Comparing different metrics on an anisotropic depth completion model

This paper discussed an anisotropic interpolation model that filling in-depth data in a largely empty region of a depth map. We consider an image with an anisotropic metric gi⁢j that incorporates spatial and photometric data. We propose a numerical implementation of our model based on the “eikonal” operator, which compute the solution of a degenerated partial differential equation (the biased Infinity Laplacian or biased Absolutely Minimizing Lipschitz Extension). This equation’s solution creates exponential cones based on the available data, extending the available depth data and completing the depth map image. Because of this, this operator is better suited to interpolating smooth surfaces. To perform this task, we assume we have at our disposal a reference color image and a depth map. We carried out an experimental comparison of the AMLE and bAMLE using various metrics with square root, absolute value, and quadratic terms. In these experiments, considered color spaces were sRGB, XYZ, CIE-L*⁢a*⁢b*, and CMY. In this document, we also presented a proposal to extend the AMLE and bAMLE to the time domain. Finally, in the parameter estimation of the model, we compared EHO and PSO. The combination of sRGB and square root metric produces the best results, demonstrating that our bAMLE model outperforms the AMLE model and other contemporary models in the KITTI depth completion suite dataset. This type of model, such as AMLE and bAMLE, is simple to implement and represents a low-cost implementation option for similar applications.

On asymptotic expansions for the fractional infinity Laplacian

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian Δ ∞ s for s ∈ ( 1 2 , 1 ). This operator has been introduced and first studied in (Comm. Pure Appl. Math. 65 (2012) 337–380). Our expansions are parametrised by the radius of the removed singularity ε, and allow for the identification of Δ ∞ s ϕ ( x ) as the ε 2 s -order coefficient of the deviation of the ε-average from the value ϕ ( x ), in the limit ε → 0 +. The averages are well posed for functions ϕ that are only Borel regular and bounded.

Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier–Stokes equations

We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow–outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier–Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow.