Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees

J Sci Comput (2018) 74:743–766 https://doi.org/10.1007/s10915-017-0460-5 Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees Jin-Jin Mei1,2 · Yiqiu Dong2 Wotao Yin3 · Ting-Zhu Huang1 · Received: 12 July 2016 / Revised: 10 May 2017 / Accepted: 17 May 2017 / Published online: 30 May 2017 © The Author(s) 2017. This article is an open access publication Abstract Image restoration is one of the essential tasks in image processing. In order to restore images from blurs and noise while also preserving their edges, one often applies total variation (TV) minimization. Cauchy noise, which frequently appears in engineering applications, is a kind of impulsive and non-Gaussian noise. Removing Cauchy noise can be achieved by solving a nonconvex TV minimization problem, which is difficult due to its nonconvexity and nonsmoothness. In this paper, we adapt recent results in the literature and develop a specific alternating direction method of multiplier to solve this problem. Theoretically, we establish the convergence of our method to a stationary point. Experimental results demonstrate that the proposed method is competitive with other methods in visual and quantitative measures. In particular, our method achieves higher PSNRs for 0.5 dB on average. This research was supported by 973 Program (2013CB329404), NSFC (61370147, 61402082, 11401081). Y. Dong: The work was supported by Advanced Grant 291405 from the European Research Council. W. Yin: The work was supported by NSF Grant ECCS-1462398 and ONR Grant N000141410683. B Yiqiu Dong Jin-Jin Mei Ting-Zhu Huang Wotao Yin 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China 2 Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark 3 Department of Mathematics, University of California, Los Angeles, CA 90025, USA 123 744 J Sci Comput (2018) 74:743–766 Keywords Nonconvex variational model · Image restoration · Total variation · Alternating direction method of multiplier · Kurdyka–Łojasiewicz 1 Introduction In many imaging applications, images inevitably contain natural non-Gaussian noises, such as impulse noise, Poisson noise, multiplicative noise, and Cauchy noise. At the same time, the images may have been blurred by the point spread function (PSF) during their acquisition. Therefore, the image restoration problem is an essential task. Researchers have proposed many methods to deblur and denoise images; see [12,16,17,27,35,36,41,54] and references therein. In this paper, we focus on recovering images corrupted by blurring and Cauchy noise. Cauchy noise usually arises in echo of radar, in the presence of low-frequency atmospheric noise, and in underwater acoustic signals [26,31,40]. According to [44,45], it follows Cauchy distribution and is impulsive. We assume that the original gray-scale image u is defined on a connected bounded domain  ⊂ R2 with a compacted Lipschitz boundary. The observed image with blurs and Cauchy noise is given as follows: f = K u + η, (1) where f ∈ L 2 () denotes the observed image, K ∈ L(L 1 (), L 2 ()) represents a known linear and continuous blurring (or convolution) operator, and η ∈ L 2 () denotes Cauchy noise. Our goal is to recover u from the observed image f . In recent years, much attention has been given to Cauchy noise removal, and several methods have been proposed. In [13], the authors applied a recursive algorithm based on the Markov random field to reconstruct images and retain sharp edges. In 2005, Achim and Kuruoǧlu utilized a bivariate maximum a posteriori estimator (BMAP) to propose a new statistical model in the complex wavelet domain for removing Cauchy noise [1]. In [34], Loza et al. proposed a statistical approach based on non-Gaussian distributions in the wavelet domain for tackling the image fusion problem. Their method achieved a significant improvement in fusion quality and noise reduction. In [46], Wan et al. developed a novel segmentation method for RGB images that are corrupted by Cauchy noise. They combined statistical methods with denoising techniques and obtained a satisfactory performance. Since TV regularization is able to preserve edges effectively while still suppressing noise satisfactorily [21], Sciacchitano et al. proposed a convex TV-based variational method for recovering images corrupted by Cauchy noise in [42]. The variational model in this method is as follows:      λ min |Du| + log γ 2 + (u − f )2 d x + αu − ũ22 , (2) u∈BV ()  2  where γ > 0 is the scale parameter of Cauchy distribution, and BV () is the space of functions of bounded variation. Here, u ∈ BV () if u ∈ L 1 () and its total variation (TV)     ∞ 2 |Du|  sup u divv d x : v ∈ C0 () , v∞ ≤ 1  (C0∞ ())2  is finite, where is the space of vector-valued functions with compact support in . The space BV () endowed with the norm u BV () = u L 1 () +  |Du| is a Banach space; see, e.g., [21]. In (2), λ denotes the positive regularization parameter, which controls the trade-off between TV regularization and the fitting to f and ũ, ũ is the result obtained by the median filter, and α is a positive penalty parameter. Note that if 8αγ 2 ≥ 1, the objective 123 J Sci Comput (2018) 74:743–766 745 functional in (2) is strictly convex and leads to a unique solution. Because of strict convexity, the model avoids the common issues of nonconvex optimization: the solutions depend on the numerical methods and how they are initialized. But the last term in (2) in fact pushes the solution close to the median filter result, and the median filter does not always provide satisfactory removals of Cauchy noise. Hence, in this paper we turn our focus back to a nonconvex model. Recently, researchers have discovered some useful convergence properties of the optimization algorithms for solving nonconvex minimization problems [24,47,48,53]. In particular, the paper [48] established the global convergence (to a stationary point) of the alternating direction method of multipliers (ADMM) for nonconvex nonsmooth optimization with linear constraints. To take advantages of the recent results, in this paper we develop the ADMM algorithm to solve the following nonconvex variational model directly for denoising and deblurring simultaneously:     λ min |Du| + log γ 2 + (K u − f )2 d x. (3) u∈BV ()  2  We prove that our algorithm starting from any initialization is globally convergent to a stationary point under certain conditions. Furthermore, we compare our proposed method to the state-of-the-art method proposed in [42] and show the effectiveness of our method in terms of restoration quality and noise reduction. The outline of the paper is summarized as follows. In the next section, we analyse some fundamental properties of Gaussian distribution, Laplace distribution and Cauchy distribution. In Sect. 3, we illustra (...truncated)