[Paper Review] Kalman Filtering with Gaussian Processes Measurement Noise

Vince Kurtz, Hai Xiang Lin|arXiv (Cornell University)|Sep 23, 2019

Target Tracking and Data Fusion in Sensor Networks14 references4 citations

TL;DR

This paper proposes using Gaussian Processes (GPs) as a non-parametric model for time-correlated measurement noise in Kalman filtering, enabling optimal estimation under realistic sensor noise. By deriving a recursive filter that accounts for GP-induced correlations, the method improves estimation accuracy and uncertainty calibration over standard Kalman filters, with simulations showing near-optimal performance using only a small window of past observations.

ABSTRACT

Real-world measurement noise in applications like robotics is often correlated in time, but we typically assume i.i.d. Gaussian noise for filtering. We propose general Gaussian Processes as a non-parametric model for correlated measurement noise that is flexible enough to accurately reflect correlation in time, yet simple enough to enable efficient computation. We show that this model accurately reflects the measurement noise resulting from vision-based Simultaneous Localization and Mapping (SLAM), and argue that it provides a flexible means of modeling measurement noise for a wide variety of sensor systems and perception algorithms. We then extend existing results for Kalman filtering with autoregressive processes to more general Gaussian Processes, and demonstrate the improved performance of our approach.

Motivation & Objective

• Address the limitation of standard Kalman filters that assume i.i.d. Gaussian measurement noise, which is often invalid in real-world robotics applications.
• Model time-correlated measurement noise—common in vision-based SLAM and perception systems—using a flexible, non-parametric Gaussian Process framework.
• Develop an efficient recursive filtering algorithm that avoids computationally expensive pre-whitening or state augmentation by leveraging matrix inversion lemmas.
• Demonstrate that accounting for temporal correlation in measurement noise leads to more accurate state estimates and better-calibrated uncertainty (error covariance).

Proposed method

• Model measurement noise as a zero-mean Gaussian Process with a user-specified covariance kernel (e.g., Matérn), capturing temporal correlations beyond i.i.d. or AR(p) assumptions.
• Derive a recursive Kalman filter variant that computes the optimal linear estimate under GP noise by treating the noise covariance as a structured, time-dependent matrix.
• Use a sliding window of size $ N $ to truncate the history of past measurements, reducing computational cost while preserving accuracy.
• Precompute the inverse of the GP noise covariance matrix over the window to enable efficient inversion using the Sherman-Morrison-Woodbury lemma, reducing complexity to $ O(N) $ per step.
• Formulate the state estimate update using a modified Kalman gain that accounts for the full covariance structure of the GP noise, ensuring minimum mean squared error estimation.
• Integrate the GP noise model directly into the measurement update step, avoiding the need for pre-whitening or state augmentation.

Experimental results

Research questions

• RQ1 Can Gaussian Processes effectively model time-correlated measurement noise observed in real-world robotic perception systems such as RGB-D SLAM?
• RQ2 How can optimal Kalman filtering be extended to handle general Gaussian Process noise without resorting to state augmentation or pre-whitening?
• RQ3 What is the trade-off between estimation accuracy and computational cost when using a finite history window in GP-based Kalman filtering?
• RQ4 Does accounting for temporal correlation in measurement noise lead to more accurate and better-calibrated uncertainty estimates compared to standard Kalman filtering?
• RQ5 Can the GP lengthscale hyperparameter be used as a heuristic for selecting the optimal window size $ N $ in the truncated filtering approach?

Key findings

• Measurement noise from vision-based SLAM systems is highly correlated in time and well-modeled by a Gaussian Process with a Matérn kernel.
• The proposed GP-based Kalman filter significantly outperforms standard Kalman filtering by reducing overconfidence in estimates and producing more stable convergence.
• Using only $ N = 5 $ past observations in the sliding window yields estimation performance nearly indistinguishable from the full-history optimal filter.
• The computational complexity of the filter scales linearly with the window size $ N $, enabling real-time operation even with moderately large $ N $, thanks to the Sherman-Morrison-Woodbury lemma.
• The error covariance $ P_t $ is better calibrated when GP noise is modeled, avoiding the overly optimistic uncertainty estimates seen in standard Kalman filters.
• The GP lengthscale hyperparameter provides a principled heuristic for selecting the window size $ N $, as observations with correlation below a threshold $ k_{\text{min}} $ can be safely neglected.