Fast and stable schemes for non-linear osmosis filtering

We consider a non-linear variant of the transport-diffusion osmosis model for solving a variety of imaging problems such as shadow/soft-light removal and compact data representation. The non-linear behaviour is encoded in terms of a general scalar diffusivity function with suitable properties, which allows to balance the diffusion intensity over different regions of the image while preventing smoothing artefacts. For the proposed model, conservation properties (average intensity and non-negativity) are proved and a variational interpretation is showed for specific choices of the diffusivity function. Upon suitable spatial discretisation, both an explicit and a semi-implicit iterative schemes are considered, for which convergence conditions and unconditional stability results are proved, respectively. To validate the proposed modelling and the computational speed of the numerical schemes considered, we report several results and comparisons with state-of-the-art methods, showing that artefact-free and computationally efficient results are obtained in comparison to standard linear and anisotropic osmosis models.