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This paper revisits the numerical shape-from-shading method proposed in early 1980s. The original problem is non-convex due to the unit norm constraint for surface normal, and the existing approaches including the original Ikeuchi and Horn's work uses approximate solution strategies for the original problem. This paper instead studies relaxation strategies for the original non-convex constraint and describes corresponding solution techniques that are built upon advanced convex optimization. We analyze the effect of the relaxations in terms of resulting accuracy and computational complexity.

This paper presents a photometric stereo method based on deep learning. One of the major difficulties in photometric stereo is designing an appropriate reflectance model that is both capable of representing real-world reflectances and computationally tractable for deriving surface normal. Unlike previous photometric stereo methods that rely on a simplified parametric image formation model, such as the Lambert's model, the proposed method aims at establishing a flexible mapping between complex reflectance observations and surface normal using a deep neural network. In addition, the proposed method predicts the reflectance, which allows us to understand surface materials and to render the scene under arbitrary lighting conditions. As a result, we propose a deep photometric stereo network (DPSN) that takes reflectance observations under varying light directions and infers the surface normal and reflectance in a per-pixel manner. To make the DPSN applicable to real-world scenes, a dataset of measured BRDFs (MERL BRDF dataset) has been used for training the network. Evaluation using simulation and real-world scenes shows the effectiveness of the proposed approach in estimating both surface normal and reflectances.