## 5 Experimental Results
### Evaluation  
The error metric mentioned in Section 4.2 could quantitatively and significantly detect the differences between the deformed surface and the target surface. We utilized the error metric (14) to calculate the errors between the target surfaces and the generated surfaces which were deformed with the \(L_{2}\) based and the \(L_{0}\) based deformation methods respectively. The error values can be found in Table 1, which includes two aforementioned scenarios (droplet, dam break). We show the results with the deformation direction from \(\psi_{1}\) to \(\psi_{2}\) and the inverse one. We can observe that our proposed \(L_{0}\) based fluid simulation method could obtain less errors than the \(L_{2}\) based method [5], and generate the desired correspondences.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fluid** **Example**} & \multirow{2}{*}{**Directions**} & \multirow{2}{*}{**Initial**} & \multicolumn{2}{c}{**The Error after Deformation**} \\ \cline{3-5}  & & & **Error** & \(\mathbf{L_{2}}\) & \(\mathbf{L_{0}}\) \\ \hline \multirow{2}{*}{Flying} & \(\psi_{1}\rightarrow\psi_{2}\) & 4.1207 & 1.1113 & 0.9440 \\ \cline{2-5}  & \(\psi_{2}\rightarrow\psi_{1}\) & 3.9823 & 1.9747 & 1.8369 \\ \hline \multirow{2}{*}{Dam Break} & \(\psi_{1}\rightarrow\psi_{2}\) & 4.0991 & 0.6768 & 0.6082 \\ \cline{2-5}  & \(\psi_{2}\rightarrow\psi_{1}\) & 5.0944 & 1.5366 & 1.3622 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Errors between the deformed surface and the target surface.  
Figure 8: A comparison of the L\({}_{2}\) based optical flow against our L\({}_{0}\) based optimization output for the dam break scenario. Left: the input \(\psi_{1}\) and \(\psi_{2}\). Middle: the interpolated results generated by using the \(L_{2}\) based method with two different directional deformations including \(\psi_{1}\rightarrow\psi_{2}\) and \(\psi_{2}\rightarrow\psi_{1}\) with the interpolation weight 1, respectively. Right: the corresponding interpolated results by using the \(L_{0}\) based method.  
Figure 9: The range of interpolation for the dambreak setup of inputs at a fixed point in time. The numbers on bottom of each image indicate the interpolation weights. Direction of the deformation is employed from \(\mathbf{\psi_{1}}\) to \(\mathbf{\psi_{2}}\).  
Our experiments used data-sets with a resolution of 64 \(\times\) 64 \(\times\) 64 \(\times\) 64. We utilized the commonly used hierarchical procedure to recursively re-sample and deform the data on coarser resolutions, including the resolutions of 16 \(\times\) 16 \(\times\) 16 \(\times\) 16, 32 \(\times\) 32 \(\times\) 32 \(\times\) 32, and 64 \(\times\) 64 \(\times\) 64 \(\times\) 64, denoted by 16\({}^{4}\), 32\({}^{4}\), and 64\({}^{4}\), respectively. The changes of resolution were very useful to reduce the inaccuracies of the optical flow solve. Table 2 lists the error between the deformed surface and the target surface, the factor (the ratio of the error of this iteration to that of last iteration), and the time to calculate the deformation during each iteration. In our experiment, we stopped the iterations when the factors for the resolution of 16\({}^{4}\), 32\({}^{4}\), and 64\({}^{4}\) were greater than 70%, 95%, and 95%, respectively. Particularly for the resolution of 16\({}^{4}\), the iterations stopping criterion should exceed three times. Note that the deformation was not refined if the gains in quality could not be yielded anymore. Thus, we did not use the deformation obtained by the last iteration. From the Table 2, we can see the execution time of the deformation by our \(L_{0}\) based optical flow method with the half quadratic splitting approach was less than that of the \(L_{2}\) based optical flow method.