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We present an approach for image denoising based on the analysis of the local Holder regularity. The method takes the point of view that denoising may be performed by increasing the Holder regularity at each point. Under the assumption that the noise is additive and white, our procedure is asymptotically minimax, provided the original signal belongs to a ball in some Besov space. Such a scheme is well adapted to the case where the image to be recovered is itself very irregular, e.g. nowhere differentiable with rapidly varying local regularity. The method is implemented through a wavelet analysis. We show an application to SAR image denoising where this technique yields good results compared to other algorithms.