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Shape-from-shading (SFS) is an important method to reconstruct three-dimensional (3D) shape of a surface in photometry and computer vision. Lambertian surface reflectance and orthographic camera projection are two fundamental assumptions which generally result in undesirable reconstructed results since inaccurate imaging model is adopted. In this paper, we propose a new fast 3D shape reconstruction approach  via  the SFS method relaxing the two assumptions. To this end, the Oren-Nayar reflectance and perspective projection models are used to establish an image irradiance equation which depicts the relationship between the 3D shape of non-Lambertian surfaces and its corresponding two-dimensional (2D) shading image. Considering the light attenuation of the near point source, the image irradiance equation is transformed into a static Hamilton-Jacobi partial differential equation (PDE) by solving a quadratic equation. The viscosity solution of the resultant Hamilton-Jacobi PDE is approximated by using optimal control theory and iterative fast marching method starting from a viscosity supersolution. The performance of the proposed approach is evaluated on both synthetic and real-world images and the experimental results demonstrate that the proposed approach is accurate and fast.

3D Reconstruction of Non-Lambertian Surfaces by Perspective Shape-From-Shading With Fast Viscosity Solution