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

TL;DR

This paper addresses state estimation for discrete-time linear systems with finite-dimensional states and infinite-dimensional measurements modeled as wide-sense stationary random fields. It derives a Kalman-like optimal linear filter with an explicit gain functional in the Fourier domain, under stationarity assumptions on the measurement noise, and proves existence, uniqueness, and stability using a Hilbert-space framework. The main contributions include a closed-form gain $\kappa_k(i)=P_k f(i)$, a d-dimensional measurement domain, a discrete-time Riccati-equation-based stability analysis, and an FFT-based algorithm with competitive computational complexity. The approach is validated through simulations of a linearized system with a pinhole camera sensor, demonstrating accurate estimation and convergence properties that align with theoretical predictions. Practical significance lies in enabling principled, scalable filtering for high- or infinite-dimensional sensor data (e.g., vision, lidar) without resorting to lumping the measurement space into finite dimensions.

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.

Figures (4)

• Figure 1: Three normalized stationary random fields, spatially discretized into a $100\times 100$ grid over the domain $[-0.5,0.5]^2$. Each field is zero-mean with a squared exponential covariance function $R(i,i')\propto e^{-\|i-i'\|_2^2(2\ell^2)^{-1}}$. The length scale $\ell$ determines how closely each element is influenced by its neighbors, with a larger length scale leading to a stronger correlation. Note that small length scales are often used to approximate ideal white noise.
• Figure 2: Diagram of the 2-D Pinhole Camera Model
• Figure 3: Two measurements of the decaying sinusoidal pattern, Fig. \ref{['fig:obsnonoise']} has no additive noise, while Fig. \ref{['fig:obsnoise']} is perturbed by the addition of a stochastic field with a squared exponential covariance function. Note that in \ref{['fig:obsnoise']} the measurements are spatially discretized for computational reasons.
• Figure 4: Fig. \ref{['fig:Position_Trajectory']} displays the true position and estimated position for a single realization of the filter over 50 time units. Fig. \ref{['fig:MSE_Trajectories']} displays the empirical position and velocity MSE, averaged over $20,000$ trials and compared with the theoretical position and velocity MSE.

Theorems & Definitions (18)

• Theorem II.1: Hilbert Projection Theorem
• Definition II.1: Multi-Dimensional Fourier Transform
• Definition II.2: Stabilizability
• Definition II.3: Detectability
• Theorem III.1: Fubini-Tonelli for w.s.s. random fields
• Theorem IV.1
• Lemma IV.2: $\mathcal{S}_k\perp M_{k-1}$
• proof
• proof
• Corollary IV.1