Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms
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| Format: | Preprint |
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2024
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| _version_ | 1866915466357768192 |
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| author | Kowalski, Matthieu Malézieux, Benoît Moreau, Thomas Repetti, Audrey |
| author_facet | Kowalski, Matthieu Malézieux, Benoît Moreau, Thomas Repetti, Audrey |
| contents | In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13276 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms Kowalski, Matthieu Malézieux, Benoît Moreau, Thomas Repetti, Audrey Optimization and Control Computer Vision and Pattern Recognition Image and Video Processing Signal Processing 90C59, 65K10, 68T07, 68U10, 94A08 In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework. |
| title | Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms |
| topic | Optimization and Control Computer Vision and Pattern Recognition Image and Video Processing Signal Processing 90C59, 65K10, 68T07, 68U10, 94A08 |
| url | https://arxiv.org/abs/2411.13276 |