## 3 Materials and Methods
### Infinity Laplacian  
The interpolator used to complete the sparse depth maps is the infinity Laplacian, or AMLEhows. The infinity Laplacian is the simplest model that holds a set of axioms for interpolators [18]. Suppose \(\Omega\subset\mathbb{R}^{2}\) is the image domain. For each incomplete depth map \(u\), we solve the following problem:  
\[\Delta_{\infty,g}u=0\in\Omega, \tag{4}\]  
where \(\Delta_{\infty,g}u\) is the infinity Laplacian, \(g\) is the metric, and \(u\) is the interpolated depth data. Let us consider the image domain \(\Omega\) as a rectangle, let \(\Omega\) be a domain in a smooth compact two-dimensional manifold \(\mathcal{M}\) embedded in \(\mathbb{R}^{3}\). The interpolated surface \(u\), or manifold, is obtained by solving the infinity Laplacian, Equation (4), in the \(\Omega\) domain. Considering solving the infinity Laplacian in this 2D surface \(\mathcal{M}\), the infinity Laplacian is stated as  
\[\Delta_{\infty,g}u:=D_{\mathcal{M}}^{2}u\bigg{(}\frac{\
abla u}{|\
abla u|}, \frac{\
abla u}{|\
abla u|}\bigg{)}, \tag{5}\]  
which is a degenerated elliptic PDE. Let us consider the depth map \(u\) as \(u:\Omega\subset\mathbb{R}^{2}\rightarrow\mathbb{R}\), and the available data are located in \(\mathcal{O}\subset\Omega\), i.e., \(u|_{\partial\mathcal{O}}=\theta\). Considering the boundary condition for the infinity Laplacian, we set the available depth data to a constant value \(\theta\) on  
Figure 4: Example of Texture+Structure decomposition. (**a**) Original color reference image. (**b**) Low spatial variation part of the image. (**c**) High spatial variation part of the image (texture component). the domain's boundary. This fact ensures that the completed depth map agrees with the available data at the boundary. As a proof of concept, we will show an example of data interpolation using the infinity Laplacian. Figure 5 shows a simple example of depth completion using the infinity Laplacian. Figure 5 shows a depth map in (**a**) with a circular hole to be completed. Furthermore, the circle has a unique depth data point in its center. Let \(\vec{0}\) be the coordinates of the depth data point in the center of the circle, and let \(\mathcal{C}\) be the circular contour of radius 60 centered at \(\vec{0}\). Then, we state that \(u\big{|}_{\vec{0}}=0.68\) and \(u\big{|}_{\mathcal{C}}=0.5\). We solve the infinity Laplacian in Equation (4), obtaining the cone shown in Figure 5c. In Figure 6, we show a second simple example where we complete a two-level depth image given random depth samples. Figure 6a shows the reference color image of this second toy example. Figure 6b shows the sampled depth map. We sampled 2% of the depth data image, which is around 1300 samples. The completed two-level depth data are plotted in 3D in Figure 6c. Equation (4) was discovered by [PERSON] in the nineteen-sixties as an Euler-Lagrange equation for the problem of the Absolutely Minimizing Lipschitz Extension (or AMLE):  
\[\min_{u}\lim_{p\to\infty}\left\{\int_{\Omega}|\
abla u|^{p}d\mathbf{x}\right\}, \tag{6}\]  
where \(u\) is the extension of the available data. Computing the Euler-Lagrange equation of Equation (6), we have  
\[\
abla\cdot(\|\
abla u(\mathbf{x})\|^{p-2}\
abla u(\mathbf{x}))=0, \tag{7}\]  
Figure 5: Example of depth completion by the infinite Laplacian. We show the reference image in (**a**). The image consists of a white rectangle, and in its center, there is a black circle. We show the depth data in (**b**). The orange region represents a constant depth of 0.5; in the center, the circle in blue represents the lack of data. In the center of the blue circle, there is only a single value equal to 0.68. We show the interpolated depth map in (**c**). The infinity Laplacian generates a cone, connecting the circular contour with the point in the circle’s center. Figure 6: Example of depth map completion. (**a**) shows a two-color reference image. (**b**) shows random samples of a two-level depth map. White values represent the value 1, and orange values represent the depth value 0.5.