Title: Optimal denoising of geometrically regular images with scattering coefficients

Abstract: Optimal suppression of additive Gaussian white noise has many image processing applications and is a key step to generate images with score diffusion algorithms. For images with edges that are piecewise regular, nearly optimal denoisers can be computed by thresholding a sparse representation in a dictionary of curvelets or bandlets. It requires to adapt the support of selected dictionary vectors to geometric image properties. In contrast, convolutional deep neural networks can implement optimal denoising algorithms with a cascade of fixed support filters and non-linearities. It also applies to much more complex images. An outstanding issue is to understand how convolutional neural networks can reach optimal results, without adapting their filter support. We show that a two layer network with fixed wavelet filters can disentangle geometric support constraints. This scattering transform explains why filter supports do not need to be adapted to the image geometry. For piecewise regular images having piecewise regular edges, numerical results are asymptotically optimal. This is supported by a characterisation of the scattering coefficient decay, but this denoising optimality remains a mathematical conjecture.