## 2 Methods
### Finite Element Method Interpolation  
The main idea of the present approach is to transform the discrete cloud of points into a mathematical framework--similar to a deformable medium--by which the strain-rate tensor can be computed. To do so, we generate a mesh \(\mathcal{T}_{h}(t)=\left\{K\right\}\) from the set of points \(\mathcal{P}\) that is composed of non-overlapping and conforming geometrical elements \(K\) of diameter \(h\). There are several methods to generate a mesh from a set of points, all which are studied in the computational geometry field [43]. Here, we apply Delaunay triangulation \(DT\left(\mathcal{P}\right)\) for several reasons: (1) the aspect ratio of the triangulated elements produce a high-quality mesh; and (2) fast Delaunay triangulation algorithms have been developed recently (see, for example, the one in [44]). The result of applying the Delaunay triangulation over the set of points is a discrete mesh \(\mathcal{T}_{h}:=DT\left(\mathcal{P}\right)\) that possess the following characteristics: it covers exactly the convex hull \(\Omega\) of the point set, no point \(p_{i}\) is isolated from the triangulation, and all the elements \(\left\{K\right\}\) are four-point tetrahedrons, which are completely defined by the position of their four corner points \(K:=\left\{\mathbf{x}_{j}\right\}\), with \(j=1,2,3,4\). The generated mesh \(\mathcal{T}_{h}=DT\left(\mathcal{P}\right)\) can be seen as a--material--domain \(\Omega\) that suffers deformations from the displacements of the points between consecutive time-steps. Since only discrete displacements between consecutive time-steps are known for the set of points, we now explain how the continuous velocity field inside the mesh is calculated. Even though the FEM has been used to perform interpolation using the point-wise data (see [41; 42] for applications in one and two-dimensional data), in this work, we apply this well-known method in a three-dimensional setting. In FEM, the finite interpolating space \(\mathbf{\mathcal{V}}_{h}\) is defined as made of continuous piece-wise polynomials \(N(\mathbf{x})\) in the mesh \(\mathcal{T}_{h}\), where the discrete approximation \(\mathbf{F}_{h}(\mathbf{x},t)\in\mathbf{\mathcal{V}}_{h}\) of any multi-dimensional function \(\mathbf{F}(\mathbf{x},t)\), \(\mathbf{x}\in\Omega\), can be written as  
\[\mathbf{F}(\mathbf{x},t)\approx \mathbf{F}_{h}(\mathbf{x},t):=\sum_{i=1}^{n}N(\mathbf{x}_{i})\mathbf{F}_{i}(t), \mathbf{x}\in\Omega. \tag{1}\]We use the simplest finite element: the tetrahedron with _linear polynomials_ and four nodes. Again, some notation is required to define the polynomials inside the element. The set of normalized coordinates \(\eta_{1},\eta_{2},\eta_{3},\eta_{4}\) in each tetrahedron \(K\) is such that the value of \(\eta_{i}\) is one at the point \(p_{i}\in K\), zero at the other three corner points, and varies linearly from that point to the opposite edges. This set of coordinates has the property that the sum of the four coordinates (each belonging to one tetrahedron point) in any location inside the tetrahedron is identically one: \(\eta_{1}(\mathbf{x}_{i})+\eta_{2}(\mathbf{x}_{i})+\eta_{3}(\mathbf{x}_{i})+\eta_{4}(\mathbf{x} _{i})=1\), with \(\mathbf{x}_{i}\in K\). Hence, the shape functions inside each linear tetrahedron are defined to be these coordinates: \(N_{i}(\mathbf{x}_{i})=\eta_{i}(\mathbf{x}_{i})\), with \(i=1,2,3,4\) denoting the corner points. The FEM interpolation in Equation (1) of a _three-dimensional vector function_, e.g.