Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus Hurst index

Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst index $H\in(0,1)$. Special attention has been paid to the impact of Hurst index on the topological properties. It is found that the clustering coefficient $C$ decreases when $H$ increases. We also found that the mean length $L$ of the shortest paths increases exponentially with $H$ for fixed length $N$ of the original time series. In addition, $L$ increases linearly with respect to $N$ when $H$ is close to 1 and in a logarithmic form when $H$ is close to 0. Although the occurrence of different motifs changes with $H$, the motif rank pattern remains unchanged for different $H$. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension $d_B$ decreases with $H$. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with the degree, which indicate that the horizontal visibility graphs are assortative. With the increase of $H$, the Pearson coefficient decreases first and then increases, in which the turning point is around $H=0.6$. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.Comment: 12 pages, 8 figure

PIXOR: Real-time 3D Object Detection from Point Clouds

We address the problem of real-time 3D object detection from point clouds in the context of autonomous driving. Computation speed is critical as detection is a necessary component for safety. Existing approaches are, however, expensive in computation due to high dimensionality of point clouds. We utilize the 3D data more efficiently by representing the scene from the Bird's Eye View (BEV), and propose PIXOR, a proposal-free, single-stage detector that outputs oriented 3D object estimates decoded from pixel-wise neural network predictions. The input representation, network architecture, and model optimization are especially designed to balance high accuracy and real-time efficiency. We validate PIXOR on two datasets: the KITTI BEV object detection benchmark, and a large-scale 3D vehicle detection benchmark. In both datasets we show that the proposed detector surpasses other state-of-the-art methods notably in terms of Average Precision (AP), while still runs at >28 FPS.Comment: Update of CVPR2018 paper: correct timing, fix typos, add acknowledgemen

Implicit 3D Orientation Learning for 6D Object Detection from RGB Images

We propose a real-time RGB-based pipeline for object detection and 6D pose estimation. Our novel 3D orientation estimation is based on a variant of the Denoising Autoencoder that is trained on simulated views of a 3D model using Domain Randomization. This so-called Augmented Autoencoder has several advantages over existing methods: It does not require real, pose-annotated training data, generalizes to various test sensors and inherently handles object and view symmetries. Instead of learning an explicit mapping from input images to object poses, it provides an implicit representation of object orientations defined by samples in a latent space. Our pipeline achieves state-of-the-art performance on the T-LESS dataset both in the RGB and RGB-D domain. We also evaluate on the LineMOD dataset where we can compete with other synthetically trained approaches. We further increase performance by correcting 3D orientation estimates to account for perspective errors when the object deviates from the image center and show extended results.Comment: Code available at: https://github.com/DLR-RM/AugmentedAutoencode

BeSpaceD: Towards a Tool Framework and Methodology for the Specification and Verification of Spatial Behavior of Distributed Software Component Systems

In this report, we present work towards a framework for modeling and checking behavior of spatially distributed component systems. Design goals of our framework are the ability to model spatial behavior in a component oriented, simple and intuitive way, the possibility to automatically analyse and verify systems and integration possibilities with other modeling and verification tools. We present examples and the verification steps necessary to prove properties such as range coverage or the absence of collisions between components and technical details

Colored anchored visibility representations in 2D and 3D space

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In a visibility representation of a graph G, the vertices are represented by nonoverlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G, the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given.Peer ReviewedPostprint (author's final draft

Efficient Cluster Algorithm for CP(N-1) Models

Despite several attempts, no efficient cluster algorithm has been constructed for CP(N-1) models in the standard Wilson formulation of lattice field theory. In fact, there is a no-go theorem that prevents the construction of an efficient Wolff-type embedding algorithm. In this paper, we construct an efficient cluster algorithm for ferromagnetic SU(N)-symmetric quantum spin systems. Such systems provide a regularization for CP(N-1) models in the framework of D-theory. We present detailed studies of the autocorrelations and find a dynamical critical exponent that is consistent with z = 0.Comment: 14 pages, 3 figure