Delaunay Graph Research Articles

3D Building Modelling from ALS Point Clouds by Delaunay Triangulation and Graph Theories

Abstract. As an important task in 3D building reconstruction, 3D roof reconstruction attracts increasing attention. Existing methods have the issue of additional errors caused by multi-step processes, resulting in relatively high uncertainty and potentially affecting the reconstruction accuracy. Deep learning-based methods to achieve the direct extraction of roof vertices and edges for reconstructing 3D roof structures tackle the issue of additional errors while leading to another problem: the reliance on large labeled datasets for training. In this study, a fully rule-based method is proposed to achieve automatic 3D roof reconstruction. Roof vertices with their edges parallel to the x-y plane are first extracted from the point clouds, and subsequently, the left roof edges connecting vertices are inferred by using Delaunay triangulation. Finally, the 3D roof structure is reconstructed by using graph analysis based on the information of roof vertices and edges. This method simplifies the process of 3D roof reconstruction, and the experiment results demonstrate that the proposed method can effectively reconstruct 3D roof structures from point clouds.

Perturbing isoradial triangulations

We consider an infinite planar Delaunay graph \mathbf{G}_{\epsilon} which is obtained by locally deforming the coordinate embedding of a general isoradial graph \mathbf{G}_{\mathrm{cr}} , with respect to a real deformation parameter \epsilon . Using Kenyon’s exact and asymptotic results for the critical Green’s function on an isoradial graph, we calculate the leading asymptotics of the first- and second-order terms in the perturbative expansion of the log-determinant of the Laplace–Beltrami operator \Delta(\epsilon) , the David–Eynard Kähler operator \mathcal{D}(\epsilon) , and the conformal Laplacian {\underline{\boldsymbol{\Delta}}}(\epsilon) on the deformed Delaunay graph \mathbf{G}_{\epsilon} . We show that the scaling limits of the second-order bi-local term for both the Laplace–Beltrami and David–Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph \mathbf{G}_{\mathrm{cr}} . Our results allow us to define a discrete analog of the stress-energy tensor for each of the three operators. Furthermore, we can identify a central charge ( c=-2 ) in the case of both the Laplace–Beltrami and David–Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c=-2 for the David–Eynard Kähler operator is, however, at odds with the value of c=-26 expected by Polyakov’s theory of 2D quantum gravity; moreover, there are problems with convergence of the scaling limit of the discrete stress-energy tensor for the David–Eynard Kähler operator. The second-order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph \mathbf{G}_{\epsilon} ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.

Coloring lines and Delaunay graphs with respect to boxes

AbstractThe goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of lines in whose intersection graph is triangle‐free of chromatic number . This improves the previously best known bound by Norin, and is also the first construction of a triangle‐free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of points in , whose Delaunay graph with respect to axis‐parallel boxes has independence number at most . This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.

The projector algorithm: A simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has found numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell (actually, even just portions of one cell) can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.

A parallel algorithm for computing Voronoi diagram of a set of spheres using restricted lower envelope approach and topology matching

We present a parallel algorithm for computing the Voronoi diagram of a set of spheres, S in R3. The spheres have varying radii and do not intersect. We compute each Voronoi cell independently using a two-stage iterative procedure, assuming the input spheres are in general position. In the first stage, an initial Voronoi cell for a sphere si is computed using an iterative lower envelope approach restricted to a subset of spheres Li⊂S. This helps to avoid defining the bisectors between all pairs of input spheres and develop a distributed memory parallel algorithm. We use the Delaunay graph of sample points from the input spheres to select the subset Li for computing each Voronoi cell. In the second stage, Voronoi cells obtained from the first stage are matched for updating the subsets. If additional spheres are added to a subset Li, the correctness of the computed vertices is verified with the bisectors of spheres newly added to Li. Results and performance of the algorithm show robustness and speed of the algorithm in handling a large set of spheres.

A parallel algorithm for computing Voronoi diagram of a set of circles using touching disc and topology matching

We present a parallel algorithm for computing the Voronoi diagram of a set of circles, C. The algorithm starts with a parallel computation of the Voronoi cell of each circle ci∈C, using a subset of circles, Ci⊂C, and a reduced touching disc method. Unlike the actual touching disc method, a reduced touching disc method does not compute antipodal discs (ADs) between all pairs of circles. We employ the Delaunay graph, D, of the center points of circles in C to set an initial Ci. For computing an initial Voronoi cell for ci∈C, it is enough to compute ADs between ci and each of the circles cj∈Ci. Computation of Voronoi cells updates initially assumed Ci, and the algorithm iteratively computes those Voronoi cells until no further updating of Ci occurs. A parallel procedure corrects the Voronoi cells, iteratively, using a topology matching approach. A reduced touching disc method and a cell-wise computation of the Voronoi diagram allow input in non-general position. Results and performance of the algorithm show robustness and speed of the algorithm in handling a large set of circles.

Hybrid Mesh Deformation for Aerodynamic-Structural Coupled Adjoint Optimization

This paper demonstrates that a discrete coupled-adjoint aeroelastic shape optimization can be made more efficient with the use of different mesh-deformation algorithms for the fluid-structure interaction (FSI) simulations and the coupled-adjoint calculations within the optimization loop. Mesh deformation using radial basis functions (RBF) with only a subset of surface points is popular due to the efficiency and mesh quality produced by the technique. However, this technique can reduce the rate of convergence of the coupled-adjoint and even prevent the coupled-adjoint equations from converging. This paper proposes a hybrid mesh-deformation strategy to improve the efficiency of coupled-adjoint optimizations: use the RBF method with a data-reduction algorithm when deforming the mesh within FSI simulations but use the Delaunay graph mapping (DGM) method in the coupled-adjoint procedure. The DGM method increases the rate of convergence of the coupled-adjoint matrix, relative to the RBF approach with a data-reduction algorithm, with the additional benefit of being a faster method. Using this hybrid approach, an optimization in which lift and pitching-moment constraints are satisfied within the FSI simulation is performed. The results of the optimization, and the effects of the hybrid approach, are presented.

A dynamic sampling approach towards computing Voronoi diagram of a set of circles

Discretizing the input curves into points or lines is a way of constructing an approximate Voronoi diagram. Sampling the input curves into points requires a fine sample density to obtain a reasonable topological and geometrical accuracy of the Voronoi diagram. In this work, it is shown that an accurate computation of the Voronoi diagram for a set of circles employing a very coarse sample set is possible. The approach proposes the selection of varying number of sample points on each circle dynamically, using the Delaunay graph of center points of circles. We then show that this approach can be used to identify neighborhood information of the circles. The touching disc method is then used to construct the Voronoi diagram with an algorithmic complexity of O(nlogn). The work also demonstrates the robustness and theoretical correctness of the proposed algorithm by considering inputs in non-general position and for a large number of circles of the order of 105.

Anchored expansion of Delaunay complexes in real hyperbolic space and stationary point processes

We give sufficient conditions for a discrete set of points in any dimensional real hyperbolic space to have positive anchored expansion. The first condition is an anchored bounded density property, ensuring not too many points can accumulate in large regions. The second is an anchored bounded vacancy condition, effectively ensuring there is not too much space left vacant by the points over large regions. These properties give as an easy corollary that stationary Poisson–Delaunay graphs have positive anchored expansion, as well as Delaunay graphs built from stationary determinantal point processes.

Heuristics for the Canadian traveler problem with neutralizations

Canadian Traveler Problem with Neutralizations (CTPN) is a recently introduced challenging graph-theoretic path-planning problem. In CTPN, traversability status of some edges in the underlying graph is dependent on an a priori probability distribution. A traveling agent has two capabilities called disambiguation and neutralization. In the disambiguation case, the true status of an ambiguous edge (traversable or untraversable) is revealed when the agent is at either end of such edges. If the neutralization capability is exploited, the edge immediately becomes traversable. These capabilities are limited and may add a cost of increased path length. The goal of the agent is to find the shortest expected path length by devising an optimal policy that dictates when and where to disambiguate or neutralize. CTPN has important and practical applications within the context of expert and intelligent systems. These include autonomous robot navigation, adaptive transportation systems, naval and land minefield countermeasures, and navigation inside disaster areas for emergency relief operations. There is a recently proposed state-of-the-art exact algorithm that solves CTPN to optimality, called CAON∗ (AO∗ with Caching and Neutralizations). CAON∗ is based on an extension of the well-known AO∗ (AND-OR search) algorithm. Even though CAON∗ has significant improvements compared to its predecessors, it still has exponential run time and space complexity and it has been shown to solve only small instances of CTPN in practice. In this study, we introduce new heuristics for CTPN based on novel strategies that can be used to solve much larger and realistic problem instances. We provide computational experiments on Delaunay graphs to assess and compare the performance of these heuristics and CAON∗, in terms of both run time and solution quality. Our computational experiments indicate that the proposed heuristics run extremely fast (well under a second in all cases) and they result in up to 58% improvement over existing heuristics with respect to expected path lengths with an overall improvement of 32% across our computational experiments.

Optimal Random Packing of Spheres and Extremal Effective Conductivity

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

On the Analysis of Spatially Constrained Power of Two Choice Policies

We consider a class of power of two choice based assignment policies for allocating users to servers, where both users and servers are located on a two-dimensional Euclidean plane. In this framework, we investigate the inherent tradeoff between the communication cost, and load balancing performance of different allocation policies. To this end, we first design and evaluate a Spatial Power of two (sPOT) policy in which each user is allocated to the least loaded server among its two geographically nearest servers sequentially. When servers are placed on a two-dimensional square grid, sPOT maps to the classical Power of two (POT) policy on the Delaunay graph associated with the Voronoi tessellation of the set of servers. We show that the associated Delaunay graph is 4-regular and provide expressions for asymptotic maximum load using results from the literature. For uniform placement of servers, we map sPOT to a classical balls and bins allocation policy with bins corresponding to the Voronoi regions associated with the second order Voronoi diagram of the set of servers. We provide expressions for the lower bound on the asymptotic expected maximum load on the servers and prove that sPOT does not achieve POT load balancing benefits. However, experimental results suggest the efficacy of sPOT with respect to expected communication cost. Finally, we propose two non-uniform server sampling based POT policies that achieve the best of both the performance metrics. Experimental results validate the effectiveness of our proposed policies.

Local routing in WSPD-based spanners

The well-separated pair decomposition (WSPD) of the complete Euclidean graph defined on points in $R^2$, introduced by Callahan and Kosaraju [JACM, 42 (1): 67-90, 1995], is a technique for partitioning the edges of the complete graph based on length into a linear number of sets. Among the many different applications of WSPDs, Callahan and Kosaraju proved that the sparse subgraph that results by selecting an arbitrary edge from each set (called WSPD-spanner) is a $1 + 8/(s − 4)$-spanner, where $s > 4$ is the separation ratio used for partitioning the edges.Although competitive local-routing strategies exist for various spanners such as Yao-graphs, $\Theta$-graphs, and variants of Delaunay graphs, few local-routing strategies are known for any WSPD-spanner. Our main contribution is a local-routing algorithm with a near-optimal competitive routing ratio of $1 + O(1/s)$ on a WSPD-spanner.Specifically, using Callahan and Kosaraju’s fair split-tree, we show how to build a WSPD-spanner with spanning ratio $1 + 4/s + 4/(s − 2)$ which is a slight improvement over $1 + 8/(s − 4)$. We then present a 2-local and a 1-local routing algorithm on this spanner with competitive routing ratios of $1 + 6/(s − 2) + 4/s$ and $1 + 8/(s − 2) + 4/s + 8/s^2$, respectively. Moreover, we prove that there exists a point set for which our WSPD-spanner has a spanning ratio of at least $1 + 8/s$, thereby proving the near-optimality of its spanning ratio and the near-optimality of the routing ratio of both our routing algorithms.

Effect of grid displacement models on sensitivity derivatives computed by the continuous adjoint method in aerodynamic and conjugate heat transfer shape optimization

ABSTRACT For aerodynamic and conjugate heat transfer problems, the work reported in this article investigated the effect of the grid displacement model on the derivatives of an objective function with respect to the design variables. A comparison was made in the first step of the study between reference sensitivity derivatives computed by finite differences and different grid displacement models (volumetric B-splines, Laplace partial differential equations, Delaunay graphs and inverse distance weighting). Then, a continuous adjoint method including the adjoint to the Laplace grid displacement model was used to recompute sensitivity derivatives. To shed more light and to support the authors' findings, an alternative continuous adjoint method for computing the objective function gradient in terms of the field integrals of grid sensitivities was also formulated and programmed. Finally, the effect of an inconsistent differentiation of the grid displacement model on the optimized solution was investigated. The choice of the grid displacement model did not affect the sensitivity derivatives in any of the cases examined.

Resilience of three-dimensional sinusoidal networks in liver tissue.

Can three-dimensional, microvasculature networks still ensure blood supply if individual links fail? We address this question in the sinusoidal network, a plexus-like microvasculature network, which transports nutrient-rich blood to every hepatocyte in liver tissue, by building on recent advances in high-resolution imaging and digital reconstruction of adult mice liver tissue. We find that the topology of the three-dimensional sinusoidal network reflects its two design requirements of a space-filling network that connects all hepatocytes, while using shortest transport routes: sinusoidal networks are sub-graphs of the Delaunay graph of their set of branching points, and also contain the corresponding minimum spanning tree, both to good approximation. To overcome the spatial limitations of experimental samples and generate arbitrarily-sized networks, we developed a network generation algorithm that reproduces the statistical features of 0.3-mm-sized samples of sinusoidal networks, using multi-objective optimization for node degree and edge length distribution. Nematic order in these simulated networks implies anisotropic transport properties, characterized by an empirical linear relation between a nematic order parameter and the anisotropy of the permeability tensor. Under the assumption that all sinusoid tubes have a constant and equal flow resistance, we predict that the distribution of currents in the network is very inhomogeneous, with a small number of edges carrying a substantial part of the flow-a feature known for hierarchical networks, but unexpected for plexus-like networks. We quantify network resilience in terms of a permeability-at-risk, i.e., permeability as function of the fraction of removed edges. We find that sinusoidal networks are resilient to random removal of edges, but vulnerable to the removal of high-current edges. Our findings suggest the existence of a mechanism counteracting flow inhomogeneity to balance metabolic load on the liver.

A unified approach towards computing Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves using touching discs

This paper proposes a unified approach towards computing geometry structures viz. Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves. It initially presents an algorithm for computing the Voronoi diagram of a set of planar freeform closed curves without approximating the curves using points, lines or biarcs. The algorithm starts by computing the minimum antipodal discs (MADs) for all pairs of curves and these MADs are systematically processed to identify all branch points. The key feature of the algorithm is that it computes a branch point without computing any of the bisectors a priori. Local computations of Voronoi segments are then done using the identified pairs of the segments of curves. The theoretical foundation of the algorithm has been first laid for a set of convex curves and then extended to non-convex curves. It has also been shown that the developed algorithm for the Voronoi diagram can also be used to compute related structures such as medial axis, Delaunay graph and α-hull. They have also been addressed without computing Voronoi edges/segments. Results of the implementation have been provided along with a detailed discussion of the algorithm.

Hamiltonicity for convex shape Delaunay and Gabriel graphs

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape C. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S, and edges defined as follows. Given p,q∈S, pq is an edge of k-DGC(S) provided there exists some homothet of C with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph, denoted k-GGC(S), is defined analogously, except that the homothets considered are restricted to be smallest homothets of C with p and q on the boundary.We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ⊆ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric C. We also improve these bounds to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for t≥10). These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs.In addition, we show lower bounds of k=3 and k=6 on the existence of a bottleneck Hamiltonian cycle in the k-order Gabriel graph for squares and hexagons, respectively. Finally, we construct a point set such that for an infinite family of regular polygons Pt, the Delaunay graph DGPt does not contain a Hamiltonian cycle.

The maximal degree in a Poisson–Delaunay graph

We investigate the maximal degree in a Poisson–Delaunay graph in $\mathbf{R}^{d}$, $d\geq 2$, over all nodes in the window $\mathbf{W}_{\rho }:=\rho^{1/d}[0,1]^{d}$ as $\rho $ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. A weaker version of this result is discussed when $d\geq 3$.

Methods for calculating walking distances

In many scientific disciplines it is often necessary to refer to geographical travel distances. While online services can provide such distances, they fail for larger distances or for distances between points not connected by roads, and they do not allow for the calculation of many distances. Here we describe two novel methods of measuring travel distances which overcome these problems. Both use waypoints of populated places from the geonames.org database. The more efficient and accurate of the two uses the Dijkstra algorithm to find the shortest path through a Delaunay graph of neighbouring​ populated places.

Robust optimization for a wing at drag divergence Mach number based on an improved PSO algorithm

In this study, an improved PSO (particle swarm optimization) algorithm is proposed and applied to the robust optimization of a wing at drag divergence Mach number. In order to reduce the number of design variables, a six-order CST (class/shape function transformation) method is employed for airfoil parameterization. For the purpose of improving the optimization efficiency, Delaunay graph mapping method is adopted for mesh deformation in each iteration of the airfoil optimization, and NURBS (non-uniform rational B-splines)-FFD (free-form deformation) method is employed for mesh deformation in each iteration of the wing optimization. For improving the standard PSO algorithm, CVTs (centroidal Voronoi tessellations) method is introduced to generate original positions of the particles more dispersedly, a second-order oscillating scheme is used and an FDR (fitness distance ratio) item is added for updating velocities and positions of the particles. By virtue of the improved PSO algorithm, single point optimization and robust optimization are conducted for both airfoil and wing. The results indicate that, comparing with the single point optimizations, the robust optimizations not only reduce drag coefficients of the airfoil and the wing at cruise Mach numbers, but also attenuate the drag increments as the Mach number increases up to drag divergence Mach numbers.