The aim of this paper is to propose a novel mapping algorithm between 2D images and a 3D volume seeking simultaneously a linear plane transformation and an in-plane dense deformation. We adopt a metric free locally over-parametrized graphical model that combines linear and deformable parameters within a coupled formulation on a 5-dimensional space. Image similarity is encoded in singleton terms, while geometric linear consistency of the solution (common/single plane) and in-plane deformations smoothness are modeled in a pair-wise term. The robustness of the method and its promising results with respect to the state of the art demonstrate the extreme potential of this approach.