【图像恢复】基于交替乘子方法(ADMM)图像恢复算法研究[固定点收敛和应用]（Matlab代码实现）

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📋📋📋本文目录如下：🎁🎁🎁

目录

💥1 概述

📚2 运行结果

🎉3 参考文献

🌈4 Matlab代码、数据、文献

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💥1 概述

文献来源：

即插即用的ADMM图像恢复算法：固定点收敛和应用

交替方向乘子方法（ADMM）是一种广泛用于解决图像恢复中约束优化问题的算法。在许多有用的特性中，ADMM算法的一个关键特点是其模块化结构，允许将任何现成的图像去噪算法插入到ADMM算法的子问题中。由于其插件式的特性，这种类型的ADMM算法被称为“即插即用ADMM”。即插即用ADMM在最近的一些论文中展示了有希望的实证结果。然而，目前尚不清楚在什么条件下以及使用哪种去噪算法才能保证其收敛性。此外，由于即插即用ADMM使用了特定的变量分割方式，对于常见的高斯和泊松图像恢复问题是否可以进行快速实现也不清楚。本文提出了一种具有可证明的固定点收敛性的即插即用ADMM算法。我们证明了对于满足一种被称为有界去噪器的渐近准则的任何去噪算法，即插即用ADMM在连续方案下收敛到一个固定点。我们还针对超分辨率和单光子成像两个图像恢复问题提出了快速实现。我们将即插即用ADMM与各类最先进的算法进行了比较，并展示了算法的有希望的实验结果。

原文摘要：

Abstract:

Alternating direction method of multiplier (ADMM) is a widely used algorithm for solving constrained optimization problems in image restoration. Among many useful features, one critical feature of the ADMM algorithm is its modular structure, which allows one to plug in any off-the-shelf image denoising algorithm for a subproblem in the ADMM algorithm. Because of the plug-in nature, this type of ADMM algorithms is coined the name “Plug-and-Play ADMM.” Plug-and-Play ADMM has demonstrated promising empirical results in a number of recent papers. However, it is unclear under what conditions and by using what denoising algorithms would it guarantee convergence. Also, since Plug-and-Play ADMM uses a specific way to split the variables, it is unclear if fast implementation can be made for common Gaussian and Poissonian image restoration problems. In this paper, we propose a Plug-and-Play ADMM algorithm with provable fixed-point convergence. We show that for any denoising algorithm satisfying an asymptotic criteria, called bounded denoisers, Plug-and-Play ADMM converges to a fixed point under a continuation scheme. We also present fast implementations for two image restoration problems on superresolution and single-photon imaging. We compare Plug-and-Play ADMM with state-of-the-art algorithms in each problem type and demonstrate promising experimental results of the algorithm.

📚2 运行结果

部分代码：

%add path to denoisers
addpath(genpath('./denoisers/BM3D/'));
addpath(genpath('./denoisers/TV/'));
addpath(genpath('./denoisers/NLM/'));
addpath(genpath('./denoisers/RF/'));

%read test image
z = im2double(imread('./data/House256.png'));

%construct A matrix, deblurring as an example
dim = size(z);
h = fspecial('gaussian',[9 9],1);
A = @(z,trans_flag) afun(z,trans_flag,h,dim);

%reset random number generator
rng(0);

%set noies level
noise_level = 10/255;

%calculate observed image
y = A(z(:),'transp') + noise_level*randn(prod(dim),1);
y = proj(y,[0,1]);
y = reshape(y,dim);

🎉3 参考文献

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🌈4 Matlab代码、数据、文献