Optimal Linear Filtering for Discrete-Time Systems with Infinite-Dimensional Measurements

TL;DR

This paper develops an optimal linear filtering approach for linear systems with infinite-dimensional measurements, extending Kalman filtering to high-dimensional sensing modalities like vision and lidar.

Contribution

It introduces a new explicit optimal filter for infinite-dimensional measurements in linear systems, with stability conditions based on finite-dimensional criteria.

Findings

Explicit optimal filter derived for infinite-dimensional measurements

Filter stability ensured by finite-dimensional conditions

Validated through simulation with a camera sensor model

Abstract

Systems equipped with modern sensing modalities such as vision and lidar gain access to increasingly high-dimensional measurements with which to enact estimation and control schemes. In this article, we examine the continuum limit of high-dimensional measurements and analyze state estimation in linear time-invariant systems with infinite-dimensional measurements but finite-dimensional states, both corrupted by additive noise. We propose a linear filter and derive the corresponding optimal gain functional in the sense of the minimum mean square error, analogous to the classic Kalman filter. By modeling the measurement noise as a wide-sense stationary random field, we are able to derive the optimal linear filter explicitly, in contrast to previous derivations of Kalman filters in distributed-parameter settings. Interestingly, we find that we need only impose conditions that are finite-dimensional in nature to ensure that the filter is asymptotically stable. The proposed filter is verified via simulation of a linearized system with a pinhole camera sensor.