Formal derivation of algorithms: The triangular sylvester equation
ACM Transactions on Mathematical Software (TOMS)
On the generalized ADI method for the matrix equation X -AXB = C
Journal of Computational and Applied Mathematics
Low rank approximate solutions to large Sylvester matrix equations
Applied Mathematics and Computation
Sylvester Tikhonov-regularization methods in image restoration
Journal of Computational and Applied Mathematics
On the ADI method for Sylvester equations
Journal of Computational and Applied Mathematics
Extended Arnoldi methods for large low-rank Sylvester matrix equations
Applied Numerical Mathematics
Applied Numerical Mathematics
On the numerical solution of large-scale sparse discrete-time Riccati equations
Advances in Computational Mathematics
Image denoising using the lyapunov equation from non-uniform samples
ICIAR'06 Proceedings of the Third international conference on Image Analysis and Recognition - Volume Part I
An efficient algorithm for solving general coupled matrix equations and its application
Mathematical and Computer Modelling: An International Journal
A Variational Approach for Sharpening High Dimensional Images
SIAM Journal on Imaging Sciences
An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation
SIAM Journal on Numerical Analysis
Optimization of the solution of the parameter-dependent Sylvester equation and applications
Journal of Computational and Applied Mathematics
Nested splitting conjugate gradient method for matrix equation AXB=C and preconditioning
Computers & Mathematics with Applications
Computers & Mathematics with Applications
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The restoration of two-dimensional images in the presence of noise by Wiener's minimum mean square error filter requires the solution of large linear systems of equations. When the noise is white and Gaussian, and under suitable assumptions on the image, these equations can be written as a Sylvester's equation \[ T_1^{-1}\hat{F}+\hat{F}T_2=C \] for the matrix $\hat{F}$ representing the restored image. The matrices $T_1$ and $T_2$ are symmetric positive definite Toeplitz matrices. We show that the ADI iterative method is well suited for the solution of these Sylvester's equations, and illustrate this with computed examples for the case when the image is described by a separable first-order Markov process. We also consider generalizations of the ADI iterative method, propose new algorithms for the generation of iteration parameters, and illustrate the competitiveness of these schemes.