  We present an anisotropic extension of the isotropic osmosis model that has
been introduced by Weickert et al.~(Weickert, 2013) for visual computing
applications, and we adapt it specifically to shadow removal applications. We
show that in the integrable setting, linear anisotropic osmosis minimises an
energy that involves a suitable quadratic form which models local directional
structures. In our shadow removal applications we estimate the local structure
via a modified tensor voting approach (Moreno, 2012) and use this information
within an anisotropic diffusion inpainting that resembles edge-enhancing
anisotropic diffusion inpainting (Weickert, 2006, Gali\'c, 2008). Our numerical
scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau
(Fehrenbach, 2014) with an exact time stepping based on highly accurate
polynomial approximations of the matrix exponential. The resulting anisotropic
model is tested on several synthetic and natural images corrupted by constant
shadows. We show that it outperforms isotropic osmosis, since it does not
suffer from blurring artefacts at the shadow boundaries.
