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Computational Geometry

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24 paper(s) to start with

preprint2017arXiv

A robust approach for tree segmentation in deciduous forests using small-footprint airborne LiDAR data

This paper presents a non-parametric approach for segmenting trees from airborne LiDAR data in deciduous forests. Based on the LiDAR point cloud, the approach collects crown information such as steepness and height on-the-fly to delineate crown boundaries, and most importantly, does not require a priori assumptions of crown shape and size. The approach segments trees iteratively starting from the tallest within a given area to the smallest until all trees have been segmented. To evaluate its performance, the approach was applied to the University of Kentucky Robinson Forest, a deciduous closed-canopy forest with complex terrain and vegetation conditions. The approach identified 94% of dominant and co-dominant trees with a false detection rate of 13%. About 62% of intermediate, overtopped, and dead trees were also detected with a false detection rate of 15%. The overall segmentation accuracy was 77%. Correlations of the segmentation scores of the proposed approach with local terrain and stand metrics was not significant, which is likely an indication of the robustness of the approach as results are not sensitive to the differences in terrain and stand structures.

preprint2017arXiv

On Planar Greedy Drawings of 3-Connected Planar Graphs

A graph drawing is $\textit{greedy}$ if, for every ordered pair of vertices $(x,y)$, there is a path from $x$ to $y$ such that the Euclidean distance to $y$ decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The $\textit{greedy embedding conjecture}$ states that every $3$-connected planar graph admits a greedy drawing. The $\textit{convex greedy embedding conjecture}$ asserts that every $3$-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every $3$-connected planar graph admits a $\textit{planar}$ greedy drawing. Apart from being a strengthening o

preprint2016arXiv

Connectivity graphs of uncertainty regions

We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty (BU), each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable Worst-Case Uncertainty (WU), the input set has worst possible objective value among all possible point locations, which are uncertain due, for example, to imprecise data. We consider these notions of uncertainty for the bottleneck spanning tree problem, giving rise to the following Best-Case Connectivity with Uncertainty (BCU) problem: Given a family of geometric regions, choose one point per region, such that the longest edge length of an associated geometric spanning tree is minimized. We show that this problem is NP-hard even for very simple scenarios in which the regions are line segments or squares. On the other hand, we give an exact solution for the case in which there are n+k regions, where k of the regions are line segments and n of the regions are fixed points. We then give approximation algorithms for cases where the regio

preprint2016arXiv

Online Square-into-Square Packing

In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself. We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to ac- commodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a 2.82...- competitive method for minimizing the required container size, and a lower bound of 1.33 . . . for the achievable factor.

preprint2017arXiv

Collecting a Swarm in a Grid Environment Using Shared, Global Inputs

This paper investigates efficient techniques to collect and concentrate an under-actuated particle swarm despite obstacles. Concentrating a swarm of particles is of critical importance in health-care for targeted drug delivery, where micro-scale particles must be steered to a goal location. Individual particles must be small in order to navigate through micro-vasculature, but decreasing size brings new challenges. Individual particles are too small to contain on-board power or computation and are instead controlled by a global input, such as an applied fluidic flow or electric field. To make progress, this paper considers a swarm of robots initialized in a grid world in which each position is either free-space or obstacle. This paper provides algorithms that collect all the robots to one position and compares these algorithms on the basis of efficiency and implementation time.

preprint2016arXiv

Connecting a Set of Circles with Minimum Sum of Radii

We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.

preprint2017arXiv

Pointer Networks

We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence and Neural Turing Machines, because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems -- finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem -- using training

preprint2016arXiv

New Geometric Algorithms for Fully Connected Staged Self-Assembly

We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature τ = 2 as well as τ = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.

preprint2016arXiv

Data driven estimation of Laplace-Beltrami operator

Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens the door to a well-founded data-driven procedure for the bandwidth selection. Our approach relies on recent results by Lacour and Massart [LM15] on the so-called Lepski's method.

preprint2016arXiv

Even $1 \times n$ Edge-Matching and Jigsaw Puzzles are Really Hard

We prove the computational intractability of rotating and placing $n$ square tiles into a $1 \times n$ array such that adjacent tiles are compatible--either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999851. (On the other hand, there is an easy $1 \over 2$-approximation.) This is the first (correct) proof of inapproximability for edge-matching and jigsaw puzzles. Along the way, we prove NP-hardness of distinguishing, for a directed graph on $n$ nodes, between having a Hamiltonian path (length $n-1$) and having at most $0.999999284 (n-1)$ edges that form a vertex-disjoint union of paths. We use this gap hardness and gap-preserving reductions to establish similar gap hardness for $1 \times n$ jigsaw and edge-matching puzzles.

preprint2016arXiv

On the Beer index of convexity and its variants

Let $S$ be a subset of $\mathbb{R}^d$ with finite positive Lebesgue measure. The Beer index of convexity $\operatorname{b}(S)$ of $S$ is the probability that two points of $S$ chosen uniformly independently at random see each other in $S$. The convexity ratio $\operatorname{c}(S)$ of $S$ is the Lebesgue measure of the largest convex subset of $S$ divided by the Lebesgue measure of $S$. We investigate the relationship between these two natural measures of convexity. We show that every set $S\subseteq\mathbb{R}^2$ with simply connected components satisfies $\operatorname{b}(S)\leqα\operatorname{c}(S)$ for an absolute constant $α$, provided $\operatorname{b}(S)$ is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of $\operatorname{b}(S)$. For $1\leq k\leq d$, the $k$-index of convexity $\operatorname{b}_k(S)$ of a set $S\subseteq\mathbb{R}^d$ is the probability that the convex hull of a $(k+1)$-tuple of points chosen uniformly independently at random from $S$ is contained in $S$. We show that for every $d\geq 2$ there is a constant $β(d)>0$ such that every set $S\

preprint2016arXiv

On-line approach to off-line coloring problems on graphs with geometric representations

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler graphs, on-line coloring algorithms using few colors or proving that no such algorithms exist. We derive upper and lower bounds on the maximum chromatic number that rectangle overlap graphs, subtree overlap graphs, and interval filament graphs (all of which generalize interval overlap graphs) can have when their clique number is bounded. The bounds are absolute for interval filament graphs and asymptotic of the form $(\log\log n)^{f(ω)}$ for rectangle and subtree overlap graphs, where $f(ω)$ is a polynomial function of the clique number and $n$ is the number of vertices. In particular, we provi

preprint2012arXiv

The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes

The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (MSE) decreases during smoothing; For tri-dominant mixed mesh, the quality of triangles always descends while that of quads ascends. Test examples show that the MDM is convergent for both planar and surface triangular, quadrilateral and tri-quad meshes.

preprint2016arXiv

Practical linear-space Approximate Near Neighbors in high dimension

The $c$-approximate Near Neighbor problem in high dimensional spaces has been mainly addressed by Locality Sensitive Hashing (LSH), which offers polynomial dependence on the dimension, query time sublinear in the size of the dataset, and subquadratic space requirement. For practical applications, linear space is typically imperative. Most previous work in the linear space regime focuses on the case that $c$ exceeds $1$ by a constant term. In a recently accepted paper, optimal bounds have been achieved for any $c>1$ \cite{ALRW17}. Towards practicality, we present a new and simple data structure using linear space and sublinear query time for any $c>1$ including $c\to 1^+$. Given an LSH family of functions for some metric space, we randomly project points to the Hamming cube of dimension $\log n$, where $n$ is the number of input points. The projected space contains strings which serve as keys for buckets containing the input points. The query algorithm simply projects the query point, then examines points which are assigned to the same or nearby vertices on the Hamming cube. We analyze in detail the query time for some standard LSH families. To illustrate our claim of practicality,

preprint2016arXiv

EgoReID: Cross-view Self-Identification and Human Re-identification in Egocentric and Surveillance Videos

Human identification remains to be one of the challenging tasks in computer vision community due to drastic changes in visual features across different viewpoints, lighting conditions, occlusion, etc. Most of the literature has been focused on exploring human re-identification across viewpoints that are not too drastically different in nature. Cameras usually capture oblique or side views of humans, leaving room for a lot of geometric and visual reasoning. Given the recent popularity of egocentric and top-view vision, re-identification across these two drastically different views can now be explored. Having an egocentric and a top view video, our goal is to identify the cameraman in the content of the top-view video, and also re-identify the people visible in the egocentric video, by matching them to the identities present in the top-view video. We propose a CRF-based method to address the two problems. Our experimental results demonstrates the efficiency of the proposed approach over a variety of video recorded from two views.

preprint2016arXiv

Stochastic closest-pair problem and most-likely nearest-neighbor search in tree spaces

Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the $\ell$-threshold probability and the expectation of the closest-pair distance of a realization of $S$. We propose the first algorithm to compute the $\ell$-threshold probability in $O(t+n\log n+ \min\{tn,n^2\})$ time for any given threshold $\ell$, which immediately results in an $O(t+\min\{tn^3,n^4\})$-time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a $(1+\varepsilon)$-approximation for the expected closest-pair distance in $O(t+\varepsilon^{-1}\min\{tn^2,n^3\})$ time, by arguing that the expected closest-pair distance can be approximated via $O(\varepsilon^{-1}n)$ threshold probability queries. For the latter, we study the $k$ most-likely nearest-neighbor search ($k$-LNN) via a notion called $k$ most-likely Voronoi Diagr

preprint2016arXiv

A Refined Analysis of LSH for Well-dispersed Data Points

Near neighbor problems are fundamental in algorithms for high-dimensional Euclidean spaces. While classical approaches suffer from the curse of dimensionality, locality sensitive hashing (LSH) can effectively solve a-approximate r-near neighbor problem, and has been proven to be optimal in the worst case. However, for real-world data sets, LSH can naturally benefit from well-dispersed data and low doubling dimension, leading to significantly improved performance. In this paper, we address this issue and propose a refined analyses for running time of approximating near neighbors queries via LSH. We characterize dispersion of data using N_b, the number of b*r-near pairs among the data points. Combined with optimal data-oblivious LSH scheme, we get a new query time bound depending on N_b and doubling dimension. For many natural scenarios where points are well-dispersed or lying in a low-doubling-dimension space, our result leads to sharper performance than existing worst-case analysis. This paper not only present first rigorous proof on how LSHs make use of the structure of data points, but also provide important insights into parameter setting in the practice of LSH beyond worst case

preprint2016arXiv

Beam Search for Learning a Deep Convolutional Neural Network of 3D Shapes

This paper addresses 3D shape recognition. Recent work typically represents a 3D shape as a set of binary variables corresponding to 3D voxels of a uniform 3D grid centered on the shape, and resorts to deep convolutional neural networks(CNNs) for modeling these binary variables. Robust learning of such CNNs is currently limited by the small datasets of 3D shapes available, an order of magnitude smaller than other common datasets in computer vision. Related work typically deals with the small training datasets using a number of ad hoc, hand-tuning strategies. To address this issue, we formulate CNN learning as a beam search aimed at identifying an optimal CNN architecture, namely, the number of layers, nodes, and their connectivity in the network, as well as estimating parameters of such an optimal CNN. Each state of the beam search corresponds to a candidate CNN. Two types of actions are defined to add new convolutional filters or new convolutional layers to a parent CNN, and thus transition to children states. The utility function of each action is efficiently computed by transferring parameter values of the parent CNN to its children, thereby enabling an efficient beam search. Ou

preprint2016arXiv

Geometric Hitting Set for Segments of Few Orientations

We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the fewest sensors (the "hitting points"). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.

preprint2016arXiv

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG $G=(V,E)$ in general position can be augmented to a 2-connected PSLG $(V,E\cup E^+)$ by adding new edges of total Euclidean length $\|E^+\|\leq 2\|E\|$, and this bound is the best possible. An optimal edge set $E^+$ can be computed in $O(|V|^4)$ time; however the problem becomes NP-hard when $G$ is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG $G=(V,E)$ into a planar straight-line cycle $G'=(V,E')$ such that $\|E'\|\leq 2\|{\rm MST}(V)\|$, and the graph remains connected with edge length below $\|E\|+\|{\rm MST}(V)\|$ at all stages. These bounds are the best possible.

preprint2016arXiv

Intersection Graphs of Rays and Grounded Segments

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, (2) not every intersection graph of rays is an intersection graph of downward rays, and (3) not every intersection graph of rays is an outer segment graph. The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.

preprint2016arXiv

No equations, no parameters, no variables: data, and the reconstruction of normal forms by learning informed observation geometries

The discovery of physical laws consistent with empirical observations lies at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters, dynamical systems theory provides, through the appropriate normal forms, an "intrinsic", prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations, without prior knowledge or understanding, they parametrize the dynamics {\em intrinsically}, without explicit reference to fundamental physical quantities.

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