Efficient algorithms for the commutative quaternion equality constrained least squares problem with applications to color image restoration

The commutative quaternion algebra increasingly serves as a powerful framework in signal processing, color image modeling, and control systems. Within this context, the commutative quaternion equality constrained least squares (CQLSE) problem is essential in modeling complex relationships involving commutative quaternionic variables under linear constraints. However, efficient and reliable computational methods for solving such problems remain underdeveloped.
This talk presents two efficient algorithms for solving the CQLSE problem using structured matrix decompositions. Based on the complex representation of commutative quaternion matrices, we study the QR decomposition and the generalized singular value decomposition (GSVD) in commutative quaternion matrices. We provide theoretical formulations together with practical algorithms derived from these decompositions. Numerical experiments demonstrate the accuracy and robustness of the proposed algorithms. Finally, we apply the developed algorithms to a commutative quaternion model of color image restoration and demonstrate their effectiveness in image restoration tasks.
This work is supported by Young Talent of Lifting Engineering for Science and Technology in Shandong, China (Grant No. SDAST2025QTB033) and by the Russian Science Foundation grant No. 23-71-30013 (https://rscf.ru/project/23-71-30013/).