Two-Level method for the total fractional-order variation model in image deblurring problem
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Two-Level method for the total fractional-order variation model in image deblurring problem Faisal Fairag1 · Adel Al-Mahdi2
· Shahbaz Ahmad1
Received: 7 September 2018 / Accepted: 5 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Image deblurring with total fractional-order variation model is used to improve the quality of the deblurred images. This model is very efficient in preserving edges and removing staircase effect. However, the regularization matrix associated with the total fractional-order model is dense which complicate developing an efficient numerical algorithm. In this research work, we present an efficient and robust TwoLevel method to overcome the dense problem. The Two-Level method started by reducing the problem to one small non-linear system with dense regularization matrix (Level-I) and one less expensive large linear system with sparse regularization matrix (Level-II). The derivation of the optimal regularization parameter of Level-II is studied and formula is presented. Numerical experiments on several images are also provided to demonstrate the efficiency of the Two-Level method. Keywords Image deblurring · Krylov subspace methods · TFOV · Two-Level method
Adel Al-Mahdi
[email protected] Faisal Fairag [email protected] Shahbaz Ahmad [email protected] 1
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
2
PYP-Math, King Fahd University of Petroleum & Minerals, Dhahran, 31261 Saudi Arabia
Numerical Algorithms
1 Introduction Important image reconstruction problems involve complicated regularizational functionals such as total fractional-order variation (TFOV). These models are very efficient in preserving edges and removing staircase effect and other nice properties. However, the associated Euler-Lagrange equations involve dense matrices which complicate developing an efficient numerical algorithm. To overcome these difficulties we introduced in this research a Two-Level method. The Two-Level discretization techniques have been used for many problems, such as for Navier-Stokes (NS) equations in [1–5] and formulation of the NS equations as a stream function in [6–10]. Other work in the Two-Level method is found in [11–14]. These methods are computationally attractive for large and ill-conditioned non-linear systems. The crux of these techniques is to deal with a small system of non-linear equations on Level-I (coarse mesh) and one linear system of equations on Level-II (fine mesh). Jintao Xu is a pioneered of Two-Level methods [11]. In this paper, we will propose and analyze a Two-Level method for the total fractional-order variation model in image deblurring problem. In the Two-Level method, at Level-I we solve a non-linear integral differential equation (image deblurring) on a coarse mesh. Also at the Level-I, we deal with the computationally expensive regularization functional (TFOV). Then we interpolate our result for a fine mesh. At Level-II, we solve a linear i
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