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25 August 2004 Physics-based computational complexity of nonlinear filters
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Abstract
Our theory is based on the mapping between two Fokker-Planck equations and two Schroedinger equations (see [1] & [2]), which is well known in physics, but which has not been exploited in filtering theory. This theory expands Brockett's Lie algebra homomorphism conjecture for characterizing finite dimensional filters. In particular, the Schroedinger equation generates a group, whereas the Zakai equation (as well as the Fokker-Planck equation) does not, owing to the lack of a smooth inverse. Simple non-pathological low-dimensional linear-Gaussian timeinvariant counterexamples show that Brockett's conjecture does not reliably predict when a nonlinear filtering problem will have an exact finite dimensional solution. That is, there are manifestly finite dimensional filters for estimation problems with infinite dimensional Lie algebras. There are three reasons that the Lie algebraic approach as originally formulated by Brockett is incomplete: (1) the Zakai equation does not generate a group; (2) Lie algebras are coordinate free, whereas separation of variables in PDEs is not coordinate free, and (3) Brockett's theory aims to characterize finite dimensional filters for any initial condition of the Zakai equation, whereas SOV for PDEs generally depends on the initial condition. We will attempt to make this paper accessible to normal engineers who do not have Lie algebras for breakfast.
© (2004) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Frederick E. Daum and Jim Huang "Physics-based computational complexity of nonlinear filters", Proc. SPIE 5428, Signal and Data Processing of Small Targets 2004, (25 August 2004); https://doi.org/10.1117/12.532611
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