Structure Preserving Algorithms for Quaternion Outer Inverses with Applications

TL;DR

This paper develops structure-preserving algorithms for quaternion outer inverses, providing explicit formulas, efficient computational methods, and applications to image deblurring and signal filtering.

Contribution

It introduces novel algorithms for quaternion outer inverses that unify and extend classical inverses, with practical applications demonstrated.

Findings

Algorithms successfully compute quaternion outer inverses with prescribed constraints.

Applications show improved color image deblurring and chaotic signal filtering.

Numerical examples validate the effectiveness of the proposed methods.

Abstract

This study investigates the theoretical and computational aspects of quaternion generalized inverses, focusing on outer inverses and {1,2}-inverses with prescribed range and/or null space constraints. In view of the non-commutative nature of quaternions, a detailed characterization of the left and right range and null spaces of quaternion matrices is presented. Explicit representations for these inverses are derived, including full rank decomposition-based formulations. We design two efficient algorithms: one leveraging the Quaternion Toolbox for MATLAB (QTFM), and the other employing a complex structure preserving approach based on the complex representation of quaternion matrices. With suitable choices of subspace constraints, these outer inverses unify and generalize several classical inverses, including the Moore-Penrose inverse, the group inverse, and the Drazin inverse. The proposed methods are validated through numerical examples and applied to two real-world tasks: quaternion-based color image deblurring, which preserves inter-channel correlations, and robust filtering of chaotic 3D signals, demonstrating their effectiveness in high-dimensional settings.