## 2 Related Work  
In this section, we briefly introduce the related work about the fluid simulation method and the motion reconstruction method with sparse constraints. Fluid simulation has a long history in computer graphics research. One of the influential works in this area is published in Foster and Metaxas [4], and the stable advection approach introduced by [3] has made a significant progress of fluid simulation for visual effects. The FLIP method [8] which combines the particle and the grid data, and the purely particle based SPH approach [9] have been widely used in movie productions. Numerous extensions and improvements have been proposed, e.g., to introduce small-scale turbulent details [10], or to address the problem emerged in reality from fluid-air and fluid-solid interactions with the surface tension force and the adhesion force [11]. Several works have used guide behaviors to make liquid surfaces follow a desired shape or motion [12; 13]. Unlike this modification approaches on existing fluid simulation, the work-flow that concentrates on align pre-computed space-time surfaces and interpolates between them for intermediate motions is mainly based on non-rigid ICP algorithms [6], and optical flow solve [5]. The non-rigid ICP algorithm has been applied to reconstruct the temporally coherent dynamic objects from data captured by the multi-view acquisition system [14] or the monocular depth  
Figure 1: This image illustrates the blending result with the \(L_{2}\) based method [5]. While final output retrieves large scale motions, the artifacts between the droplet and surface are noticeable. camera [15]. This registration technique has also been applied to the tracking of changing topology shapes such as liquid surfaces [16]. These method will run into the problem of local minima that is associated with non-rigid ICP. [PERSON] et al. [6] roughly prevents the problem with user-guided correspondences. Regular optical flow aims at recovering the motion of dynamic objects, especially for consecutive frames. Up to now, the Horn-Schunck algorithm [17] is extended to estimate high quality flow field. The connection between the optical flow and fluid flow are established for typical flow visualizations [18]. The optical flow techniques have also been used to acquire the fluid velocities based on physical priors [19]. Optical flow estimation commonly minimizes an energy function to obtain an optimal result, where the smoothness regularization is incorporated as a convex regularizer. The optical flow deformation approach [5] is essentially similar to the non-rigid iterative closest point algorithm that incorporates user guidance [6] in term of the registration process. Likewise, this optical flow deformation approach robustly recovers a match among the inputs based on local smoothness constraints and obtains a deformation field to align the surfaces. The \(L_{2}\) based smoothness regularization [5] may cause the error propagation and the accumulated errors. In this paper, we utilize a specific sparse regularization as a constraint to improve the results. In fact, sparse representation [20] has successfully applied in field of image processing [21], computer vision [22], and graphics [15]. Especially, the sparsity priors have been utilized in the optical flow estimation and motion reconstruction. For example, a new sparsity prior of the flow field in certain domains is introduced to estimate dense flow fields, and a stronger additional sparsity constraint on the flow gradients is incorporated into the model to cope the measurement noises [22]. While [23] decomposes the motion field into sparse and non-sparse components to restrain the over-sparsity. In the motion tracking [15], the \(L_{0}\) based motion regularizer is incorporated into the non-rigid deformation to propose the \(L_{0}\)-\(L_{2}\) non-rigid motion tracking method to efficiency stop the error propagation.