A new class of anisotropic diffusion models is proposed for image processing which can be viewed either as a novel kind of regularization of the classical Perona--Malik model or, as advocated by the authors, as a new independent model. The models are diffusive in nature and are characterized by the presence of both forward and backward regimes. In contrast to the Perona--Malik model, in the proposed model the backward regime is confined to a bounded region, and gradients are only allowed to grow up to a large but tunable size, thus effectively preventing indiscriminate singularity formation, i.e., staircasing. Extensive numerical experiments demonstrate that the method is a viable denoising/deblurring tool. The method is significantly faster than competing state-of-the-art methods and appears to be particularly effective for simultaneous denoising and deblurring. An application to satellite image enhancement is also presented.