Computationally Efficient Adaptive Spline Filters for Nonlinear State Estimation

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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.