Incremental 2D Grid Map Generation from RGB-D Images

In this paper, we extend RGB-D SLAM to address the problem that sparse map-building RGB-D SLAM cannot directly generate maps for indoor navigation and propose a SLAM system for fast generation of indoor planar maps. The system uses RGBD images to generate positional information while converting the corresponding RGBD images into 2D planar lasers for 2D grid navigation map reconstruction of indoor scenes under the condition of limited computational resources, solving the problem that the sparse point cloud maps generated by RGB-D SLAM cannot be directly used for navigation. Meanwhile, the pose information provided by RGB-D SLAM and scan matching respectively is fused to obtain a more accurate and robust pose, which improves the accuracy of map building. Furthermore, we demonstrate the function of the proposed system on the ICL indoor dataset and evaluate the performance of different RGB-D SLAM. The method proposed in this paper can be generalized to RGB-D SLAM algorithms, and the accuracy of map building will be further improved with the development of RGB-D SLAM algorithms.

Energy Minimisers with Prescribed Jacobian

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity $$\lambda [f]$$ λ [ f ] which controls the symmetry, uniqueness and regularity of minimisers: if $$\lambda [f]\le 1$$ λ [ f ] ≤ 1 then minimisers are symmetric and unique; if $$\lambda [f]$$ λ [ f ] is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if $$\lambda [f]$$ λ [ f ] is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by $${\mathbb {R}}^2$$ R 2 and boundary conditions are not prescribed.

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ for $$\kappa '$$ κ ′ in (4, 8) that is drawn on an independent $$\gamma $$ γ -LQG surface for $$\gamma ^2=16/\kappa '$$ γ 2 = 16 / κ ′ . The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$ CLE κ for $$\kappa $$ κ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ independently into two colors with respective probabilities p and $$1-p$$ 1 - p . This description was complete up to one missing parameter $$\rho $$ ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$ κ ′ . It shows in particular that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$ FK q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if $$q=4\cos ^2(4\pi / \kappa ')$$ q = 4 cos 2 ( 4 π / κ ′ ) . This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ represent the scaling limits of $$\hbox {FK}_q$$ FK q percolation and the q-Potts model when q and $$\kappa '$$ κ ′ are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$ ρ ( p , κ ′ ) is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

Quenched Local Convergence of Boltzmann Planar Maps

Stephenson (2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit.