An Event-Based Approach for the Conservative Compression of Covariance Matrices IEEE Transactions on Automatic Control, vol. 70, pp. 3213–3225, May, 2025.
This work introduces a flexible and versatile method for the data-efficient yet conservative transmission of covariance matrices, where a matrix element is only transmitted if a triggering condition is satisfied for the element. Here, triggering conditions can be parametrized on a per-element basis, applied simultaneously to yield combined triggering conditions or applied only to certain subsets of elements. This allows, e.g., to specify transmission accuracies for individual elements or to constrain the bandwidth available for the transmission of subsets of elements. The method is simple to implement, computationally efficient, and thus suitable for resource-constrained systems. Additionally, a methodology for learning triggering condition parameters from an application-specific dataset is presented. The performance of the proposed approach is quantitatively assessed in terms of data reduction and conservativeness using estimate data derived from real-world vehicle trajectories from the InD-dataset, demonstrating substantial data reduction with minimal over-conservativeness. The feasibility of learning triggering condition parameters is demonstrated.
@article{TAC25_Funk,
title = {{An Event-Based Approach for the Conservative Compression of Covariance Matrices}},
author = {Christopher Funk and Benjamin Noack},
doi = {10.1109/TAC.2024.3494672},
issue = {5},
journal = {IEEE Transactions on Automatic Control},
month = may,
pages = {3213--3225},
volume = {70},
year = {2025}
}
Christopher Funk, Benjamin Noack, Uwe D. Hanebeck
Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion Sensors, vol. 21, no. 9, April, 2021.
Information fusion in networked systems poses challenges with respect to both theory and implementation. Limited available bandwidth can become a bottleneck when high-dimensional estimates and associated error covariance matrices need to be transmitted. Compression of estimates and covariance matrices can endanger desirable properties like unbiasedness and may lead to unreliable fusion results. In this work, quantization methods for estimates and covariance matrices are presented and their usage with the optimal fusion formulas and covariance intersection is demonstrated. The proposed quantization methods significantly reduce the bandwidth required for data transmission while retaining unbiasedness and conservativeness of the considered fusion methods. Their performance is evaluated using simulations, showing their effectiveness even in the case of substantial data reduction.
@article{Sensors21_Funk,
title = {{Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion}},
author = {Christopher Funk and Benjamin Noack and Uwe D. Hanebeck},
doi = {10.3390/s21093059},
issn = {1424-8220},
journal = {Sensors},
month = apr,
number = {9},
url = {https://www.mdpi.com/1424-8220/21/9/3059},
volume = {21},
year = {2021}
}
Conference Proceedings
Christopher Funk, Taruna Sukhama Tiwari, Benjamin Noack
An Improved Bound on the Accuracy of Scan-Based Emitter Passive Positioning (accepted) Proceedings of the 29th International Conference on Information Fusion (FUSION 2026), Trondheim, Norway, June, 2026.
BibTeX
@inproceedings{Fusion26_Funk,
title = {{An Improved Bound on the Accuracy of Scan-Based Emitter Passive Positioning (accepted)}},
author = {Christopher Funk and Taruna Sukhama Tiwari and Benjamin Noack},
booktitle = {Proceedings of the 29th International Conference on Information Fusion (FUSION 2026)},
address = {Trondheim, Norway},
month = jun,
year = {2026}
}
Taruna Sukhama Tiwari, Christopher Funk, Shuo Li, Benjamin Noack
Sensor Placement in a Passive Sensor Network for Scan-Based Ship Localization Proceedings of the 17th Symposium Sensor Data Fusion: Trends, Solutions and Applications, Bonn, Germany, November, 2025.
Maritime safety remains vulnerable to GNSS spoofing, limited detection ranges, and environmental uncertainties, even with the use of positioning systems such as AIS and shore-based radar. To address these challenges, previous work introduced SeaSentry, a passive shore-based sensor network designed to detect and monitor vessels. The placement of sensors within the network significantly impacts the accuracy of vessel localization and tracking, as both the number and spatial configuration of sensors determine positioning performance. This study proposes a grid-based search for optimizing sensor placement in scan-based vessel localization. A network of fixed sensors is analyzed to identify the optimal positions for additional sensors, using the Cramér-Rao Lower Bound (CRLB) to quantify the minimum achievable localization error. Experiments conducted in the Neuburgweier region of Germany demonstrate the potential of the proposed grid-based search approach to establish a robust optimization technique for practical deployment. The findings also underscore the importance of strategic sensor placement and validate the approach through empirical results.
@inproceedings{SDF25_Tiwari,
title = {{Sensor Placement in a Passive Sensor Network for Scan-Based Ship Localization}},
author = {Taruna Sukhama Tiwari and Christopher Funk and Shuo Li and Benjamin Noack},
booktitle = {Proceedings of the 17th Symposium Sensor Data Fusion: Trends, Solutions and Applications},
address = {Bonn, Germany},
doi = {10.1109/SDF67080.2025.11330546},
month = nov,
year = {2025}
}
Taruna Sukhama Tiwari, Shuo Li, Christopher Funk, Benjamin Noack, Christian Steger, Hilko Wiards, Matthias Steidel, Florian Schiegg, Nhat Ming Hoang, Mohit Mittal, Vesa Klumpp, Jörn Beschnidt
SeaSentry: Maritime Real-Time Positioning
in a Passive Radar-Detector Network Proceedings of the 28th International Conference on Information Fusion (FUSION 2025), Rio de Janeiro, Brazil, July, 2025.
Maritime transport and vessel monitoring rely on
multiple systems for positioning, such as the Automatic Identifi-
cation System, electro-optical systems, and shore-based radar sys-
tems, to improve safety and efficiency in vessel tracking. However,
each system has inherent limitations, including coverage gaps,
reliance on vessel compliance, and limited real-time monitoring
capabilities. As a complementary approach to existing methods
and systems, this paper presents the SeaSentry system, a passive
sensor network designed to detect, position, and track vessels in
real time, thus eliminating the need for onboard installations.
The sensors detect radar pulses emitted by the vessels’ rotating
radar antennas and compute time stamps as the radar beams
pass over them. Geometric constraints can be derived from time
differences of arrival to localize the vessels, with time error and
synchronization demands in the millisecond range. Along with
some initial results, this paper discusses the SeaSentry setup and
data processing pipeline.
@inproceedings{Fusion25_Tiwari,
title = {{SeaSentry: Maritime Real-Time Positioning
in a Passive Radar-Detector Network}},
author = {Taruna Sukhama Tiwari and Shuo Li and Christopher Funk and Benjamin Noack and Christian Steger and Hilko Wiards and Matthias Steidel and Florian Schiegg and Nhat Ming Hoang and Mohit Mittal and Vesa Klumpp and J{\"o}rn Beschnidt},
booktitle = {Proceedings of the 28th International Conference on Information Fusion (FUSION 2025)},
address = {Rio de Janeiro, Brazil},
doi = {10.23919/FUSION65864.2025.11124109},
month = jul,
year = {2025}
}
Christopher Funk, Benjamin Noack
Conservative Compression of Information Matrices using Event-Triggering and Robust Optimization Proceedings of the 27th International Conference on Information Fusion (FUSION 2024), Venice, Italy, July, 2024.
Distributed sensor fusion requires the transmission of intermediate fusion results, consisting of point estimates and associated error covariance or information matrices. Bandwidth constraints necessitate data compression techniques for error covariance and information matrices, which typically dominate data volume. To ensure the safe use of the fusion results for decision-making, these techniques must be conservative, i.e., not lead to the compressed error covariance or information matrices underestimating the true estimate error. This work introduces a novel approach for the conservative compressed transmission of information matrices, that builds on a previous event-based method for covariance matrices. The proposed method allows the entire sensor fusion pipeline to operate in information space, facilitating efficient fusion operations without the need to compute corresponding covariance matrices. Contributions include an event-trigger for information matrices and a robust-optimization-based bounding mechanism ensuring conservativeness. The proposed approach is evaluated in the context of transmitting error information matrices generated by extended information filter SLAM to a receiver for further processing.
@inproceedings{Fusion24_Funk,
title = {{Conservative Compression of Information Matrices using Event-Triggering and Robust Optimization}},
author = {Christopher Funk and Benjamin Noack},
booktitle = {Proceedings of the 27th International Conference on Information Fusion (FUSION 2024)},
address = {Venice, Italy},
doi = {10.23919/FUSION59988.2024.10706472},
month = jul,
year = {2024}
}
Christopher Funk, Benjamin Noack
Conservative Data Reduction for Covariance Matrices Using Elementwise Event Triggers Proceedings of the 26th International Conference on Information Fusion (Fusion 2023), Charleston, South Carolina, USA, July, 2023.
Decentralized data fusion algorithms are fundamentally built on the exchange of estimates and covariance matrices between the individual components. This leads to a high volume of data, mainly caused by the covariance matrices, which can be problematic, especially in environments with limited bandwidth. In order to guarantee the proper functioning of decentralized estimation algorithms, data reduction methods for covariance matrices must ensure that the reduced matrices are conservative, i.e., do not underestimate the actual uncertainty. Motivated by these considerations, this paper presents an elementwise event-triggered method for the data-reduced transmission of covariance matrices that takes into account the aforementioned condition concerning uncertainty. For this purpose, several event triggers are proposed and, based on the event data and diagonal dominance, upper bounds for the actual covariance matrices are derived. An investigation of the data reduction and its influence on the estimation results is performed in a decentralized tracking scenario. The results show that substantial data reduction is possible with only minor losses in estimation quality.
@inproceedings{Fusion23_Funk,
title = {{Conservative Data Reduction for Covariance Matrices Using Elementwise Event Triggers}},
author = {Christopher Funk and Benjamin Noack},
booktitle = {Proceedings of the 26th International Conference on Information Fusion (Fusion 2023)},
address = {Charleston, South Carolina, USA},
doi = {10.23919/FUSION52260.2023.10224214},
month = jul,
year = {2023}
}
Christopher Funk, Benjamin Noack
Graduated Moving Window Optimization as a Flexible Framework for Multi-Object Tracking Proceedings of the 2023 American Control Conference (ACC 2023), San Diego, California, USA, June, 2023.
Continuous optimization methods for multiple object tracking allow to jointly estimate continuous object trajectories and perform implicit data association. However, the local minima that arise from including data association in a continuous optimization problem pose challenges. In addition, optimization is usually performed either over a fixed or an indefinitely growing time frame. This either discards valuable past information or is computationally unsustainable. Hence, in this work, a flexible continuous optimization based framework for multiple object tracking that accounts for these issues is proposed. The framework provides a unified approach to not only include data association, but also multiple motion models and temporary interactions between objects in a continuous optimization problem. It leverages the concept of graduated optimization, a heuristic, which allows avoiding local minima. The proposed framework's performance is benchmarked on a synthetic dataset, showing its capabilities and indicating areas of possible improvement.
@inproceedings{ACC23_Funk,
title = {{Graduated Moving Window Optimization as a Flexible Framework for Multi-Object Tracking}},
author = {Christopher Funk and Benjamin Noack},
booktitle = {Proceedings of the 2023 American Control Conference (ACC 2023)},
address = {San Diego, California, USA},
doi = {10.23919/ACC55779.2023.10156509},
month = jun,
year = {2023}
}
Haibin Zhao, Christopher Funk, Benjamin Noack, Uwe D. Hanebeck, Michael Beigl
Kalman Filtered Compressive Sensing Using Pseudo-Measurements Proceedings of the 2021 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2021), Karlsruhe, Germany, September, 2021.
In this paper, we combine the Kalman filter and compressive sensing using pseudo-measurements in order to reduce the number of measurements usually required by the Kalman filter. To overcome the non-sparsity of the measurement vectors, we make use of the change of their coefficients when represented in a certain basis, reduce the dimensionality of the coefficients, and learn a sparse basis for the measurement vectors. We further improve our proposed method by introducing dynamic weighting of the pseudo-measurements, by aiding compressive measurement reconstruction with Kalman filter estimates and by employing iterative versions of this process. Simulations show that our approach achieves a 37 percent improvement with respect to the mean-square error compared to the traditional Kalman filter with the same number of measurements. Our approach yields better results when the measurement noise is relatively large compared to the system noise, and it significantly improves the accuracy of state estimation in sensor networks with low sensor precision.
@inproceedings{MFI21_Zhao,
title = {{Kalman Filtered Compressive Sensing Using Pseudo-Measurements}},
author = {Haibin Zhao and Christopher Funk and Benjamin Noack and Uwe D. Hanebeck and Michael Beigl},
booktitle = {Proceedings of the 2021 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2021)},
address = {Karlsruhe, Germany},
doi = {10.1109/MFI52462.2021.9591186},
month = sep,
year = {2021}
}
Christopher Funk, Benjamin Noack, Uwe D. Hanebeck
Conservative Quantization of Fast Covariance Intersection Proceedings of the 2020 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2020), Virtual, September, 2020.
Sensor data fusion in wireless sensor networks poses challenges with respect to both theory and implementation. Unknown cross-correlations between estimates distributed across the network need to be addressed carefully as neglecting them leads to overconfident fusion results. In addition, limited processing power and energy supply of the sensor nodes prohibit the use of complex algorithms and high-bandwidth communication. In this work, fast covariance intersection using both quantized estimates and quantized covariance matrices is considered. The proposed method is computationally efficient and significantly reduces the bandwidth required for data transmission while retaining unbiasedness and conservativeness of fast covariance intersection. The performance of the proposed method is evaluated with respect to that of fast covariance intersection, which proves its effectiveness even in the case of substantial data reduction.
@inproceedings{MFI20_Funk,
title = {{Conservative Quantization of Fast Covariance Intersection}},
author = {Christopher Funk and Benjamin Noack and Uwe D. Hanebeck},
booktitle = {Proceedings of the 2020 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2020)},
address = {Virtual},
doi = {10.1109/MFI49285.2020.9235249},
month = sep,
year = {2020}
}
Benjamin Noack, Christopher Funk, Susanne Radtke, Uwe D. Hanebeck
State Estimation with Event-Based Inputs Using Stochastic Triggers Proceedings of the 21st IFAC World Congress (IFAC 2020), Berlin, Germany, July, 2020.
Event-based communication and state estimation offer the potential to improve resource utilization in networked sensor and control systems significantly. Sensor nodes can trigger transmissions when data are deemed useful for the remote estimation units. To improve the estimation performance, the remote estimator can exploit the implicit information conveyed by the event trigger even if no transmission is triggered. The implicit information is typically incorporated into the measurement update of a remote Kalman filter. In this paper, event-triggered transmissions of input data are investigated that enter the prediction step of the remote estimator.
By employing a stochastic trigger, the implicit input information
remains Gaussian and can easily be incorporated into the remote Kalman filter. The proposed event-based scheme is evaluated in remote tracking scenarios, where system inputs are transmitted aperiodically.
@inproceedings{IFAC20_Noack,
title = {{State Estimation with Event-Based Inputs Using Stochastic Triggers}},
author = {Benjamin Noack and Christopher Funk and Susanne Radtke and Uwe D. Hanebeck},
booktitle = {Proceedings of the 21st IFAC World Congress (IFAC 2020)},
address = {Berlin, Germany},
doi = {10.1016/j.ifacol.2020.12.2491},
month = jul,
year = {2020}
}
Other
Christopher Funk, Benjamin Noack
An Event-Based Approach for the Conservative Compression of Covariance Matrices arXiv, 2024.
This work introduces a flexible and versatile method for the data-efficient yet conservative transmission of covariance matrices, where a matrix element is only transmitted if a so-called triggering condition is satisfied for the element. Here, triggering conditions can be parametrized on a per-element basis, applied simultaneously to yield combined triggering conditions or applied only to certain subsets of elements. This allows, e.g., to specify transmission accuracies for individual elements or to constrain the bandwidth available for the transmission of subsets of elements. Additionally, a methodology for learning triggering condition parameters from an application-specific dataset is presented. The performance of the proposed approach is quantitatively assessed in terms of data reduction and conservativeness using estimate data derived from real-world vehicle trajectories from the InD-dataset, demonstrating substantial data reduction ratios with minimal over-conservativeness. The feasibility of learning triggering condition parameters is demonstrated.
@misc{arXiv2024_Funk,
title = {{An Event-Based Approach for the Conservative Compression of Covariance Matrices}},
author = {Christopher Funk and Benjamin Noack},
copyright = {Creative Commons Attribution 4.0 International},
doi = {10.48550/ARXIV.2403.05977},
keywords = {Robotics (cs.RO), Artificial Intelligence (cs.AI), FOS: Computer and information sciences, FOS: Computer and information sciences},
publisher = {arXiv},
year = {2024}
}
Christopher Funk, Ofer Dagan, Benjamin Noack, Nisar R. Ahmed
Exploiting Structure for Optimal Multi-Agent Bayesian Decentralized Estimation arXiv, 2023.
A key challenge in Bayesian decentralized data fusion is the `rumor propagation' or `double counting' phenomenon, where previously sent data circulates back to its sender. It is often addressed by approximate methods like covariance intersection (CI) which takes a weighted average of the estimates to compute the bound. The problem is that this bound is not tight, i.e. the estimate is often over-conservative. In this paper, we show that by exploiting the probabilistic independence structure in multi-agent decentralized fusion problems a tighter bound can be found using (i) an expansion to the CI algorithm that uses multiple (non-monolithic) weighting factors instead of one (monolithic) factor in the original CI and (ii) a general optimization scheme that is able to compute optimal bounds and fully exploit an arbitrary dependency structure. We compare our methods and show that on a simple problem, they converge to the same solution. We then test our new non-monolithic CI algorithm on a large-scale target tracking simulation and show that it achieves a tighter bound and a more accurate estimate compared to the original monolithic CI.
@misc{arXiv2023_Funk,
title = {{Exploiting Structure for Optimal Multi-Agent Bayesian Decentralized Estimation}},
author = {Christopher Funk and Ofer Dagan and Benjamin Noack and Nisar R. Ahmed},
copyright = {Creative Commons Attribution 4.0 International},
doi = {10.48550/ARXIV.2307.10594},
keywords = {Robotics (cs.RO), Artificial Intelligence (cs.AI), FOS: Computer and information sciences, FOS: Computer and information sciences},
publisher = {arXiv},
year = {2023}
}
Ofer Dagan, Christopher Funk, Nisar R. Ahmed, Benjamin Noack
Exploiting Structure for Optimal Multi-Agent Bayesian Decentralized Estimation Workshop on Inference and Decision Making for Autonomous Vehicles as Robotics: Science and Systems (RSS 2023), Daegu, Republic of Korea, July, 2023.
BibTeX
@inproceedings{RSS23_Dagan,
title = {{Exploiting Structure for Optimal Multi-Agent Bayesian Decentralized Estimation}},
author = {Ofer Dagan and Christopher Funk and Nisar R. Ahmed and Benjamin Noack},
booktitle = {Workshop on Inference and Decision Making for Autonomous Vehicles as Robotics: Science and Systems (RSS 2023)},
address = {Daegu, Republic of Korea},
month = jul,
year = {2023}
}
Last Modification: 01.08.2026 - Contact Person: Christopher Funk
