Anisotropic osmosis filtering for shadow removal in images

Abstract

We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al. [38] 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 [24] and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting [39,13]. Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau [10] 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.

Variational Osmosis for Non-Linear Image Fusion

“…Image osmosis: A more flexible model allowing for multiplicative image variations and applicable to many more problems (shadow removal, image decompression and many more) is the parabolic image osmosis model originally proposed in [33] and later considered in few other works, e.g. [7], [27], [32]. Given a final stopping time T > 0, two positive images u 0 , v : Ω → R + , a so called drift vector field defined as…”

“…and when W = I, the diffusivity tensor is non-constant and favours anisotropic diffusion and transport, making it amenable for directional image completion, i.e. inpainting, see [27].…”

“…Furthermore, the image osmosis model shows high flexibility when applied to other type of problems such as image compression and shadow removal, and is amenable for efficient numerical schemes guaranteeing mass and positivity preservation of the solution, see [7], [32]. In order to overcome the Laplace smoothing of the osmosis model when applied to this latter problem, recently, an anisotropic variant of the osmosis model driving image diffusion along coherent directions has been considered in [27]. In this model the two different steps, reference drift estimation and image reconstruction, are performed in a disjoint way.…”

“…From a numerical point of view, explicit, implicit [32], and semi-implicit schemes based on the dimensional decomposition of the space-discretisation operators [7] have been considered for the efficient computation of the solution of the isotropic model. In [27], suitable anisotropic stencils and efficient time-integration schemes have been introduced to further provide a discrete theory for the anisotropic case, exploiting ideas of Lattice Basis Reduction (LBR) models proposed in [13] for pure diffusion processes and adapted to deal with the convection part as well. For all these schemes, the discrete conservation of mass and non-negativity properties, along with convergence results, were proved.…”

“…for u 0 ∈ R S + being the discretised version of u 0 (x). Note that in [27] the anisotropic tensor W (6) is computed with respect to the given image u 0 only, thus making the model linear. Local coherence direction estimation is performed in [27] in terms of the multi-scale tensor voting technique proposed in [24] to make structure tensor approaches more robust.…”