This thesis deals with the elliptic osmosis equation and several of its applications. One application in particular is the texturation of 3D models with multi-date satellite images.

The osmosis equation is similar to Poisson equation but with an illumination-invariant data term. It was first introduced for image editing in a parabolic formulation for a domain with Neumann boundary conditions.

We propose an elliptic formulation of the model and several of its associated boundary-value problems: Dirichlet, Neumann and mixed boundary-value conditions. We prove the existence and uniqueness of solutions to these problems for regular domains in the continuous and discrete case. For the discrete case we propose a graph formulation that allows more flexibility for the manipulation of the different boundary conditions. We extend these results to the case of manifolds with one local chart and triangular meshes. This model is then applied as a tool for seamless cloning, shadow removal and the fusion of more than two images.

These applications illustrate the utility of having an illumination invariant term. They also illustrate the appeal of mixed boundary conditions for some applications.

We also present more concrete applications. In particular we show how to digitally restore with censored medieval illuminations using infrared reflectogram. We propose a pipeline to texture a given 3D model from multi-date satellite images. We automatically detect the shadows, distinguishing the cast and attached shadows. The final texture is a PDE based fusion of the satellite images weighted by the presence of shadows and the orientation of the satellite sensor.