In this work, we show that neural networks aimed at solving various image restoration tasks can be successfully trained on fully synthetic data. In order to do so, we rely on a generative model of images, the scaling dead leaves model, which is obtained by superimposing disks whose size distribution is scale-invariant. Pairs of clean and corrupted synthetic images can then be obtained by a careful simulation of the degradation process. We show on various restoration tasks that such a synthetic training yields results that are only slightly inferior to those obtained when the training is performed on large natural image databases. This implies that, for restoration tasks, the geometric contents of natural images can be nailed down to only a simple generative model and a few parameters. This prior can then be used to train neural networks for specific modality, without having to rely on demanding campaigns of natural images acquisition. We demonstrate the feasibility of this approach on difficult restoration tasks, including the denoising of smartphone RAW images and the full development of low-light images.