QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python
This provides a practical tool for researchers and practitioners working with multi-channel signals, such as in image processing or time series analysis, though it is incremental as it extends existing linear algebra methods to quaternions.
The authors tackled the lack of mature software for quaternion-valued representations by developing QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, which includes core operations, decompositions, iterative solvers, and domain-specific routines, as demonstrated in applications like image deblurring and completion.
Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $μ$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials.