We consider the inpainting problem for noisy images. It is very challenge to suppress noise when image inpainting is processed. An image patches based nonlocal variational method is proposed to simultaneously inpainting and denoising in this paper. Our approach is developed on an assumption that the small image patches should be obeyed a distribution which can be described by a high dimension Gaussian Mixture Model. By a maximum a posteriori (MAP) estimation, we formulate a new regularization term according to the log-likelihood function of the mixture model. To optimize this regularization term efficiently, we adopt the idea of the Expectation Maximum (EM) algorithm. In which, the expectation step can give an adaptive weighting function which can be regarded as a nonlocal connections among pixels. Using this fact, we built a framework for non-local image inpainting under noise. Moreover, we mathematically prove the existence of minimizer for the proposed inpainting model. By using a spitting algorithm, the proposed model are able to realize image inpainting and denoising simultaneously. Numerical results show that the proposed method can produce impressive reconstructed results when the inpainting region is rather large.

中文翻译：

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用于图像修复的基于非局部补丁的高斯混合模型

我们考虑噪声图像的修复问题。在处理图像修复时抑制噪声是非常具有挑战性的。本文提出了一种基于图像块的非局部变分方法同时进行修复和去噪。我们的方法是在假设小图像块应该服从可以由高维高斯混合模型描述的分布的基础上开发的。通过最大后验（MAP）估计，我们根据混合模型的对数似然函数制定新的正则化项。为了有效地优化这个正则化项，我们采用了期望最大值 (EM) 算法的思想。其中，期望步骤可以给出一个自适应权重函数，可以看作是像素之间的非局部连接。利用这个事实，我们为噪声下的非局部图像修复构建了一个框架。此外，我们在数学上证明了所提出的修复模型的最小化器的存在。通过使用吐出算法，所提出的模型能够同时实现图像修复和去噪。数值结果表明，当修复区域相当大时，所提出的方法可以产生令人印象深刻的重建结果。