Quaternion Domain Super MDS for 3D Localization

2025-04-24

AI Summary

This paper addresses the low localization accuracy of three-dimensional (3D) positioning in wireless sensor networks under significant measurement errors. To this end, we propose a low-complexity quaternion-domain semi-metric multidimensional scaling (SMDS) algorithm. Our method innovatively extends conventional real-domain SMDS to the quaternion domain, enabling unified representation of 3D coordinates, Euclidean distances, and directional phase information. We construct a phase-integrated rank-1 Gram edge kernel (GEK) matrix leveraging quaternion algebra. This marks the first incorporation of quaternion representations into the SMDS framework, allowing the GEK to jointly encode both relative distances and directional relationships. Noise robustness is further enhanced via singular value decomposition (SVD)-based low-rank truncation. Experimental results demonstrate that the proposed approach achieves substantially higher localization accuracy than traditional SMDS under large measurement errors, while maintaining low computational complexity.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

We propose a novel low-complexity three-dimensional (3D) localization algorithm for wireless sensor networks, termed quaternion-domain super multidimensional scaling (QD-SMDS). This algorithm reformulates the conventional SMDS, which was originally developed in the real domain, into the quaternion domain. By representing 3D coordinates as quaternions, the method enables the construction of a rank-1 Gram edge kernel (GEK) matrix that integrates both relative distance and angular (phase) information between nodes, maximizing the noise reduction effect achieved through low-rank truncation via singular value decomposition (SVD). The simulation results indicate that the proposed method demonstrates a notable enhancement in localization accuracy relative to the conventional SMDS algorithm, particularly in scenarios characterized by substantial measurement errors.

Problem

• Develops low-complexity 3D localization for wireless sensor networks
• Reformulates SMDS in quaternion domain for noise reduction
• Improves accuracy in high-measurement-error scenarios

Innovation

• Reformulates SMDS in quaternion domain
• Uses quaternions for 3D coordinate representation
• Constructs rank-1 GEK matrix with SVD