"Project FLEXABLE lies in the field of 3D Computer Vision, which seeks to recover depth or the 3D shape of the observed environment from images. One of the most successful and mature techniques in 3D Computer Vision is Shape-from-Motion which is based on the well-established theory of Multiple-View Geometry. This uses multiple images and assumes that the environment is rigid.The world is however made of objects which move and undergo deformations. Researchers have tried to extend Shape-from-Motion to a deformable environment for about a decade, yet with only very limited success to date. We believe that there are two main reasons for this. Firstly there is still a lack of a solid theory for Deformable Shape-from-Motion. Fundamental questions, such as what kinds of deformation can facilitate unambiguous 3D reconstruction, are not yet answered. Secondly practical solutions have not yet come about: for accurate and dense 3D shape results, the Motion cue must be combined with other visual cues, since it is certainly weaker in the deformable case. It may require strong object-specific priors, needing one to bridge the gap with object recognition.This project develops these two key areas. It includes three main lines of research: theory, its computational implementation, and its real-world application. Deformable Multiple-View Geometry will generalize the existing rigid theory and will provide researchers with a rigorous mathematical framework that underpins the use of Motion as a proper visual cue for Deformable 3D Reconstruction. Our theory will require us to introduce new mathematical tools from differentiable projective manifolds. Our implementation will study and develop new computational means for solving the difficult inverse problems formulated in our theory. Finally, we will develop cutting-edge applications of our framework specific to Minimally Invasive Surgery, for which there is a very high need for 3D computer vision."