Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters

XUE Yu; XU Lei

doi:10.11999/JEIT250759

Article Contents

Article Navigation > Journal of Electronics & Information Technology > 2025 >

XUE Yu, XU Lei. Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250759

Citation:

XUE Yu, XU Lei. Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250759

XUE Yu, XU Lei. Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250759

Citation:

XUE Yu, XU Lei. Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250759

PDF( 2586 KB)

Optimal Federated Average Fusion of Gaussian Mixture–Probability Hypothesis Density Filters

doi: 10.11999/JEIT250759 cstr: 32379.14.JEIT250759

XUE Yu^,,
XU Lei

Shanghai Electro-Mechanical Engineering Institute, Shanghai 201109, China

Funds: The National Natural Science Foundation of China (62071386)

Received Date: 2025-08-19
Rev Recd Date: 2025-09-29

Available Online: 2025-10-22

Abstract

Abstract

Objective To realize optimal decentralized fusion tracking of uncertain targets, this study proposes a federated average fusion algorithm for Gaussian Mixture–Probability Hypothesis Density (GM-PHD) filters, designed with a hierarchical structure. Each sensor node operates a local GM-PHD filter to extract multi-target state estimates from sensor measurements. The fusion node performs three key tasks: (1) maintaining a master filter that predicts the fusion result from the previous iteration; (2) associating and merging the GM-PHDs of all filters; and (3) distributing the fused result and several parameters to each filter. The association step decomposes multi-target density fusion into four categories of single-target estimate fusion. We derive the optimal single-target estimate fusion both in the absence and presence of missed detections. Information assignment applies the covariance upper-bounding theory to eliminate correlation among all filters, enabling the proposed algorithm to achieve the accuracy of Bayesian fusion. Simulation results show that the federated fusion algorithm achieves optimal tracking accuracy and consistently outperforms the conventional Arithmetic Average (AA) fusion method. Moreover, the relative reliability of each filter can be flexibly adjusted. Methods The multi-sensor multi-target density fusion is decomposed into multiple groups of single-target component merging through the association operation. Federated filtering is employed as the merging strategy, which achieves the Bayesian optimum owing to its inherent decorrelation capability. Section 3 rigorously extends this approach to scenarios with missed detections. To satisfy federated filtering’s requirement for prior estimates, a master filter is designed to compute the predicted multi-target density, thereby establishing a hierarchical architecture for the proposed algorithm. In addition, auxiliary measures are incorporated to compensate for the observed underestimation of cardinality. Results and Discussions modified Mahalanobis distance (Fig.3). The precise association and the single-target decorrelation capability together ensure the theoretical optimality of the proposed algorithm, as illustrated in Fig. 2. Compared with conventional density fusion, the Optimal Sub-Pattern Assignment (OSPA) error is reduced by 8.17% (Fig. 4). The advantage of adopting a small average factor for the master filter is demonstrated in Figs. 5 and 6. The effectiveness of the measures for achieving cardinality consensus is also validated (Fig. 7). Another competitive strength of the algorithm lies in the flexibility of adjusting the average factors (Fig. 8). Furthermore, the algorithm consistently outperforms AA fusion across all missed detection probabilities (Fig. 9). Conclusions This paper achieves theoretically optimal multi-target density fusion by employing federated filtering as the merging method for single-target components. The proposed algorithm inherits the decorrelation capability and single-target optimality of federated filtering. A hierarchical fusion architecture is designed to satisfy the requirement for prior estimates. Extensive simulations demonstrate that: (1) the algorithm can accurately associate filtered components belonging to the same target, thereby extending single-target optimality to multi-target fusion tracking; (2) the algorithm supports flexible adjustment of average factors, with smaller values for the master filter consistently preferred; and (3) the superiority of the algorithm persists even under sensor malfunctions and high missed detection rates. Nonetheless, this study is limited to GM-PHD filters with overlapping Fields Of View (FOVs). Future work will investigate its applicability to other filter types and spatially non-overlapping FOVs.
- Decentralized tracking fusion,
- Uncertain targets,
- Average fusion,
- Optimal estimate fusion,
- Covariance upper-bounding theory

FullText(HTML)

References(35)

References

[1]	李克勇, 李楚晨, 蔡云泽. 群对群拦截协同态势感知关键技术[J]. 空天防御, 2025, 8(1): 1–9. doi: 10.3969/j.issn.2096-4641.2025.01.001. LI Keyong, LI Chuchen, and CAI Yunze. Key technologies for collaborative situational awareness based on swarm-to-swarm interception[J]. Air & Space Defense, 2025, 8(1): 1–9. doi: 10.3969/j.issn.2096-4641.2025.01.001.
[2]	CHONG C Y, MORI S, BARKER W H, et al. Architectures and algorithms for track association and fusion[J]. IEEE Aerospace and Electronic Systems Magazine, 2000, 15(1): 5–13. doi: 10.1109/62.821657.
[3]	BAR-SHALOM Y and CAMPO L. The effect of the common process noise on the two-sensor fused-track covariance[J]. IEEE Transactions on Aerospace and Electronic Systems, 1986, AES-22(6): 803–805. doi: 10.1109/TAES.1986.310815.
[4]	王张夫, 汤显峰. 基于协方差交叉融合的多传感器数据融合研究[J]. 电子测量技术, 2024, 47(8): 78–85. doi: 10.19651/j.cnki.emt.2415396. WANG Zhangfu and TANG Xianfeng. Research on multi-sensor data fusion based on covariance cross fusion[J]. Electronic Measurement Technology, 2024, 47(8): 78–85. doi: 10.19651/j.cnki.emt.2415396.
[5]	CARLSON N A. Federated square root filter for decentralized parallel processors[J]. IEEE Transactions on Aerospace and Electronic Systems, 1990, 26(3): 517–525. doi: 10.1109/7.106130.
[6]	WANG Ziyi, LI Ning, WANG Zhao, et al. An adaptive federated filter based on variational Bayes with application to multisource navigation[J]. IEEE Sensors Journal, 2023, 23(9): 9859–9870. doi: 10.1109/JSEN.2023.3258932.
[7]	薛昱, 冯西安. 一种纯方位多目标跟踪的联合多高斯混合概率假设密度滤波器[J]. 电子与信息学报, 2024, 46(11): 4295–4304. doi: 10.11999/JEIT240201. XUE Yu and FENG Xi'an. Joint multi-Gaussian mixture probability hypothesis density filter for bearings-only multi-target tracking[J]. Journal of Electronics & Information Technology, 2024, 46(11): 4295–4304. doi: 10.11999/JEIT240201.
[8]	WANG Liu, CHEN Guifen, ZHANG Lei, et al. Stochastic outlier selection via GM-CPHD fusion for multitarget tracking using sensors with different fields of view[J]. IEEE Sensors Journal, 2024, 24(6): 9148–9161. doi: 10.1109/JSEN.2024.3355903.
[9]	DAVIES E S and GARCÍA-FERNÁNDEZ Á F. Information exchange track-before-detect multi-Bernoulli filter for superpositional sensors[J]. IEEE Transactions on Signal Processing, 2024, 72: 607–621. doi: 10.1109/TSP.2024.3349769.
[10]	LIU Jiacheng, LIANG Meiling, MA Jinxuan, et al. Microbubble tracking based on partial smoothing-based adaptive generalized labelled Multi-Bernoulli filter for super-resolution imaging[J]. Ultrasonics, 2025, 145: 107455. doi: 10.1016/j.ultras.2024.107455.
[11]	YANG Chaoqun, CAO Xianghui, and SHI Zhiguo. Road-map aided Gaussian mixture labeled multi-Bernoulli filter for ground multi-target tracking[J]. IEEE Transactions on Vehicular Technology, 2023, 72(6): 7137–7147. doi: 10.1109/TVT.2023.3240740.
[12]	LIU Yang, XU Yong, WU Peipei, et al. Labelled non-zero diffusion particle flow SMC-PHD filtering for multi-speaker tracking[J]. IEEE Transactions on Multimedia, 2024, 26: 2544–2559. doi: 10.1109/TMM.2023.3301221.
[13]	BAILEY T, JULIER S, and AGAMENNONI G. On conservative fusion of information with unknown non-Gaussian dependence[C]. 2012 15th International Conference on Information Fusion, Singapore, Singapore, 2012: 1876–1883.
[14]	MAHLER R. Toward a theoretical foundation for distributed fusion[M]. HALL D, CHONG C Y, LLINAS J, et al. Distributed Data Fusion for Network-Centric Operations. Boca Raton: CRC Press, 2013.
[15]	ÜNEY M, CLARK D E, and JULIER S J. Distributed fusion of PHD filters via exponential mixture densities[J]. IEEE Journal of Selected Topics in Signal Processing, 2013, 7(3): 521–531. doi: 10.1109/JSTSP.2013.2257162.
[16]	GUNAY M, ORGUNER U, and DEMIREKLER M. Chernoff fusion of Gaussian mixtures based on sigma-point approximation[J]. IEEE Transactions on Aerospace and Electronic Systems, 2016, 52(6): 2732–2746. doi: 10.1109/TAES.2016.150403.
[17]	MAHLER R P S. Optimal/robust distributed data fusion: A unified approach[C]. Proceedings of SPIE 4052, Signal Processing, Sensor Fusion, and Target Recognition IX, Orlando, USA, 2000: 128–138. doi: 10.1117/12.395064. (查阅网上资料,未找到本条文献页码信息,请确认).
[18]	ABBAS A E. A Kullback-Leibler view of linear and log-linear pools[J]. Decision Analysis, 2009, 6(1): 25–37. doi: 10.1287/deca.1080.0133.
[19]	ÜNEY M, HOUSSINEAU J, DELANDE E, et al. Fusion of finite-set distributions: Pointwise consistency and global cardinality[J]. IEEE Transactions on Aerospace and Electronic Systems, 2019, 55(6): 2759–2773. doi: 10.1109/TAES.2019.2893083.
[20]	GAO Lin, BATTISTELLI G, and CHISCI L. Multiobject fusion with minimum information loss[J]. IEEE Signal Processing Letters, 2020, 27: 201–205. doi: 10.1109/LSP.2019.2963817.
[21]	LI Tiancheng, CORCHADO J M, and SUN Shudong. On generalized covariance intersection for distributed PHD filtering and a simple but better alternative[C]. 2017 20th International Conference on Information Fusion (Fusion), Xi’an, China, 2017: 1–8. doi: 10.23919/ICIF.2017.8009732.
[22]	GOSTAR A K, HOSEINNEZHAD R, and BAB-HADIASHAR A. Cauchy-Schwarz divergence-based distributed fusion with Poisson random finite sets[C]. 2017 International Conference on Control, Automation and Information Sciences (ICCAIS), Chiang Mai, Thailand, 2017: 112–116. doi: 10.1109/ICCAIS.2017.8217559.
[23]	DING Changwen, SHAO Chuntao, ZHOU Siteng, et al. Distributed multi-target tracking with labeled multi-Bernoulli filter considering efficient label matching[J]. Frontiers of Information Technology & Electronic Engineering, 2025, 26(3): 400–414. doi: 10.1631/FITEE.2400582.
[24]	LI Tiancheng, CORCHADO J M, and SUN Shudong. Partial consensus and conservative fusion of Gaussian mixtures for distributed PHD fusion[J]. IEEE Transactions on Aerospace and Electronic Systems, 2019, 55(5): 2150–2163. doi: 10.1109/TAES.2018.2882960.
[25]	LI Tiancheng. Arithmetic average density fusion—Part II: Unified derivation for unlabeled and labeled RFS fusion[J]. IEEE Transactions on Aerospace and Electronic Systems, 2024, 60(3): 3255–3268. doi: 10.1109/TAES.2024.3359592.
[26]	YANG Hao, LI Tiancheng, YAN Junkun, et al. Hierarchical average fusion with GM-PHD filters against FDI and DoS attacks[J]. IEEE Signal Processing Letters, 2024, 31: 934–938. doi: 10.1109/LSP.2024.3356823.
[27]	XUE Yu and FENG Xi’an. An optimal hierarchical arithmetic average fusion of GM-PHD filters[J]. IEEE Signal Processing Letters, 2025, 32: 1605–1609. doi: 10.1109/LSP.2025.3554142.
[28]	LI Tiancheng, LIANG Haozhe, LI Guchong, et al. Arithmetic average density fusion—Part IV: Distributed heterogeneous fusion of RFS and LRFS filters via variational approximation[J]. IEEE Transactions on Signal Processing, 2025, 73: 1454–1469. doi: 10.1109/TSP.2025.3550157.
[29]	李天成, 李固冲, 李文硕, 等. 均值融合跟踪概念、理论与方法[J/OL]. 中国科学: 信息科学. https://www.sciengine.com/SSI/doi/10.1360/SSI-2025-0080. LI Tiancheng, LI Guchong, LI Wenshuo, et al. Concepts, theories and methods for average fusion tracking[J/OL]. Scientia Sinica Informationis. https://www.sciengine.com/SSI/doi/10.1360/SSI-2025-0080. (查阅网上资料,未找到本条文献引用日期信息,请补充).
[30]	MAHLER R. Approximate multisensor CPHD and PHD filters[C]. 2010 13th International Conference on Information Fusion, Edinburgh, UK, 2010: 1–8. doi: 10.1109/ICIF.2010.5711984.
[31]	SUN Shuli. Distributed optimal linear fusion estimators[J]. Information Fusion, 2020, 63: 56–73. doi: 10.1016/j.inffus.2020.05.006.
[32]	WEI Jingxin, LUO Feng, CHEN Shichao, et al. Robust fusion of GM-PHD filters based on geometric average[J]. Signal Processing, 2023, 206: 108912. doi: 10.1016/j.sigpro.2022.108912.
[33]	LI Tiancheng, SONG Yan, SONG Enbin, et al. Arithmetic average density fusion-Part I: Some statistic and information-theoretic results[J]. Information Fusion, 2024, 104: 102199. doi: 10.1016/j.inffus.2023.102199.
[34]	李寿鹏, 陶贞吉, 张晓宇, 等. 基于分布式信息滤波的集群导弹协同定位方法[J]. 空天防御, 2024, 7(2): 36–41. doi: 10.3969/j.issn.2096-4641.2024.02.006. LI Shoupeng, TAO Zhenji, ZHANG Xiaoyu, et al. Distributed information filtering-based cooperative localization method for cluster missiles[J]. Air & Space Defense, 2024, 7(2): 36–41. doi: 10.3969/j.issn.2096-4641.2024.02.006.
[35]	SCHUHMACHER D, VO B T, and VO B N. A consistent metric for performance evaluation of multi-object filters[J]. IEEE Transactions on Signal Processing, 2008, 56(8): 3447–3457. doi: 10.1109/TSP.2008.920469.