  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.
