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  • A Practical Algorithm with Performance Guarantees for the Art~Gallery Problem
    arXiv.cs.CG Pub Date : 2020-07-14
    Simon Hengeveld; Tillmann Miltzow

    Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems

    更新日期:2020-07-15
  • Predicates of the 3D Apollonius Diagram
    arXiv.cs.CG Pub Date : 2020-07-13
    Manos Kamarianakis

    In this thesis we study one of the fundamental predicates required for the construction of the 3D Apollonius diagram (also known as the 3D Additively Weighted Voronoi diagram), namely the EDGECONFLICT predicate: given five sites $S_i, S_j,S_k,S_l,S_m$ that define an edge $e_{ijklm}$ in the 3D Apollonius diagram, and a sixth query site $S_q$, the predicate determines the portion of $e_{ijklm}$ that

    更新日期:2020-07-15
  • Conformal mapping in linear time
    arXiv.cs.CG Pub Date : 2020-07-13
    Christopher J. Bishop

    Given any $\epsilon >0$ and any planar region $\Omega$ bounded by a simple n-gon $P$ we construct a ($1 + \epsilon)$-quasiconformal map between $\Omega$ and the unit disk in time $C(\epsilon)n$. One can take $ C(\epsilon) = C + C \log (1/\epsilon) \log \log (1/\epsilon)$.

    更新日期:2020-07-15
  • City Guarding with Limited Field of View
    arXiv.cs.CG Pub Date : 2020-07-12
    Ovidiu Daescu; Hemant Malik

    Drones and other small unmanned aerial vehicles are starting to get permission to fly within city limits. While video cameras are easily available in most cities, their purpose is to guard the streets at ground level. Guarding the aerial space of a city with video cameras is a problem that so far has been largely ignored. In this paper, we present bounds on the number of cameras needed to guard the

    更新日期:2020-07-14
  • Extending Nearly Complete 1-Planar Drawings in Polynomial Time
    arXiv.cs.CG Pub Date : 2020-07-08
    Eduard Eiben; Robert Ganian; Thekla Hamm; Fabian Klute; Martin Nöllenburg

    The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, the extension problem asks whether $\mathcal{H}$ can be extended into a drawing of $G$ while maintaining some desired property of the drawing (e.g., planarity)

    更新日期:2020-07-13
  • An Interior Point Method Solving Motion Planning Problems with Narrow Passages
    arXiv.cs.CG Pub Date : 2020-07-09
    Jim Mainprice; Nathan Ratliff; Marc Toussaint; Stefan Schaal

    Algorithmic solutions for the motion planning problem have been investigated for five decades. Since the development of A* in 1969 many approaches have been investigated, traditionally classified as either grid decomposition, potential fields or sampling-based. In this work, we focus on using numerical optimization, which is understudied for solving motion planning problems. This lack of interest in

    更新日期:2020-07-10
  • Incidences with curves in three dimensions
    arXiv.cs.CG Pub Date : 2020-07-07
    Micha Sharir; Noam Solomon

    We study incidence problems involving points and curves in $R^3$. The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz, requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies, by algebraic curves of some constant degree

    更新日期:2020-07-09
  • Computing a maximum clique in geometric superclasses of disk graphs
    arXiv.cs.CG Pub Date : 2020-07-07
    Nicolas Grelier

    In the 90's Clark, Colbourn and Johnson wrote a seminal paper, where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of (unit) d-dimensional balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation

    更新日期:2020-07-08
  • Skeletonization via Local Separators
    arXiv.cs.CG Pub Date : 2020-07-07
    Andreas Bærentzen; Eva Rotenberg

    We propose a new algorithm for curve skeleton computation which differs from previous algorithms by being based on the notion of local separators. The main benefits of this approach are that it is able to capture relatively fine details and that it works robustly on a range of shape representations. Specifically, our method works on shape representations that can be construed as a spatially embedded

    更新日期:2020-07-08
  • Streaming Complexity of SVMs
    arXiv.cs.CG Pub Date : 2020-07-07
    Alexandr Andoni; Collin Burns; Yi Li; Sepideh Mahabadi; David P. Woodruff

    We study the space complexity of solving the bias-regularized SVM problem in the streaming model. This is a classic supervised learning problem that has drawn lots of attention, including for developing fast algorithms for solving the problem approximately. One of the most widely used algorithms for approximately optimizing the SVM objective is Stochastic Gradient Descent (SGD), which requires only

    更新日期:2020-07-08
  • Improved flat-back 3D gadgets in origami extrusions completely downward compatible with the conventional pyramid-supported 3D gadgets
    arXiv.cs.CG Pub Date : 2020-07-02
    Mamoru Doi

    An origami extrusion is a folding of a 3D object in the middle of a flat piece of paper, using 3D gadgets which create faces with solid angles. In this paper we focus on 3D gadgets which create a top face parallel to the ambient paper and two side faces sharing a ridge, with two outgoing simple pleats, where a simple pleat is a pair of a mountain fold and a valley fold. There are two such types of

    更新日期:2020-07-07
  • Detail Preserved Point Cloud Completion via Separated Feature Aggregation
    arXiv.cs.CG Pub Date : 2020-07-05
    Wenxiao Zhang; Qingan Yan; Chunxia Xiao

    Point cloud shape completion is a challenging problem in 3D vision and robotics. Existing learning-based frameworks leverage encoder-decoder architectures to recover the complete shape from a highly encoded global feature vector. Though the global feature can approximately represent the overall shape of 3D objects, it would lead to the loss of shape details during the completion process. In this work

    更新日期:2020-07-07
  • Radial Intersection Count Image: a Clutter Resistant 3D Shape Descriptor
    arXiv.cs.CG Pub Date : 2020-07-05
    Bart Iver van Blokland; Theoharis Theoharis

    A novel shape descriptor for cluttered scenes is presented, the Radial Intersection Count Image (RICI), and is shown to significantly outperform the classic Spin Image (SI) and 3D Shape Context (3DSC) in both uncluttered and, more significantly, cluttered scenes. It is also faster to compute and compare. The clutter resistance of the RICI is mainly due to the design of a novel distance function, capable

    更新日期:2020-07-07
  • TilinGNN: Learning to Tile with Self-Supervised Graph Neural Network
    arXiv.cs.CG Pub Date : 2020-07-05
    Hao Xu; Ka Hei Hui; Chi-Wing Fu; Hao Zhang

    We introduce the first neural optimization framework to solve a classical instance of the tiling problem. Namely, we seek a non-periodic tiling of an arbitrary 2D shape using one or more types of tiles: the tiles maximally fill the shape's interior without overlaps or holes. To start, we reformulate tiling as a graph problem by modeling candidate tile locations in the target shape as graph nodes and

    更新日期:2020-07-07
  • Computational LEGO Technic Design
    arXiv.cs.CG Pub Date : 2020-07-05
    Hao Xu; Ka-Hei Hui; Chi-Wing Fu; Hao Zhang

    We introduce a method to automatically compute LEGO Technic models from user input sketches, optionally with motion annotations. The generated models resemble the input sketches with coherently-connected bricks and simple layouts, while respecting the intended symmetry and mechanical properties expressed in the inputs. This complex computational assembly problem involves an immense search space, and

    更新日期:2020-07-07
  • A Complete List of All Convex Polyhedra Made by Gluing Regular Pentagons
    arXiv.cs.CG Pub Date : 2020-07-03
    Elena Arseneva; Stefan Langerman; Boris Zolotov

    We give a complete description of all convex polyhedra whose surface can be constructed from several congruent regular pentagons by folding and gluing them edge to edge. Our method of determining the graph structure of the polyhedra from a gluing is of independent interest and can be used in other similar settings.

    更新日期:2020-07-06
  • Sublinear Explicit Incremental Planar Voronoi Diagrams
    arXiv.cs.CG Pub Date : 2020-07-03
    Elena Arseneva; John Iacono; Grigorios Koumoutsos; Stefan Langerman; Boris Zolotov

    A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms. This is the first result to achieve

    更新日期:2020-07-06
  • Universality of the Bottleneck Distance for Extended Persistence Diagrams
    arXiv.cs.CG Pub Date : 2020-07-03
    Ulrich Bauer; Magnus Bakke Botnan; Benedikt Fluhr

    The extended persistence diagram is an invariant of piecewise linear functions, introduced by Cohen-Steiner, Edelsbrunner, and Harer. The bottleneck distance has been introduced by the same authors as an extended pseudometric on the set of extended persistence diagrams, which is stable under perturbations of the function. We address the question whether the bottleneck distance is the largest possible

    更新日期:2020-07-06
  • Medial Axis Isoperimetric Profiles
    arXiv.cs.CG Pub Date : 2020-07-02
    Paul Zhang; Daryl Deford; Justin Solomon

    Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations

    更新日期:2020-07-06
  • volesti: Volume Approximation and Sampling for Convex Polytopes in R
    arXiv.cs.CG Pub Date : 2020-07-03
    Apostolos Chalkis; Vissarion Fisikopoulos

    Sampling from high dimensional distributions and volume approximation of convex bodies are fundamental operations that appear in optimization, finance, engineering and machine learning. In this paper we present volesti, a C++ package with an R interface that provides efficient, scalable algorithms for volume estimation, uniform and Gaussian sampling from convex polytopes. volesti scales to hundreds

    更新日期:2020-07-06
  • Surface Denoising based on Normal Filtering in a Robust Statistics Framework
    arXiv.cs.CG Pub Date : 2020-07-02
    Sunil Kumar Yadav; Martin Skrodzki; Eric Zimmermann; Konrad Polthier

    During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as points-set or triangulated mesh). The noise-removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterwards. Therefore

    更新日期:2020-07-03
  • Multi-Axis Support-Free Printing of Freeform Parts with Lattice Infill Structures
    arXiv.cs.CG Pub Date : 2020-06-30
    Yamin Li; Kai Tang; Dong He; Xiangyu Wang

    In additive manufacturing, infill structures are commonly used to reduce the weight and cost of a solid part. Currently, most infill structure generation methods are based on the conventional 2.5-axis printing configuration, which, although able to satisfy the self-supporting condition on the infills, suffer from the well-known stair-case effect on the finished surface and the need of extensive support

    更新日期:2020-07-02
  • Future Urban Scenes Generation Through Vehicles Synthesis
    arXiv.cs.CG Pub Date : 2020-07-01
    Alessandro Simoni; Luca Bergamini; Andrea Palazzi; Simone Calderara; Rita Cucchiara

    In this work we propose a deep learning pipeline to predict the visual future appearance of an urban scene. Despite recent advances, generating the entire scene in an end-to-end fashion is still far from being achieved. Instead, here we follow a two stages approach, where interpretable information is included in the loop and each actor is modelled independently. We leverage a per-object novel view

    更新日期:2020-07-02
  • Deep Feature Space: A Geometrical Perspective
    arXiv.cs.CG Pub Date : 2020-06-30
    Ioannis Kansizoglou; Loukas Bampis; Antonios Gasteratos

    One of the most prominent attributes of Neural Networks (NNs) constitutes their capability of learning to extract robust and descriptive features from high dimensional data, like images. Hence, such an ability renders their exploitation as feature extractors particularly frequent in an abundant of modern reasoning systems. Their application scope mainly includes complex cascade tasks, like multi-modal

    更新日期:2020-07-02
  • Vertex guarding for dynamic orthogonal art galleries
    arXiv.cs.CG Pub Date : 2020-06-30
    Debangshu Banerjee; R. Inkulu

    Given an orthogonal polygon with orthogonal holes, we devise a dynamic algorithm for guarding with vertex guards, i.e., whenever orthogonal polygon is modified, algorithm updates the set of vertex guards and their positions for guarding the modified orthogonal polygon. Our algorithm modifies the guard placement locally while ensuring the updated orthogonal polygon with $h$ holes and $n$ vertices, is

    更新日期:2020-07-01
  • Subspace approximation with outliers
    arXiv.cs.CG Pub Date : 2020-06-30
    Amit Deshpande; Rameshwar Pratap

    The subspace approximation problem with outliers, for given $n$ points in $d$ dimensions $x_{1},\ldots, x_{n} \in R^{d}$, an integer $1 \leq k \leq d$, and an outlier parameter $0 \leq \alpha \leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-\alpha)n$ points. More generally, the $\ell_{p}$ subspace approximation problem with

    更新日期:2020-07-01
  • Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots
    arXiv.cs.CG Pub Date : 2020-06-28
    Jared Coleman; Evangelos Kranakis; Oscar Morales-Ponce; Jaroslav Opatrny; Jorge Urrutia; Birgit Vogtenhuber

    In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions

    更新日期:2020-06-30
  • Social Distancing is Good for Points too!
    arXiv.cs.CG Pub Date : 2020-06-28
    Alejandro Flores-Velazco

    The nearest-neighbor rule is a well-known classification technique that, given a training set P of labeled points, classifies any unlabeled query point with the label of its closest point in P. The nearest-neighbor condensation problem aims to reduce the training set without harming the accuracy of the nearest-neighbor rule. FCNN is the most popular algorithm for condensation. It is heuristic in nature

    更新日期:2020-06-30
  • Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
    arXiv.cs.CG Pub Date : 2020-06-26
    Esther M. Arkin; Rathish Das; Jie Gao; Mayank Goswami; Joseph S. B. Mitchell; Valentin Polishchuk; Csaba D. Toth

    Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different

    更新日期:2020-06-29
  • Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries
    arXiv.cs.CG Pub Date : 2020-06-25
    Akash Kumar; Adish Singla; Yisong Yue; Yuxin Chen

    We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $\phi$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher

    更新日期:2020-06-29
  • An Efficient, Practical Algorithm and Implementation for Computing Multiplicatively Weighted Voronoi Diagrams
    arXiv.cs.CG Pub Date : 2020-06-25
    Martin Held; Stefan de Lorenzo

    We present a simple wavefront-like approach for computing multiplicatively weighted Voronoi diagrams of points and straight-line segments in the Euclidean plane. If the input sites may be assumed to be randomly weighted points then the use of a so-called overlay arrangement [Har-Peled&Raichel, Discrete Comput. Geom. 53:547-568, 2015] allows to achieve an expected runtime complexity of $O(n\log^4 n)$

    更新日期:2020-06-26
  • A linear time algorithm for constructing orthogonal floor plans with minimum number of bends
    arXiv.cs.CG Pub Date : 2020-06-25
    Pinki; Krishnendra Shekhawat

    Let G = (V, E) be a planar triangulated graph (PTG) having every face triangular. A rectilinear dual or an orthogonal floor plan (OFP) of G is obtained by partitioning a rectangle into \mid V \mid rectilinear regions (modules) where two modules are adjacent if and only if there is an edge between the corresponding vertices in G. In this paper, a linear-time algorithm is presented for constructing an

    更新日期:2020-06-26
  • Distance bounds for high dimensional consistent digital rays and 2-D partially-consistent digital rays
    arXiv.cs.CG Pub Date : 2020-06-24
    Man-Kwun Chiu; Matias Korman; Martin Suderland; Takeshi Tokuyama

    We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with $\Theta(\log N)$ error

    更新日期:2020-06-26
  • A Linear-Time Algorithm for Discrete Radius Optimally Augmenting Paths in a Metric Space
    arXiv.cs.CG Pub Date : 2020-06-24
    Haitao Wang; Yiming Zhao

    Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of $P$. Previously, the "continuous" version of the problem where a center may be a point in the interior of an edge of the graph was studied and a linear-time algorithm

    更新日期:2020-06-26
  • A Parameterized Family of Meta-Submodular Functions
    arXiv.cs.CG Pub Date : 2020-06-23
    Mehrdad Ghadiri; Richard Santiago; Bruce Shepherd

    Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function

    更新日期:2020-06-25
  • Disjointness through the Lens of Vapnik-Chervonenkis Dimension: Sparsity and Beyond
    arXiv.cs.CG Pub Date : 2020-06-24
    Anup Bhattacharya; Sourav Chakraborty; Arijit Ghosh; Gopinath Mishra; Manaswi Paraashar

    The disjointness problem - where Alice and Bob are given two subsets of $\{1, \dots, n\}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be $\Theta(n)$, it is also known that if the sets are assumed to be drawn from some restricted set systems

    更新日期:2020-06-25
  • OMiCroN -- Oblique Multipass Hierarchy Creation while Navigating
    arXiv.cs.CG Pub Date : 2020-06-23
    Vinícius da Silva; Claudio Esperança; Ricardo Marroquim

    Rendering large point clouds ordinarily requires building a hierarchical data structure for accessing the points that best represent the object for a given viewing frustum and level-of-detail. The building of such data structures frequently represents a large portion of the cost of the rendering pipeline both in terms of time and space complexity, especially when rendering is done for inspection purposes

    更新日期:2020-06-25
  • A Family of Constant-Areas Deltoids Associated with the Ellipse
    arXiv.cs.CG Pub Date : 2020-06-23
    Ronaldo Garcia; Dan Reznik; Hellmuth Stachel; Mark Helman

    The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp deltoid which is the affine image of the Steiner Curve. Over all M the family has invariant area and displays an array of interesting properties.

    更新日期:2020-06-24
  • Analytic Solution to the Piecewise Linear Interface Construction Problem and its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes
    arXiv.cs.CG Pub Date : 2020-06-23
    Moritz Lehmann; Stephan Gekle

    The plane-cube intersection problem has been around in literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding compute time, so a faster, analytic solution to the plane-cube intersection would greatly reduce

    更新日期:2020-06-24
  • Duality-based approximation algorithms for depth queries and maximum depth
    arXiv.cs.CG Pub Date : 2020-06-22
    Dror Aiger; Haim Kaplan; Micha Sharir

    We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in higher dimensions. We then use this structure to find a point of an approximate maximum depth in $\mathcal{A}(S)$. Specifically, given an error parameter $\epsilon>0$

    更新日期:2020-06-23
  • Improved Bounds for Metric Capacitated Covering Problems
    arXiv.cs.CG Pub Date : 2020-06-22
    Sayan Bandyapadhyay

    In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to the balls in $\mathcal{B}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most

    更新日期:2020-06-23
  • The space of sections of a smooth function
    arXiv.cs.CG Pub Date : 2020-06-22
    Gunnar Carlsson; Benjamin Filippenko

    Given a compact manifold $X$ with boundary and a submersion $f : X \rightarrow Y$ whose restriction to the boundary of $X$ has isolated critical points with distinct critical values and where $Y$ is $[0,1]$ or $S^1$, the connected components of the space of sections of $f$ are computed from $\pi_0$ and $\pi_1$ of the fibers of $f$. This computation is then leveraged to provide new results on a smoothed

    更新日期:2020-06-23
  • Cycle-based formulations in Distance Geometry
    arXiv.cs.CG Pub Date : 2020-06-20
    Leo Liberti; Gabriele Iommazzo; Carlile Lavor; Nelson Maculan

    The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the

    更新日期:2020-06-23
  • Universal Geometric Graphs
    arXiv.cs.CG Pub Date : 2020-06-19
    Fabrizio Frati; Michael Hoffmann; Csaba D. Tóth

    We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric

    更新日期:2020-06-22
  • Efficient Planar Two-Center Algorithms
    arXiv.cs.CG Pub Date : 2020-06-18
    Jongmin Choi; Hee-Kap Ahn

    We consider the planar Euclidean two-center problem in which given $n$ points in the plane find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n \log n)$-time algorithm for the case that the centers of the two optimal disks are close together, that is, the overlap of the two optimal disks is a constant fraction of the disk area. This improves the previous

    更新日期:2020-06-19
  • Cyclic space-filling curves and their clustering property
    arXiv.cs.CG Pub Date : 2020-06-18
    Igor V. Netay

    In this paper we introduce an algorithm of construction of cyclic space-filling curves. One particular construction provides a family of space-filling curves in all dimensions (H-curves). They are compared here with the Hilbert curve in the sense of clustering properties, and it turns out that the constructed curve is very close and sometimes a bit better than the Hilbert curve. At the same time, its

    更新日期:2020-06-19
  • Fair k-Means Clustering
    arXiv.cs.CG Pub Date : 2020-06-17
    Mehrdad Ghadiri; Samira Samadi; Santosh Vempala

    We show that the popular $k$-means clustering algorithm (Lloyd's heuristic), used for a variety of scientific data, can result in outcomes that are unfavorable to subgroups of data (e.g., demographic groups). Such biased clusterings can have deleterious implications for human-centric applications such as resource allocation. We present a fair $k$-means objective and algorithm to choose cluster centers

    更新日期:2020-06-19
  • Robust Persistence Diagrams using Reproducing Kernels
    arXiv.cs.CG Pub Date : 2020-06-17
    Siddharth Vishwanath; Kenji Fukumizu; Satoshi Kuriki; Bharath Sriperumbudur

    Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations

    更新日期:2020-06-18
  • Bad Projections of the PSD Cone
    arXiv.cs.CG Pub Date : 2020-06-17
    Yuhan Jiang; Bernd Sturmfels

    The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with

    更新日期:2020-06-18
  • Distinct distances in the complex plane
    arXiv.cs.CG Pub Date : 2020-06-16
    Adam Sheffer; Joshua Zahl

    We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of points in $P$ have complex distance 0.

    更新日期:2020-06-16
  • Dense Non-Rigid Structure from Motion: A Manifold Viewpoint
    arXiv.cs.CG Pub Date : 2020-06-15
    Suryansh Kumar; Luc Van Gool; Carlos E. P. de Oliveira; Anoop Cherian; Yuchao Dai; Hongdong Li

    Non-Rigid Structure-from-Motion (NRSfM) problem aims to recover 3D geometry of a deforming object from its 2D feature correspondences across multiple frames. Classical approaches to this problem assume a small number of feature points and, ignore the local non-linearities of the shape deformation, and therefore, struggles to reliably model non-linear deformations. Furthermore, available dense NRSfM

    更新日期:2020-06-15
  • High-precision Wasserstein barycenters in polynomial time
    arXiv.cs.CG Pub Date : 2020-06-14
    Jason M. Altschuler; Enric Boix-Adsera

    Computing Wasserstein barycenters is a fundamental geometric problem with widespread applications in machine learning, statistics, and computer graphics. However, it is unknown whether Wasserstein barycenters can be computed in polynomial time, either exactly or to high precision (i.e., with $\textrm{polylog}(1/\varepsilon)$ runtime dependence). This paper answers these questions in the affirmative

    更新日期:2020-06-14
  • A Generalized Asymmetric Dual-front Model for Active Contours and Image Segmentation
    arXiv.cs.CG Pub Date : 2020-06-14
    Da~Chen; Jack Spencer; Jean-Marie Mirebeau; Ke Chen; Ming-Lei Shu; Laurent D. Cohen

    The geodesic distance-based dual-front curve evolution model is a powerful and efficient solution to the active contours and image segmentation issues. In its basic formulation, the dual-front model regards the meeting interfaces of two adjacent Voronoi regions as the evolving curves in the course of curve evolution. One of the most crucial ingredients for the construction of Voronoi regions or Voronoi

    更新日期:2020-06-14
  • Defending Support Vector Machines against Poisoning Attacks: the Hardness and Algorithm
    arXiv.cs.CG Pub Date : 2020-06-14
    Hu Ding; Fan Yang; Jiawei Huang

    Adversarial machine learning has attracted a great amount of attention in recent years. In a poisoning attack, the adversary can inject a small number of specially crafted samples into the training data which make the decision boundary severely deviate and cause unexpected misclassification. Due to the great importance and popular use of support vector machines (SVM), we consider defending SVM against

    更新日期:2020-06-14
  • Learning Manifolds for Sequential Motion Planning
    arXiv.cs.CG Pub Date : 2020-06-13
    Isabel M. Rayas Fernández; Giovanni Sutanto; Peter Englert; Ragesh K. Ramachandran; Gaurav S. Sukhatme

    Motion planning with constraints is an important part of many real-world robotic systems. In this work, we study manifold learning methods to learn such constraints from data. We explore two methods for learning implicit constraint manifolds from data: Variational Autoencoders (VAE), and a new method, Equality Constraint Manifold Neural Network (ECoMaNN). With the aim of incorporating learned constraints

    更新日期:2020-06-13
  • Application of Mean Curvature Flow for Surface Parametrizations
    arXiv.cs.CG Pub Date : 2020-06-12
    Ka Wai Wong

    This is an expository article describing the conformalized mean curvature flow, originally introduced by Kazhdan, Solomon, and Ben-Chen. We are interested in applying mean curvature flow to surface parametrizations. We discuss our own implementation of their algorithm and some limitations.

    更新日期:2020-06-12
  • On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds
    arXiv.cs.CG Pub Date : 2020-06-12
    Frank Nielsen

    We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis entropy related to the conformal flattening of the Fisher-Rao geometry. We prove that the Voronoi diagrams of the Fisher-Rao

    更新日期:2020-06-12
  • Planar Rectilinear Drawings of Outerplanar Graphs in Linear Time
    arXiv.cs.CG Pub Date : 2020-06-12
    Fabrizio Frati

    We show how to test in linear time whether an outerplanar graph admits a planar rectilinear drawing, both if the graph has a prescribed plane embedding that the drawing has to respect and if it does not. Our algorithm returns a planar rectilinear drawing if the graph admits one.

    更新日期:2020-06-12
  • Unsupervised Learning of 3D Point Set Registration
    arXiv.cs.CG Pub Date : 2020-06-11
    Lingjing Wang; Xiang Li; Yi Fang

    Point cloud registration is the process of aligning a pair of point sets via searching for a geometric transformation. Recent works leverage the power of deep learning for registering a pair of point sets. However, unfortunately, deep learning models often require a large number of ground truth labels for training. Moreover, for a pair of source and target point sets, existing deep learning mechanisms

    更新日期:2020-06-11
  • The nearest-colattice algorithm
    arXiv.cs.CG Pub Date : 2020-06-10
    Thomas Espitau; Paul Kirchner

    In this work, we exhibit a hierarchy of polynomial time algorithms solving approximate variants of the Closest Vector Problem (\CVP). Our first contribution is a heuristic algorithm achieving the same distance tradeoff as \HSVP{} algorithms, namely $\approx \beta^{\frac{n}{2\beta}}\textrm{covol}(\Lambda)^{\frac{1}{n}}$ for a random lattice $\Lambda$ of rank $n$. Compared to the so-called Kannan's embedding

    更新日期:2020-06-10
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