On 'A Kalman Filter-Based Algorithm for IMU-Camera Calibration: Observability Analysis and Performance Evaluation' (1311.4769v2)

Abstract: The above-mentioned work [1] in IEEE-TR'08 presented an extended Kalman filter for calibrating the misalignment between a camera and an IMU. As one of the main contributions, the locally weakly observable analysis was carried out using Lie derivatives. The seminal paper [1] is undoubtedly the cornerstone of current observability work in SLAM and a number of real SLAM systems have been developed on the observability result of this paper, such as [2, 3]. However, the main observability result of this paper [1] is founded on an incorrect proof and actually cannot be acquired using the local observability technique therein, a fact that is apparently not noticed by the SLAM community over a number of years.

• The paper critically analyzes and refutes key observability proofs (Lemma 1, Corollary 1, Lemma 3) in a widely cited 2008 Kalman filter-based IMU-camera calibration paper by Mirzaei and Roumeliotis.
• It argues that the original proofs were based on faulty assumptions regarding the system's observability conditions, particularly concerning rotations required for calibration.
• The analysis provides revised conditions (Corollary 1c, Lemma 3c) and questions the validity of the observability matrix full rank under certain scenarios, impacting the theoretical understanding and practical implementation of SLAM systems.

An Analysis of "A Kalman Filter-Based Algorithm for IMU-Camera Calibration: Observability Analysis and Performance Evaluation"

The paper critically examines the seminal work in the field of Sensor Fusion, specifically addressing the Kalman filter-based approach for IMU-camera calibration, which has deeply influenced the observability analysis in simultaneous localization and mapping (SLAM) systems. This exploration explores the observability contentions in the original paper by Mirzaei and Roumeliotis, published in 2008, which laid the groundwork for various SLAM implementations but, as this analysis proposes, was based on faulty assumptions and proofs.

The central thesis of the examined paper involves questioning the calculable observability of a sensor fusion system composed of an inertial measurement unit (IMU) and a camera, through an extended Kalman filter-based approach. The original work claimed to have resolved the observability conditions necessary for such sensor setups, primarily under conditions of system rotation about multiple axes. However, the paper under review argues that the original claims (specifically Lemma 1, Corollary 1, and Lemma 3) were grounded on incorrect proofs. The observability conclusions drawn, including the necessity of rotation about at least two different axes for calibration, are more appropriately associated with misinterpreted assumptions.

In this analysis, it is asserted that local weak observability cannot be reliably proven through the methods outlined in the original paper. The investigation uses Lie derivatives to detail inconsistencies that arise from the original observability proofs. A critical finding is that the understanding of the difference between the number of nonzero components of a rotational velocity vector, and the degrees of freedom available during rotation, led to inaccuracies. The paper clarifies this by providing revised conditions (Corollary 1c and Lemma 3c) that rightfully adjust the observability prerequisites to those rotations characterized by at least two nonzero angular velocity components, challenging the system's actual observability under previous claims.

Furthermore, the broader implications of these findings are significant. They question the legitimacy of the observability matrix full rank condition under certain assumptions that had gone largely unchallenged in the sensor fusion community until now. By highlighting these inconsistencies and logically revising the conditions under which IMU-camera systems can be deemed observable, this analysis could influence the theoretical and practical implementations of future SLAM systems, prompting a reassessment of the assumptions in high-dimension sensor calibration.

The contradictions noted, such as that found with the parallel axis scenario leading to unobservable parameters despite full rank conditions being met, prompt further inquiry, which underscores the need for continued conversation and investigation in this domain. It invites researchers to revisit foundational theories in sensor fusion and observability, encouraging derivations and methods that are meticulous and robust to ensure valid and applicable outcomes.

In conclusion, this analysis provides a crucial audit of a widely accepted Kalman filter-based sensor calibration approach, urging the scientific community to examine long-held assumptions within the sphere of nonlinear observability. This work stands to recalibrate the understanding of system calibrations in both theoretical development and practical system design, potentially sparking further innovations in the capabilities and reliability of SLAM systems.