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Proximity algorithms for image models: denoising. (English) Zbl 1216.94015
Summary: This paper introduces a novel framework for the study of the total-variation model for image denoising. In the model, the denoised image is the proximity operator of the total-variation evaluated at a given noisy image. The total-variation can be viewed as the composition of a convex function (the $\ell^1$ norm for the anisotropic total-variation or the $\ell^2$ norm for the isotropic total-variation) with a linear transformation (the first-order difference operator). These two facts lead us to investigate the proximity operator of the composition of a convex function with a linear transformation. Under the assumption that the proximity operator of a given convex function (e.g., the $\ell^1$ norm or the $\ell^2$ norm) can be readily obtained, we propose a fixed-point algorithm for computing the proximity operator of the composition of the convex function with a linear transformation. We then specialize this fixed-point methodology to the total-variation denoising models. The resulting algorithms are compared with the Goldstein-Osher split-Bregman denoising algorithm. An important advantage of the fixed-point framework leads us to a convenient analysis for convergence of the proposed algorithms as well as a platform for us to develop efficient numerical algorithms via various fixed-point iterations. Our numerical experience indicates that the methods proposed here perform favorably.

##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) 68T45 Machine vision and scene understanding
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