Solving nonlinear matrix equations via novel fixed point techniques: Application to denoising in Diffusion Tensor Imaging

This paper focuses on solving a specific class of nonlinear matrix equations (NMEs) by employing novel fixed point techniques. We introduce a novel α−βL-Geraghty-type contraction mapping, defined through a generalized non-negative real-valued continuous function Φ which takes the form of a convex combination: Φ(t1,t2,…,t6)=a1t1+a2t2+a3t3+…+a6t6, with coefficients satisfying ∑i=16ai=1 and ai∈[0,1] for all i=1,…,6. This formulation offers a broad generalization, integrating and extending existing concepts such as Geraghty, ΦE, α−βE, and α−βM Geraghty contractions. The existence and uniqueness of fixed points are established for α−βL-Geraghty-type contraction mapping in complete b-metric spaces. The developed theoretical framework is then rigorously applied to demonstrate the existence and uniqueness of solutions for this class of NMEs. A robust and practical application is provided for Diffusion Tensor Imaging (DTI) tensor denoising, for which meticulously developed numerical algorithms are presented.