## 5 Experimental Results
### Deformation Results  
We evaluate the quality of the deformation through the visual quality of the interpolated results, where the uni-directional nearest-neighbor interpolation is used, such as the deformation direction from \(\psi_{1}\) to \(\psi_{2}\). We consider two flying droplets with different initial shapes and different positions. They drop in the pool at different times to create different water ripples. We try to deform the source one to the target one by our proposed fluid simulation method and state-of-the-art method [5] respectively, with the interpolation weight as 1. Figure 4 shows the interpolated results of the \(L_{2}\) based and the \(L_{0}\) based deformation methods respectively. In order to easily observe, the sectional view images at a fixed frame are shown along with the interpolated results. We can observe that our results are closer to the target input \(\psi_{2}\) and with less artifacts than those of the \(L_{2}\) based method. In addition,  
Figure 3: The convergence analysis of flying dr oplets with variable parameter \(\kappa\). X label means iteration number, Y label means the error metric \(error\). the interpolated results of these two different methods with the deformation from the input \(\psi_{2}\) to the input \(\psi_{1}\) are shown in Figure 5. The \(L_{2}\) based deformation method may create some ambiguous correspondences between the input \(\psi_{1}\) to the input \(\psi_{2}\), and the undesired ripples appear at the surface due to the suboptimal matching. Our proposed method can alleviate the artificial motions brought by the \(L_{2}\) based method effectively. Three examples of interpolated results at different frames for flying droplets are demonstrated in Figure 6, where the droplets were falling into the pool and causing the varying splashes. Figure 7 demonstrates the interpolation results of the droplets falling into the pool and the varying splashes. We observe that our interpolated results could preserve the large-scale structures and recover the small-scale details. Our \(L_{0}\) based method could reduce the artificial motions brought by the \(L_{2}\) based method, which demonstrates that our \(L_{0}\) based method could calculate moderate correspondences and a good deformation field and could further reduce the undesired errors of complex surfaces of the liquid. We take the dam break as an example to demonstrate the accuracy of our optimization method. In this set of animations, a block of water was released at one side of a pool, and this formed a wave that swept the pool. As shown in Figure 8, we produced fluid animations for two different widths for this block of water. The interpolated results were generated with the interpolation weight as 1 to compare the generated surfaces with the target surfaces visually. We compared the results of our proposed \(L_{0}\) based method with those of [5]. As shown in the right columns of Figure 8, the \(L_{2}\) based method caused the undesired ripples at the surface, and the \(L_{0}\) optimization method captured the accurate motions of the waves. Therefore, the surface of our results based on the \(L_{0}\) regularization was closer to the target one than that based on the \(L_{2}\) optimization. Figure 9 shows the whole range of the interpolation results of dam break deformed from \(\psi_{1}\) to \(\psi_{2}\) with different interpolation weights. The inputs had inconsistent behaviors in this static frame. Compared with the \(L_{2}\) based regularizer which restricted the deformation locally across the surface region, the \(L_{0}\) based regularizer effectively recovered the small-scale details. The animation simulation of all cases by using the \(L_{2}\) based method [5] and our \(L_{0}\) methods can be found in the supplemental video. The accompanying video shows a direct comparison to illustrate the difference between the results of [5] and our method. Figure 4: The visual quality comparison of flying droplets via different deformation methods based on \(L_{2}\) and \(L_{0}\) smooth regularization.