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Asymptotically Optimal Finite-Dimensional Approximations for Linear Filtering with Infinite-Dimensional Measurements | IEEE Conference Publication | IEEE Xplore

Asymptotically Optimal Finite-Dimensional Approximations for Linear Filtering with Infinite-Dimensional Measurements


Abstract:

This work proposes a novel approach to approximate optimal linear filters for discrete-time linear Gaussian systems with infinite-dimensional measurements and finite- dim...Show More

Abstract:

This work proposes a novel approach to approximate optimal linear filters for discrete-time linear Gaussian systems with infinite-dimensional measurements and finite- dimensional states. Assuming scalar-valued states for simplicity, we formulate the problem in terms of optimally selecting N points at which to sample the infinite-dimensional measurement, in order to minimize the mean-squared filtering error. We show that for large N, this problem can be expressed using the notion of an asymptotic point density function from the field of high-resolution quantization theory. To the best of the authors' knowledge, this method has not been considered in infinite- dimensional filtering previously. This leads to a characterization in terms of an Urysohn integral equation, which can be solved numerically to yield an asymptotically optimal N-point filter. The mean-squared approximation error is proportional to N-4, which is faster than the typical N-2 decay of high-resolution quantization and suggests that this approximation method will be useful even for moderate or small N. These properties are verified by simulations based on a linearized pinhole camera measurement model.
Date of Conference: 13-15 December 2023
Date Added to IEEE Xplore: 19 January 2024
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Conference Location: Singapore, Singapore

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