## 4 Method
### \(L_{0}\) Based Optical Flow Deformation  
The goal of optical flow in this work is to compute deformations and to recover the discontinuous fluid motions. In the following, a sparsity constraint on the deformation gradients is incorporated into the optical flow estimation model to reduce the undesirable errors.  
The \(L_{2}\) based optical flow solve [5] for the certain inputs produced by semi-Lagrangian advection algorithm cannot robustly and accurately match surfaces in the nearest-neighbor fashion. Especially in the inputs with little similarities because of the surface flickering, the deformations will try to match pieces in proximity but not necessarily the whole target shape. We propose an \(L_{0}\) based deformation regularizer over the existing optical flow solve to implicitly define the appropriate correspondences and to prevent motion changes on other regions.  
Specifically, the deformation is denoted by \(\mathbf{u}\) to transform the source surface closer into the target surface. We formulate the \(L_{0}\) based deformation by minimizing the following energy function  
\[\underset{\mathbf{u}}{\mathbf{min}}\,E_{data}(\mathbf{u})+\beta_{T}E_{Tikhonov }(\mathbf{u})+\beta_{S}E^{{}^{\prime}}_{smooth}(\mathbf{u}) \tag{5}\]  
where a new smoothness regularizer \(E^{{}^{\prime}}_{smooth}\) is introduced to the general optical flow solve to replace the \(L_{2}\) norm smoothness term. \(E^{{}^{\prime}}_{smooth}\) measures the deformation differences between all pairs of neighboring points, and is given by  
\[E^{{}^{\prime}}_{smooth}=S(\mathbf{u}) \tag{6}\]  
\[S(\mathbf{u})=\#\left\{\ p\ ||\partial_{x}\mathbf{u}^{p}|+|\partial_{y}\mathbf{u} ^{p}|+|\partial_{z}\mathbf{u}^{p}|+|\partial_{t}\mathbf{u}^{p}|\
eq 0\right\} \tag{7}\]  
where \(S(\mathbf{u})\) is the \(L_{0}\) norm for the deformation gradient \(\
abla\mathbf{u}\). The gradient \(\
abla\mathbf{u}^{p}=\{\partial_{x}\mathbf{u}^{p},\partial_{y}\mathbf{u}^{p}, \partial_{z}\mathbf{u}^{p},\partial_{t}\mathbf{u}^{p}\}\) for each point \(p\) of the deformation field is calculated based on the two  
Figure 2: The pipeline of the registration and interpolation scheme. The inputs animation show two crown splashes. Our proposed \(L_{0}\) based optimization is applied into the optical flow solve.  
4D SDFs between neighboring points in space and time. The gradient magnitude \(|\partial\mathbf{u}^{p}|\) is defined as the sum of gradient magnitudes over four dimensions. Our smoothness constraint with \(L_{0}\) norm regularization is used to control the number of non-zero gradients globally and present the sparsity of deformation gradients.  
We estimated the optimized deformation by minimizing the energy function  
\[\underset{\mathbf{u}}{\mathbf{min}}\left\{\sum_{p}|\boldsymbol{\psi}_{r}^{p}+ \
abla\boldsymbol{\psi}^{p}\cdot\ \mathbf{u}^{p}|^{2}+\beta_{T}|\mathbf{u}^{p}|^{2}+\beta_{S}S(\mathbf{u})\right\} \tag{8}\]  
The \(E_{smooth}^{{}^{\prime}}\) term of \(L_{0}\) norm regularizer in (5) is a non-convex problem. Due to the computationally intractability of the \(L_{0}\) norm, the half-quadratic splitting strategy utilized in image smoothing achieves a high qualitative effect [21]. Inspired by the strategy described in [21], we split the objective function (8) into two sub-problems by introducing auxiliary variables into the energy function. We argue that this method found appropriate matches and retrieved the surface motions robustly.