Convergence properties of Gauss-Newton iterative algorithms in nonlinear image restoration

Recently, some research has been devoted to finding the explicit forms of the Î·-Hermitian and Î·-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding Î·-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining Î·-Hermitian and Î·-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the Î·-Hermitian and Î·-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the Î·-Hermitian and Î·-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

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Volterra-type nonlinear image restoration of medical imagery using principal dynamic modes

2004 2nd IEEE International Symposium on Biomedical Imaging: Macro to Nano (IEEE Cat No. 04EX821) ◽

10.1109/isbi.2004.1398649 ◽

2005 ◽

Author(s):

S. Do ◽

D. Shin ◽

J.-W. Jeong ◽

T.-S. Kim ◽

V.Z. Marmarelis

Keyword(s):

Image Restoration ◽

Volterra Type ◽

Medical Imagery ◽

Principal Dynamic Modes ◽

Nonlinear Image Restoration

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A Fast Alternating Minimization Algorithm for Nonlocal Vectorial Total Variational Multichannel Image Denoising

Mathematical Problems in Engineering ◽

10.1155/2014/731272 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Rubing Xi ◽

Zhengming Wang ◽

Xia Zhao ◽

Meihua Xie

Keyword(s):

Complexity Analysis ◽

Iterative Algorithms ◽

Linear Convergence ◽

Convergence Properties ◽

Theoretical Derivation ◽

Variational Models ◽

Fixed Point Algorithm ◽

Alternating Minimization Algorithm ◽

Uniqueness Of The Solution ◽

Nonlocal Regularization

The variational models with nonlocal regularization offer superior image restoration quality over traditional method. But the processing speed remains a bottleneck due to the calculation quantity brought by the recent iterative algorithms. In this paper, a fast algorithm is proposed to restore the multichannel image in the presence of additive Gaussian noise by minimizing an energy function consisting of anl2-norm fidelity term and a nonlocal vectorial total variational regularization term. This algorithm is based on the variable splitting and penalty techniques in optimization. Following our previous work on the proof of the existence and the uniqueness of the solution of the model, we establish and prove the convergence properties of this algorithm, which are the finite convergence for some variables and theq-linear convergence for the rest. Experiments show that this model has a fabulous texture-preserving property in restoring color images. Both the theoretical derivation of the computation complexity analysis and the experimental results show that the proposed algorithm performs favorably in comparison to the widely used fixed point algorithm.

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