## 2 IIR for Oren-Nayar Reflectance  
Under the basis that the image plane of the camera is an \(x-y\) plane and the camera's optical axis is a \(z\)-axis, the SFS problem is able to be expressed as reconstructing a 3D surface, \(z(x,y)\), which satisfies the partial differential IIR [1]:  
\[I(x,y)=R(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y})=R(p(x,y),q(x,y)), \tag{1}\]  
where \(I(x,y)\) is the image irradiance that is considered to be equal to the image intensity, \(p(x,y)\) and \(q(x,y)\) respectively stand for the first partial derivatives of the surface \(z(x,y)\) with respect to \(x\) and \(y\), and \(R\) is the reflectance map derived from the reflection model. Assuming that the camera performs an orthographic projection of a surface which is irradiated by a single point light source placed distantly from the surface, we can adopt the formulation as follows to describe the normal vector \(\mathbf{n}\) at a 3D surface point \((x,y,z(x,y))\),  
\[\mathbf{n}=[p,q,-1]. \tag{2}\]  
The reflected radiance of an ideal diffuse surface obeys [PERSON]'s law, which can be expressed by  
\[L_{r}(\theta_{i})=\frac{\rho}{\pi}I_{0}\cos\theta_{i}, \tag{3}\]  
where \(\theta_{i}\), as illustrated in Figure 1, is the angle between the normal vector \(\mathbf{n}\), and the incident direction of the point light source \(\mathbf{L}\), \(\rho\) is the albedo of diffuse surfaces that is constant in this paper, and \(I_{0}\) is the radiant intensity of the light source. However, for a typical real diffuse surface, the Lambertian reflection model is tested to be an inexact approximation of the diffuse component of the surface reflection [17; 25]. To eliminate the inexactness for diffuse reflection deriving from the assumption of the Lambertian model, [PERSON] and [PERSON] [17] produced an advanced reflection model for rough diffuse surfaces where the roughness model has been used. The roughness model considers that the surface is constituted by extended symmetric V-shaped cavities, where every V-cavity consists of two planar facets which are considered to obey the Lambert's law. The roughness of the surface is determined by a probability function of the planar facet orientations. Moreover, using a Gaussian distribution, with the reflection geometry shown in Figure 1, a formulation for the surface reflected radiance \(L_{r}\) is addressed:  
\[L_{r}(\varphi_{i},\theta_{i};\varphi_{r},\theta_{i};\sigma)=\frac{\rho}{\pi}I _{0}\cos\theta_{i}(A(\sigma)+B(\sigma)\sin\alpha\tan\beta\text{max}[\cos( \varphi_{r}-\varphi_{i}),0]), \tag{4}\]  
where \(\alpha=\text{max}[\theta_{i},\theta_{r}]\), \(\beta=\text{min}[\theta_{i},\theta_{r}]\), \(A(\sigma)=1-0.5\sigma^{2}/(\sigma^{2}+0.33)\) and \(B(\sigma)=0.45\sigma^{2}/(\sigma^{2}+0.09)\). The parameter \(\sigma\) is funded as a measure of the surface roughness, and it represents the standard deviation of the Gaussian distribution. Considering that the direction of the point light source \(\mathbf{L}\) coincides with the camera's direction \(\mathbf{V}\), we have \(\varphi_{i}=\varphi_{r},\theta_{i}=\theta_{r}\), and \(\alpha=\beta\). Consequently, Equation (4) can be simplified to  
\[L_{r}(\theta_{i};\sigma)=\frac{\rho}{\pi}I_{0}\Big{(}A(\sigma)\cos\theta_{i}+ B(\sigma)\sin^{2}\theta_{i}\Big{)}. \tag{5}\]  
Generally speaking, Equation (5) can only be established under the assumptions that are a single distant point light source and an orthographic camera projection whose direction coincides with the light source or a nearby point light source attached to the projection center of the camera performing a perspective projection. If the camera's direction does not coincide with the light source, our approach is no longer suitable, in which case one can refer to the work of [6; 23]. Defining that the unit vectors of \(\mathbf{L}\) and \(\mathbf{V}\) are all \([0,0,-1]\) and since \(\theta_{i}\) is the angle between the normal vector \(\mathbf{n}\) and \(\mathbf{L}\), we have  
\[