## 4 Experimental Results
### Experimental Results on Synthetic Images  
A synthetic image of the sphere which was produced using the following formulation (20) was employed in our experiments:  
\[z(x,y)=\sqrt{R^{2}-x^{2}-y^{2}}, \tag{20}\]  
where \((x,y)\in[-63,64]\times[-63,64]\) and \(R=50\) is the sphere's radius. Another synthetic image of the case which was produced using the following formulation (21) was also employed:  
\[z(x,y)=\sqrt{f(y)^{2}-x^{2}}, \tag{21}\]  
where \(f(y)=-0.025(6y-1)(2y+1)(2y-1)^{2}(3y+2)^{2}+0.15\) and \((x,y)\in[-0.5,0.5]\times[-0.5,0.5]\). We can map the \(x\) and \(y\) values to \([-63,64]\) and scale \(z(x,y)\) by a factor of 128. This yields a maximum height value of approximately 36.55. Their original height maps are demonstrated in Figure 2.  
The two images are generated by the model in this paper with the direction vectors of the point light source and the camera all being \([0,0,-1]^{T}\). The variant approach of [PERSON] and [PERSON] [8] and the approach of [PERSON] and [PERSON] [20] are used to compare with our presented approach. It has been noted that the approach of [PERSON] and [PERSON] [20] applied the Lax-Friedrichs sweeping scheme to solve the Oren-Nayar SFS, and that the existing approach of [PERSON] and [PERSON] [8] used the first-order Godunov-based scheme to solve the Lambertian SFS and therefore cannot directly solve the Oren-Nayar SFS. However, it is able to solve our partial differential IIR (10) directly. We define the first-order Godunov-based scheme that is accelerated by an alternating sweeping strategy for the Oren-Nayar SFS as the variant approach of [PERSON] and [PERSON] [8]. The experimental results of the sphere are illustrated in Figure 3. Figure 3a shows the synthetic intensity image which is adopted for the 3D reconstruction with the parameter set: \(\sigma=0.2\). Figure 3b demonstrates the reconstructed height map using the variant approach of [PERSON] and [PERSON] [8] and the known Dirichlet boundary condition \(z=\varphi\) on \(\partial\Omega\), while the error map with the original height map is shown in Figure 3c. Figure 3d demonstrates the reconstructed height map using the approach of [PERSON] and [PERSON] [20], while the error map with the original height map is shown in Figure 3e. Figure 3f demonstrates the reconstructed height map using our presented approach with the known Dirichlet boundary condition \(z=\varphi\) on \(\partial\Omega\), while the error map with the original height map is shown in Figure 3g. Finally, Figure 4 illustrates the corresponding experimental results of the synthetic vase. Figures 3 and 4 show that the variant approach of [PERSON] and [PERSON] [8], the approach of [PERSON] and [PERSON] [20], and our presented approach can all obtain satisfactory reconstructed results of the diffuse surfaces. Moreover, we can further see that our presented approach gets a higher accuracy and smaller errors than the variant approach of [PERSON] and [PERSON] [8] and the approach of [PERSON] and [PERSON] [20]. To further demonstrate the effectiveness of our presented approach, a quantitative comparison between the variant approach of [PERSON] and [PERSON] [8], the approach of [PERSON] and [PERSON] [20], and our presented approach is adopted by using the CPU time, the Mean Absolute Error (MAE), and the Root Mean Square Error (RMSE). Table 1 lists the comparisons of the three approaches for the synthetic sphere and vase images. As is directly seen, our presented approach shows better performance in the reconstructed errors than the variant approach of [PERSON] and [PERSON] [8], as well as than the approach of [PERSON] and [PERSON] [20] in all the images which we performed. The MAE and RMSE of our proposed approach are about one-fiftieth and one-twentieth of the variant approach of [PERSON] and [PERSON] [8] for the sphere, and about one-seventh and one-fourth for the vase, respectively. The approach of [PERSON] and [PERSON] [20] shows a slightly worse performance; maybe it is difficult to find a good estimate for the artificial viscosity term.