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

Nonlocal Patches based Gaussian Mixture Model for Image Inpainting