Variational formulation of a hybrid perspective shape from shading model

Abstract

Over a long period of time, it is challenging to solve the completed shape from shading (SFS) problem which has no limitations on light position and surface roughness. Many existing researches developed Hamilton–Jacobi equation (HJE)-based method of such SFS problem. However, HJE-based methods have one major weakness of boundary condition requirement. Finding a proper algorithm for calculating direct depth in a completed SFS model is desirable. In this paper, a completed hybrid perspective shape from shading model based on Cook–Torrance BRDF reflectance model is established. The depth is expressed and calculated directly on the Cartesian coordinates rather than spherical coordinates, which prevents exponential explosion risks caused by substitution of natural exponential. A direct variational formulation method is employed to solve the complex partial differential equation (PDE) of the image irradiance model. The major contribution of the method is that depth is considered as the only variable to deduce iterative equation. After discretizing the partial derivative terms of the iterative equation, the variational method could be programed conveniently and intuitively. Simulations and experiments demonstrate the accuracy and robustness of the method.