181 research outputs found

    Generalized Forward-Backward Splitting

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    This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form F+i=1nGiF + \sum_{i=1}^n G_i, where FF has a Lipschitz-continuous gradient and the GiG_i's are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than n=1n = 1 non-smooth function, our method generalizes it to the case of arbitrary nn. Our method makes an explicit use of the regularity of FF in the forward step, and the proximity operators of the GiG_i's are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of FF. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

    First order algorithms in variational image processing

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    Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form D(Ku)+αR(u)minu{\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u ; where the functional D{\cal D} is a data fidelity term also depending on some input data ff and measuring the deviation of KuKu from such and R{\cal R} is a regularization functional. Moreover KK is a (often linear) forward operator modeling the dependence of data on an underlying image, and α\alpha is a positive regularization parameter. While D{\cal D} is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or 1\ell_1-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
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