## 1 Introduction  
3D shape reconstruction from images has been an important topic in the field of robot vision. Shape-From-shading (SFS), initiated by [PERSON] [1; 2], is a classical method for determining the shape of a 3D surface from a one intensity image. He described the SFS problem by the so-called Image Irradiance Equation (IIR), which is a nonlinear Partial Differential Equation (PDE) of the first-order, and demonstrates the relationship between the 3D surface shape and its relevant 2D image intensity variations. There are primarily two steps when processing an SFS problem. The first step is that of setting up a partial differential IIR under certain assumptions. The assumptions are used to model the image formation process and are dependent on the reflection attribute of the surface, the light source, and the camera. The second step is that of building a numerical scheme to acquire a solution of the IIR, which is the 3D shape of the known intensity image. Since the pioneering work of [PERSON], the majority of the SFS works focuses on how to build a numerical scheme with the simple Lambertian model; thus, many SFS algorithms have been reported (one can see the extensive list of the references in the three surveys [3; 4; 5]). According to [PERSON] et al. [4], [PERSON] et al. [5], and [PERSON] and [PERSON] [6], these algorithms are roughly divided into two categories: PDE-based approaches and optimization approaches. The early work of [PERSON] [PERSON] (1974) based on the method of characteristics and the viscosity solution theories-based methods [PERSON] (1992); [PERSON] and [PERSON] (1992); [PERSON] (1992); [PERSON] and [PERSON] (1992); [PERSON] (1992) all belong to the first category. It is worth mentioning that [PERSON] and [PERSON] [PERSON] and [PERSON] (1992) first formulated the SFS problem as a Hamilton-Jacobi-Bellman (HJB) PDE, whose solution is obtained through the viscosity solution theories. [PERSON] and [PERSON] [PERSON] and [PERSON] (1992) converted the partial differential IIR with Lambertian reflection model into an eikonal PDE. They adopted the fast-marching method [PERSON] and [PERSON] (1992) to compute its viscosity solution. These approaches [PERSON] (1974); [PERSON] and [PERSON] (1992); [PERSON] and [PERSON] (1992) are based on an orthographic camera projection. For a perspective camera projection, [PERSON] and [PERSON] [PERSON] and [PERSON] (2000) associated the partial differential IIR with a Hamiltonian and approximated its viscosity solution using optimal control strategy on the basis of the work [PERSON] and [PERSON] (1992). [PERSON] et al. [PERSON] et al. (2000); [PERSON] et al. (2000) studied analytically and numerically the perspective SFS model introduced by [PERSON] and [PERSON] [PERSON] and [PERSON] (2000) and the related Hamilton-Jacobi (HJ) PDE. Also, they proved the convergence of a finite-difference scheme and a semi-Lagrangian scheme for the HJ PDE. The second category includes the minimization methods for the variational problem. [PERSON] and [PERSON] [PERSON] and [PERSON] (2001) presented an iterative SFS method under the perspective camera projection. They approximated a smooth surface by the union of triangular surface patches which involved only the height variables. The shape was recovered by linearizing the reflectance map and minimizing a quadratic cost functional. [PERSON] et al. [PERSON] et al. (2000) introduced some form of regularization, by formulating a PDE-based weak perspective or perspective IIR in terms of a spline basis. We want to mention the work by [PERSON] et al. [PERSON] et al. [PERSON] et al. (2001), who combined the advantages of PDE-based approaches and optimization approaches, and created a generic variational solution based on PDEs that is suitable for SFS under natural illumination and can handle a variety of scenarios for the camera and the lighting. Finally, in addition to these work, we should also mention that a latest work by [PERSON] et al. [PERSON] et al. (2001), who gave a novel graph theoretical framework built on cycle bases.