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Quaternion-Domain Super MDS for Robust 3D Localization

Alessio Lukaj, Keigo Masuoka, Takumi Takahashi, Giuseppe Thadeu Freitas de Abreu, Hideki Ochiai

TL;DR

Simulation results demonstrate that the proposed QD-SMDS algorithm significantly improves localization accuracy compared to the original SMDS algorithm, especially in scenarios with substantial measurement errors, and achieves comparable localization accuracy without requiring SVD.

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.

Quaternion-Domain Super MDS for Robust 3D Localization

TL;DR

Simulation results demonstrate that the proposed QD-SMDS algorithm significantly improves localization accuracy compared to the original SMDS algorithm, especially in scenarios with substantial measurement errors, and achieves comparable localization accuracy without requiring SVD.

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.

Paper Structure

This paper contains 35 sections, 79 equations, 15 figures, 2 tables, 4 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Basics of Quaternion Algebra
  4. Quaternion Singular Value Decomposition (QSVD)
  5. Problem Formulation and the SMDS Algorithm
  6. Problem Formulation
  7. SMDS Algorithm Recap
  8. Classical Localization Models and Link to SMDS and its Quaternion Variants
  9. Quaternion-Domain Super MDS
  10. QD-SMDS Algorithm
  11. Derivation of the Proposed Algorithm
  12. Construction of the Quaternion-Domain GEK Matrix
  13. Handling Incomplete Data
  14. Performance Assessment
  15. Simulation Conditions
  16. ...and 20 more sections

Figures (15)

  • Figure 1: Illustration of the parameters required to construct quaternion-domain GEK matrix $\bm{K}_\mathrm{q}$.
  • Figure 2: Parameters that can be obtained using a planar antenna.
  • Figure 3: High-level architecture of the proposed quaternion-domain localization pipeline.
  • Figure 4: Comparison of average estimation error between the SMDS and QD-SMDS algorithms in Scenario I.
  • Figure 5: Comparison of the empirical CDF between the SMDS and QD-SMDS algorithms in Scenario I.
  • ...and 10 more figures