ACCELERATED PROXIMAL GRADIENT ALGORITHM FOR FRAME BASED IMAGE RESTORATIONS Zouwei Shen 1 , Kim-Chuan Toh 1 and Sangwoon Yun 2 1) Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, SINGAPORE 2) Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, KOREA Corresponding Author : Sangwoon Yun,
[email protected] ABSTRACT Frame-based image restoration have been developed over the last decade. Many recently developed algorithms for image restoration can be viewed as an acceleration of the proximal forward-backward splitting algorithm. Accelerated proximal gradient algorithms studied by Nesterov, Nemirovski, and others have been demonstrated as efficient methods to solve various regularized convex optimization problems arising in compressive sensing, machine learning, and control. In this paper, we adapt the accelerated proximal gradient algorithm to solve the balanced approach model, in frame-based image restoration, which can√be formulated as the `1 -regularized least squares problem. This algorithm terminates in O(L/ ) iterations with an -optimal solution and Lipschitz constant L, and gives a set of new frame-based image restoration algorithms that can cover several topics in image restorations, such as image deblurring, denoising, inpainting, and cartoon-texture image decomposition. The numerical results suggest that our algorithm is efficient and robust in solving large-scale image restoration problems. The algorithms we implemented are able to restore 512 × 512 images in various image restoration problems in less than 50 seconds on a modest PC. We also compare the numerical performance of our proposed algorithms applied to image problems by using one frame-based system with that by using cartoon and texture systems for image deblurring, denoising, and inpainting. REFERENCES 1.
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