A first-order image denoising model for staircase reduction
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A first-order image denoising model for staircase reduction Wei Zhu1 Received: 16 October 2018 / Accepted: 23 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we consider a total variation–based image denoising model that is able to alleviate the well-known staircasing phenomenon possessed by the Rudin-OsherFatemi model (Rudin et al., Phys. D 60, 259–268, 1992). To minimize this variational model, we employ augmented Lagrangian method (ALM). Convergence analysis is established for the proposed algorithm. Numerical experiments are presented to demonstrate the features of the proposed model and also show the efficiency of the proposed numerical method. Keywords Image denoising · Augmented Lagrangian method · Variational model Mathematics Subject Classification (2010) 94A08 · 65K10 · 65M32
1 Introduction Image denoising aims to remove the noise part of a given image in order to get a clean one inside which meaningful signals should be largely retained. This is a typical inverse problem, and to treat such a problem, an appropriate regularizer should often be employed. In the literature, one of the most famous variational models for this problem was given by Rudin, Osher, and Fatemi (ROF) [30], where the total variation was used as the regularizer. The appealing feature of this regularizer lies in its ability of allowing discontinuous solution of the variational model, and thus object boundary can be captured. Ever since this remarkable work, the total variation–based regularizer has been adopted to dealing with different imaging tasks [8, 14, 15, 17, 19]. Moreover, numerous variational models have been proposed by using different
Communicated by: Raymond H. Chan Wei Zhu
[email protected] 1
Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487, USA
W. Zhu
forms of regularizers in the field of mathematical imaging [3, 16, 18, 23, 24, 27, 28]. Despite its effectiveness of removing noise while keeping object boundaries, the ROF model bears several undesirable properties. For instance, in [25], Meyer pointed out that the ROF model cannot preserve image contrast and object corners, and also in [7], Bellettini, Caselles, and Novaga studied what kind of shapes can be maintained by the ROF model, which suggests that the ROF model will smear object corners. More importantly, the ROF model also suffers from the staircase effect, that is, the denoised image presents blocky pieces or piecewise constant regions, even for originally smooth parts inside the given image, which surely leads to visually unpleasant denoised results. To fix the staircasing phemomenon, quite a few of higher order variational models have been developed [9, 18, 23, 31, 35], which is due to the fact that piecewise constant functions could lead to large magnitude of their second-order derivatives. Precisely, in [9], Bredies et al. developed a novel term called total generalized variation as a regularizer for dealing with inverse problems in mathematical imaging, includi
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