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.

在很长的一段时间内，要解决从遮光（SFS）到完整形状的挑战，这对光位置和表面粗糙度没有限制。现有的许多研究开发了基于Hamilton-Jacobi方程（HJE）的此类SFS问题的方法。然而，基于HJE的方法具有边界条件要求的一大弱点。期望找到合适的算法来计算完整的SFS模型中的直接深度。本文基于库克-托伦斯BRDF反射模型，从阴影模型建立了完整的混合透视图形状。深度直接在直角坐标系上而不是在球面坐标系上表示和计算，从而避免了由自然指数替代所引起的指数爆炸危险。采用直接变分公式化方法来求解图像辐照度模型的复偏微分方程（PDE）。该方法的主要作用是将深度视为推导迭代方程的唯一变量。在离散迭代方程的偏导数项之后，可以方便直观地编程变分方法。仿真和实验证明了该方法的准确性和鲁棒性。