A fixed-point algorithm for second-order total variation models in image denoising
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A fixed-point algorithm for second-order total variation models in image denoising Tianling Gao1 · Xiaofei Wang1 · Qiang Liu1
· Zhiguang Zhang1
Received: 10 September 2018 / Accepted: 8 November 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we construct fixed-point algorithms for the second-order total variation models through discretization models and the subdifferential and proximity operators. Particularly, we focus on the convergence conditions of our algorithms by analyzing the eigenvalues of the difference matrix. The algorithms are tested on various images to verify our proposed convergence conditions. The experiments compared with the split Bregman algorithms demonstrate that fixed-point algorithms could solve the second-order functional minimization problem stably and effectively. Keywords Fixed-point algorithm · Convergence · High-order total variation · Image denoising Mathematics Subject Classification 68U10 · 65K10
1 Introduction Image denoising is an important work throughout image processing, whose main goal is to recover a true image (the unknown) from an observation (the data). Specifically, let be a rectangular domain in R2 and u be a true image defined on , which assumes that u has been corrupted by some additive Gaussian white noise η. Then, the image denoising problem is to recover u from the data f = u + η. Classically, it was usually understood as a typical example of an inverse problem, and was attacked by solving a related variational problem by regularization methods. To remove noise while preserving edges, Rudin et al. (1992) first proposed a framework for image denoising based on total variation. Specially, the image denoising problem is originally formulated to the following optimization problem (the ROF model):
Supported by NSF of China (11431015), NSF of Guangdong (2016A030313048), ministry of education in Guangdong for excellent young teachers.
B 1
Qiang Liu [email protected] College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, People’s Republic of China
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T. Gao et al.
1 min Er o f (u) = u 2
|u − f | dxdy + μ 2
(u 2x
1 + u 2y ) 2
,
(1.1)
where μ > 0 is a tuning parameter. The first term is a fidelity term which lets the recovered image u be close to the given data f . The second term is the total variation of u, a regularization term which allows for discontinuities and eliminates oscillations. Informally, when μ is large, the recovered image u will be more regular and approximate to a piecewise constant function. Furthermore, when μ is small enough, the recovered image u will hold more information about f . The ROF model may be the most successful variational model on image denoising. At the same time, it was also extended to the other extensive image processing tasks, such as image inpainting and image classification. While the ROF model suffers the staircasing faults, it results in denoising image with many blocks Chan et al. (2005). In the last decade, many new mo
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