Weighted Nuclear Norm Minimization-Based Regularization Method for Image Restoration
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Weighted Nuclear Norm Minimization‑Based Regularization Method for Image Restoration Yu‑Mei Huang1 · Hui‑Yin Yan1,2 Received: 28 July 2019 / Revised: 25 March 2020 / Accepted: 18 May 2020 © Shanghai University 2020
Abstract Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem. Different assumptions or priors on images are applied in the construction of image regularization methods. In recent years, matrix low-rank approximation has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. Low-rank matrix minimization is an NP-hard problem and it is often replaced with the matrix’s weighted nuclear norm minimization (WNNM). The assumption that an image contains an extensive amount of self-similarity is the basis for the construction of the matrix low-rank approximation-based image denoising method. In this paper, we develop a model for image restoration using the sum of block matching matrices’ weighted nuclear norm to be the regularization term in the cost function. An alternating iterative algorithm is designed to solve the proposed model and the convergence analyses of the algorithm are also presented. Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities. Keywords Image restoration · Regularization method · Weighted nuclear norm · Alternating iterative method Mathematics Subject Classification 68U10 · 94A08
This work is supported by the National Natural Science Foundation of China nos. 11971215 and 11571156, MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China. * Yu‑Mei Huang [email protected] Hui‑Yin Yan [email protected] 1
Key Laboratory of Applied Mathematics and Complex Systems, School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, Henan Province, China
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
1 Introduction Image restoration is a fundamental problem in image processing. Images obtained from the imaging recording system are often degraded ones. Noise and blur are two factors being regarded as the major causes of image distortions. Mathematically, the recorded image g obtained through the imaging system can be modeled as
g = Hf + v,
(1)
where g, f , v ∈ ℝmn are vectors corresponding to the m-by-n observed image, original image and additive zero-mean white Gaussian noise, respectively. H ∈ ℝmn×mn is the blurring matrix representing the blurring degradation operation. If the blurring distortion does not appear in images, H will be an identity matrix and the problem becomes a pure image denoising problem. The purpose of image restoration is to invert process (1) so that an estimate of the original image f can be obtained by utilizing
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