Quaternion-Domain Super MDS for Robust 3D Localization

TL;DR

This paper introduces a quaternion-based 3D localization algorithm for wireless sensor networks that enhances robustness and reduces computational complexity by reformulating existing methods in the quaternion domain.

Contribution

It presents a novel quaternion-domain reformulation of SMDS, improving robustness and efficiency in 3D localization tasks.

Findings

Significantly improves localization accuracy under measurement errors

Achieves comparable accuracy without SVD by leveraging quaternion matrix structure

Reduces computational complexity with a SVD-free variant

Abstract

This paper proposes a novel low-complexity three-dimensional (3D) localization algorithm for wireless sensor networks, termed quanternion-domain super multi-dimensional scaling (QD-SMDS). The algorithm is based on a reformulation of the SMDS, originally developed in the real domain, using quaternion algebra. By representing 3D coordinates as quaternions, the method constructs a rank-1 Gram edge kernel (GEK) matrix that integrates both relative distance and angular information between nodes, which enhances the noise reduction effect achieved through low-rank truncation employing singular value decomposition (SVD), thereby improving robustness against information loss. To further reduce computational complexity, we also propose a variant of QD-SMDS that eliminates the need for the computationally expensive SVD by leveraging the inherent structure of the quaternion-domain GEK matrix. This alternative directly estimates node coordinates using only matrix multiplications within the quaternion domain. Simulation results demonstrate that the proposed method significantly improves localization accuracy compared to the original SMDS algorithm, especially in scenarios with substantial measurement errors. The proposed method also achieves comparable localization accuracy without requiring SVD.