Iterative optimal solutions of linear matrix equations for Hyperspectral and Multispectral image fusing

TL;DR

This paper introduces an iterative method combining conjugate gradient, Lanczos, and More-Sorensen techniques to solve inhomogeneous linear matrix equations efficiently, with applications in hyperspectral and multispectral image fusion.

Contribution

It develops a universal, efficient iterative approach for solving various linear matrix equations, including Sylvester, Lyapunov, and Stein types, with adaptations for structured and sparse data.

Findings

Demonstrates efficiency for dense and small matrices

Shows adaptability to sparse and structured matrices

Validates method in hyperspectral and multispectral image fusion

Abstract

For a linear matrix function $f$ in $X \in \R^{m\times n}$ we consider inhomogeneous linear matrix equations $f(X) = E$ for $E \neq 0$ that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos' methods in combination with the More-Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution.