Summary: Nonlinear anisotropic diffusion filters have been introduced in the field of image processing for image denoising and image restoration. They are based on the solution of partial differential equations involving a nonlinear anisotropic diffusion operator. From a mathematical point of view, these filters enjoy attractive properties, such as minimum-maximum principle, and an inherent decomposition of the images in different scales. We investigate in this study how these filters can be applied to help solving data-fitting inverse problems. We focus on seismic imaging using the full waveform, a well known nonlinear instance of such inverse problems. In this context, we show how the filters can be applied directly to the solution space, to enhance the structural coherence of the parameters representing the subsurface mechanical properties and accelerate the convergence. We also show how they can be applied to the seismic data itself. In the latter case, the method results in an original low-frequency data enhancement technique making it possible to stabilize the inversion process when started from an initial model away from the basin of attraction of the global minimizer. Numerical results on a 2D realistic synthetic full waveform inversion case study illustrate the interesting properties of both approaches.