  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 function g with suitable properties, which allows to balance the
diffusion intensity on the different regions of the image while preventing
smoothing artefacts. For the proposed model, conservation properties (intensity
and non-negativity) are proved and a variational interpretation is showed for
specific choices of g. Upon suitable spatial discretisation, both an explicit
and a semi-implicit iterative scheme are considered, for which convergence
restrictions and unconditional stability are proved, respectively. To validate
the proposed modelling and the computational speed of the numerical schemes
considered, we report several results and comparisons for the problem of
shadow/light-spot removal and compact data representation, showing that
artefact-free and computationally efficient results are obtained in comparison
to standard linear and anisotropic models, and state-of-the art approaches.
