Designing an unsupervised image denoising approach in practical applications is a challenging task due to the complicated data acquisition process. In the realworld case, the noise distribution is so complex that the simplified additive white Gaussian (AWGN) assumption rarely holds, which significantly deteriorates the Gaussian denoisers’ performance. To address this problem, we apply a deep neural network that maps the noisy image into a latent space in which the AWGN assumption holds, and thus any existing Gaussian denoiser is applicable. More specifically, the proposed neural network consists of the encoder-decoder structure and approximates the likelihood term in the Bayesian framework. Together with a Gaussian denoiser, the neural network can be trained with the input image itself and does not require any pre-training in other datasets. Extensive experiments on real-world noisy image datasets have shown that the combination of neural networks and Gaussian denoisers improves the performance of the original Gaussian denoisers by a large margin. In particular, the neural network+BM3D method significantly outperforms other unsupervised denoising approaches and is competitive with supervised networks such as DnCNN, FFDNet, and CBDNet.

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The harmonic Bergman kernel$Q$_{Ω}for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator$F$_{Ω}║$F$_{Ω}║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' ($z$)=1+ψ ($z$) where ψ(0)=0. We show that if the function ψ belongs to a space$D$_{$s$}($D$),$s$>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for$Q$_{Ω}also converges pointwise in $$\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every$s$>0 there exists a constant$C$_{$s$}>0 such that if ║ψ║_{$D$}_{s}($D$)≤$C$_{$s$}, then the biharmonic Green function for Ω=ø ($D$) is positive.

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