Quaternion-Domain Super MDS for Robust 3D Localization

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.

Quaternion Domain Super MDS for 3D Localization

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.