A differential approach to shape from polarization

© 2017. The copyright of this document resides with its authors. State-of-the-art formulations of the Shape from Polarisation problem consist of several steps based on merging physical principles that prevent this problem being described by a single mathematical framework. In addition, specular and diffuse reflections need to be separately considered, making the three-dimensional shape reconstruction not easily applicable to heterogeneous scenes consisting of different materials. In this work we derive a unified specular/diffuse reflection parametrisation of the Shape from Polarisation problem based on a linear partial differential equation capable of recovering the level-set of the surface. The inherent ambiguity of the Shape from Polarization problem becomes evident through the impossibility of reconstructing the whole surface with this differential approach. To overcome this limitation, we consider shading information elegantly embedding this new formulation into a two-lights calibrated photometric stereo approach. Thus we derive an albedo independent and well-posed differential model based on a system of hyperbolic PDEs capable of reconstructing the shape with no ambiguity. We validate the geometrical properties of the new differential model for the Shape from Polarisation problem using synthetic and real data by computing the isocontours of the shape under observation. Lastly, we show the suitability of this new model to elegantly fit into a variational solver that is able to provide 3D shape reconstructions from synthetic and real data