A Single-Lobe Photometric Stereo Approach for Heterogeneous Material

Shape from shading with multiple light sources is an active research area and a diverse range of approaches have been proposed in the last decades. However, devising a robust reconstruction technique still remains a challenging goal as the image acquisition process is highly non-linear. Recent Photometric Stereo variants rely on simplifying assumptions in order to make the problem solvable: light propagation is still commonly assumed to be uniform and the Bidirectional Reflectance Distribution Function is assumed to be diffuse, with limited interest for specular materials. In this work, we introduce a well-posed formulation based on Partial Differential Equations for a unified reflectance function that can model both diffuse and specular reflections. We base our derivation on ratios of images, which make the model independent from photometric invariants and yield a well-posed differential problem based on a system of quasi-linear PDEs with discontinuous coefficients. In addition, we directly solve a differential problem for the unknown depth thus avoiding the intermediate step of approximating the normal field. A variational approach is presented ensuring robustness to noise and outliers (such as black shadows) and this is confirmed with a wide range of experiments on both synthetic and real data, where we compare favorably to the state of the art.