A Family of Iterative Methods for Computing Generalized Inverses of Quaternion Matrices and its Applications

TL;DR

This paper introduces three new iterative algorithms for computing the Moore-Penrose pseudoinverse of quaternion matrices, demonstrating improved efficiency and stability for applications in signal processing and image analysis.

Contribution

The paper proposes three novel quaternion iterative methods with convergence guarantees, and shows their effectiveness as preconditioners and in practical applications.

Findings

Algorithms achieve accuracy comparable or superior to existing methods.

Significant reduction in iteration count and runtime for large-scale systems.

Successful application to image completion and signal filtering tasks.

Abstract

The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper presents three efficient quaternion iterative algorithms for computing the Moore-Penrose pseudoinverse: (i) the quaternion rapid iterative method (QRAPID), (ii) the quaternion strong approximate inverse (QSAI), and (iii) the quaternion hyperpower iterative method of order nineteen (QHPI19). Convergence theorems and perturbation bounds are established to ensure numerical stability and robustness. The QSAI method is further employed as a preconditioner for quaternion Krylov subspace solvers, resulting in substantial reductions in the iteration count and runtime for large-scale linear systems. Comprehensive numerical experiments demonstrate that the proposed algorithms achieve an accuracy comparable to or better than existing approaches, including quaternion SVD, quaternion Newton-Schulz, and classical hyperpower schemes, while offering significant computational savings. The practical utility of the framework is illustrated through two representative applications: image completion via CUR decomposition and signal filtering, which confirm its scalability and effectiveness in real-world multidimensional data applications.