Conjugate gradient method preconditioned with modified block SSOR iteration for multiplicative half-quadratic image rest
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Conjugate gradient method preconditioned with modified block SSOR iteration for multiplicative half‑quadratic image restoration Pei‑Pei Zhao1 · Yu‑Mei Huang1 Received: 18 July 2019 / Revised: 30 July 2020 / Accepted: 19 August 2020 / Published online: 31 August 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed to solve the nonlinear system of equations resulted from the optimization problem for image restoration. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. The preconditioned conjugate gradient method with the proposed modified block SSOR (symmetric successive over-relaxation) preconditioner is applied to solve this linear system of equations. The condition number of the preconditioned matrix is estimated and numerical experiments are also implemented for image restoration. Both theoretical and numerical results show that the modified block SSOR preconditioned PCG methods can greatly improve the computation efficiency when solving the multiplicative half-quadratic regularized image restoration problem. Keywords Image restoration · Multiplicative half-quadratic regularization · Newton method · Preconditioners Mathematics Subject Classification 65C20 · 65F10 · 94A08
* Yu‑Mei Huang [email protected] Pei‑Pei Zhao [email protected] 1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China
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P.-P. Zhao, Y.-M. Huang
1 Introduction Image restoration problem exists in numerous imaging applications such as medical and astronomical imaging, film restoration, image and video coding [12] etc. Image formation process is usually mathematically formulated as:
b = Ax + 𝜂, where A ∈ ℝmn×mn is the blur matrix, x, b, 𝜂 ∈ ℝmn are vectors stacked from the m × n matrices of the original image, contaminated image and the additive zeromean Gaussian white noise column by column, respectively. The goal of image restoration is to get an estimate x̂ which is as close as possible to the original image x. Such image restoration problem usually can be solved by minimizing a cost function J ∶ ℝmn → ℝ as follows:
x̂ = argminx∈ℝmn J(x)
(1)
with
J(x) = ‖Ax − b‖22 + 𝛽𝛷(x), where 𝛽 > 0 is the regularization parameter. In the cost function J(x) , ‖Ax − b‖22 is the data-fitting term and the regularization term 𝛷 is of the form
𝛷(x) =
l ∑
𝜙(gTi x),
i=1
where 𝜙 ∶ ℝ → ℝ is a continuously differentiable function, and gTi ∶ ℝmn → ℝ (i = 1, 2, … , l) are linear operators, in particular, gTi is the first or the second-order difference operators. We use G ∈ ℝl×mn to denote the matr
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