are sensor-agnostic and can be adaptive by varying knots spatially and temporally. Significant theoretical extensions to the more realistic state estimation problems with multiple targets, false alarms, missed detections and constraints are proposed in this work.