A Fast Algorithm for a Mean Curvature Based Image Denoising Model Using Augmented Lagrangian Method
Recently, many variational models using high order derivatives have been proposed to accomplish advanced tasks in image processing. Even though these models are effective in fulfilling those tasks, it is very challenging to minimize the associated high or
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Department of Mathematics, University of Alabama, 870350, Tuscaloosa, AL 35487, USA [email protected] Department of Mathematics, University of Bergen, 5007, Bergen, Norway [email protected] Office of the President, Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowlon, Hong Kong [email protected]
Abstract. Recently, many variational models using high order derivatives have been proposed to accomplish advanced tasks in image processing. Even though these models are effective in fulfilling those tasks, it is very challenging to minimize the associated high order functionals. In [33], we focused on a recently proposed mean curvature based image denoising model and developed an efficient algorithm to minimize it using augmented Lagrangian method, where minimizers of the original high order functional can be obtained by solving several low order functionals. Specifically, these low order functionals either have closed form solutions or can be solved using FFT. Since FFT yields exact solutions to the associated equations, in this work, we consider to use only approximations to replace these exact solutions in order to reduce the computational cost. We thus employ the Gauss-Seidel method to solve those equations and observe that the new strategy produces almost the same results as the previous one but needs less computational time, and the reduction of the computational time becomes salient for images of large sizes.
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Introduction
Image denoising is to remove noise while keeping meaningful vision information such as object edges and boundaries. It is a crucial step in image processing with a wide range of applications in medical image analysis, video monitoring, and others. During the last three decades, numerous models have been proposed to deal with this problem [3,4,7,19–21,23–25,28]. One of the most popular variational models was proposed by Rudin, Osher, and Fatemi in their seminal work (ROF model) [25], where the cleaned image corresponds to the minimizer of the following functional |∇u| + (f − u)2 , (1) E(u) = λ Ω
Ω
A. Bruhn et al. (Eds.): Global Optimization Methods, LNCS 8293, pp. 104–118, 2014. c Springer-Verlag Berlin Heidelberg 2014 DOI: 10.1007/978-3-642-54774-4 5,
A Fast Algorithm for a Mean Curvature Based Image Denoising Model
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where f : Ω → R is a given noisy image defined on Ω (always a rectangle in R2 ) and λ > 0 is a positive tuning parameter controlling how much noise will be removed. The remarkable feature of the ROF model lies in its effectiveness in preserving object edges while removing noise. This is due to the total variation based regularizer. In fact, the total variation has been widely employed in accomplishing other image tasks such as deblurring, segmentation, and registration. However, as pointed out in [6], the ROF model has several unfavorable features. The main caveat is the stair case effect, that is, the resulting clean image would present blocks even though the desired image could be smooth, such as human face. Other undesirable properties includ
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