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PEMesh: a Graphical Framework for the Analysis of the InterplayBetween Geometry and PEM Solvers arXiv.cs.CG Pub Date : 2021-02-23 Daniela Cabiddu; Giuseppe Patanè; Michela Spagnuolo
Partial differential equations can be solved on general polygonal and polyhedral meshes, through Polytopal Element Methods (PEMs). Unfortunately, the relation between geometry and analysis is still unknown and subject to ongoing research in order to identify weaker shape-regularity criteria under which PEMs can reliably work. We propose PEMesh, a graphical framework to support the analysis of the relation
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Functional norms, condition numbers and numerical algorithms in algebraic geometry arXiv.cs.CG Pub Date : 2021-02-23 Felipe Cucker; Alperen A. Ergür; Josué Tonelli-Cueto
In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand in order to optimize accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty}$. This classical
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A System for 3D Reconstruction Of Comminuted Tibial Plafond Bone Fractures arXiv.cs.CG Pub Date : 2021-02-23 Pengcheng Liu; Nathan Hewitt; Waseem Shadid; Andrew Willis
High energy impacts at joint locations often generate highly fragmented, or comminuted, bone fractures. Current approaches for treatment require physicians to decide how to classify the fracture within a hierarchy fracture severity categories. Each category then provides a best-practice treatment scenario to obtain the best possible prognosis for the patient. This article identifies shortcomings associated
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No-Reference Quality Assessment for 360-degree Images by Analysis of Multi-frequency Information and Local-global Naturalness arXiv.cs.CG Pub Date : 2021-02-22 Wei Zhou; Jiahua Xu; Qiuping Jiang; Zhibo Chen
360-degree/omnidirectional images (OIs) have achieved remarkable attentions due to the increasing applications of virtual reality (VR). Compared to conventional 2D images, OIs can provide more immersive experience to consumers, benefitting from the higher resolution and plentiful field of views (FoVs). Moreover, observing OIs is usually in the head mounted display (HMD) without references. Therefore
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Cut Locus Realizations on Convex Polyhedra arXiv.cs.CG Pub Date : 2021-02-22 Joseph O'Rourke; Costin Vîlcu
We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuum of polyhedra P each realizing T for some x on P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem
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Wallpaper group kirigami arXiv.cs.CG Pub Date : 2021-02-22 Lucy Liu; Gary P. T. Choi; L. Mahadevan
Kirigami, the art of paper cutting, has become the subject of study in mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns, the design of which has to date largely relied on inspirations from art, nature, and intuition, embedded in a choice of the underlying pattern symmetry. Here we complement these approaches by clarifying
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On guarding polygons with holes arXiv.cs.CG Pub Date : 2021-02-20 Sharareh Alipour
There is an old conjecture by Shermer \cite{sher} that in a polygon with $n$ vertices and $h$ holes, $\lfloor \dfrac{n+h}{3} \rfloor$ vertex guards are sufficient to guard the entire polygon. The conjecture is proved for $h=1$ by Shermer \cite{sher} and Aggarwal \cite{aga} seperately. In this paper, we prove a theorem similar to the Shermer's conjecture for a special case where the goal is to guard
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Curvy: An Interactive Design Tool for Varying Density Support Structures arXiv.cs.CG Pub Date : 2021-02-19 Erva Ulu; Nurcan Gecer Ulu; Jiahao Li; Walter Hsiao
We introduce Curvy-an interactive design tool to generate varying density support structures for 3D printing. Support structures are essential for printing models with extreme overhangs. Yet, they often cause defects on contact areas, resulting in poor surface quality. Low-level design of support structures may alleviate such negative effects. However, it is tedious and unintuitive for novice users
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Reviews: Topological Distances and Losses for Brain Networks arXiv.cs.CG Pub Date : 2021-02-17 Moo K. Chung; Alexander Smith; Gary Shiu
Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it
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On Greedily Packing Anchored Rectangles arXiv.cs.CG Pub Date : 2021-02-16 Christoph Damerius; Dominik Kaaser; Peter Kling; Florian Schneider
Consider a set P of points in the unit square U, one of them being the origin. For each point p in P you may draw a rectangle in U with its lower-left corner in p. What is the maximum area such rectangles can cover without overlapping each other? Freedman [1969] posed this problem in 1969, asking whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu and T\'oth [2011] achieved
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Euler Characteristic Surfaces arXiv.cs.CG Pub Date : 2021-02-16 Gabriele Beltramo; Rayna Andreeva; Ylenia Giarratano; Miguel O. Bernabeu; Rik Sarkar; Primoz Skraba
We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data
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On Ray Shooting for Triangles in 3-Space and Related Problems arXiv.cs.CG Pub Date : 2021-02-15 Esther Ezra; Micha Sharir
We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra,
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IMF: An Iterative Max-Flow Method for Localizability Detection arXiv.cs.CG Pub Date : 2021-02-14 Haodi Ping
Determining whether nodes can be localized, called localizability detection, is a basic work in network localization. We present a novel approach for localizability detection given a set of nodes and their connectivity relationships. The key idea is transforming the problem of detecting three disjoint paths into a max-flow problem, which can be efficiently solved by existing algorithms. Furthermore
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A Topological Approach for Motion Track Discrimination arXiv.cs.CG Pub Date : 2021-02-10 Tegan Emerson; Sarah Tymochko; George Stantchev; Jason A. Edelberg; Michael Wilson; Colin C. Olson
Detecting small targets at range is difficult because there is not enough spatial information present in an image sub-region containing the target to use correlation-based methods to differentiate it from dynamic confusers present in the scene. Moreover, this lack of spatial information also disqualifies the use of most state-of-the-art deep learning image-based classifiers. Here, we use characteristics
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Translation Invariant Fréchet Distance Queries arXiv.cs.CG Pub Date : 2021-02-11 Joachim Gudmundsson; André van Renssen; Zeinab Saeidi; Sampson Wong
The Fr\'echet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Fr\'echet distance between them. This variant is called the Translation Invariant Fr\'echet distance, and algorithms to compute it are well studied. The query version, however, is much less well understood. We study Translation Invariant
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Sweeps, polytopes, oriented matroids, and allowable graphs of permutations arXiv.cs.CG Pub Date : 2021-02-11 Arnau Padrol; Eva Philippe
A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Mimicking the fact that sweep polytopes of point configurations
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Approximation Algorithms for Generalized Multidimensional Knapsack arXiv.cs.CG Pub Date : 2021-02-11 Arindam Khan; Eklavya Sharma; K. V. N. Sreenivas
We study a generalization of the knapsack problem with geometric and vector constraints. The input is a set of rectangular items, each with an associated profit and $d$ nonnegative weights ($d$-dimensional vector), and a square knapsack. The goal is to find a non-overlapping axis-parallel packing of a subset of items into the given knapsack such that the vector constraints are not violated, i.e., the
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A better lower bound for Lower-Left Anchored Rectangle Packing arXiv.cs.CG Pub Date : 2021-02-10 Ruben Hoeksma; Matthew Maat
Given any set of points $S$ in the unit square that contains the origin, does a set of axis aligned rectangles, one for each point in $S$, exist, such that each of them has a point in $S$ as its lower-left corner, they are pairwise interior disjoint, and the total area that they cover is at least 1/2? This question is also known as Freedman's conjecture (conjecturing that such a set of rectangles does
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Throwing a Sofa Through the Window arXiv.cs.CG Pub Date : 2021-02-08 Dan Halperin; Micha Sharir; Itay Yehuda
We study several variants of the problem of moving a convex polytope $K$, with $n$ edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically: $\bullet$ We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms
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The Maximum Exposure Problem arXiv.cs.CG Pub Date : 2021-02-06 Neeraj Kumar; Stavros Sintos; Subhash Suri
Given a set of points $P$ and axis-aligned rectangles $\mathcal{R}$ in the plane, a point $p \in P$ is called \emph{exposed} if it lies outside all rectangles in $\mathcal{R}$. In the \emph{max-exposure problem}, given an integer parameter $k$, we want to delete $k$ rectangles from $\mathcal{R}$ so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible
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Classification based on Topological Data Analysis arXiv.cs.CG Pub Date : 2021-02-07 Rolando Kindelan; José Frías; Mauricio Cerda; Nancy Hitschfeld
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML) methods. This paper proposes an algorithm that applies TDA directly to multi-class classification problems, even imbalanced datasets, without any further ML stage. The
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Foldover-free maps in 50 lines of code arXiv.cs.CG Pub Date : 2021-02-05 Vladimir GaranzhaCCRAS, MIPT; Igor KaporinCCRAS, MIPT; Liudmila KudryavtsevaCCRAS, MIPT; François Protais; Nicolas Ray; Dmitry Sokolov
Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is the most fundamental problem in geometry processing.In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers.In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to
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Length of a Full Steiner Tree as a Function of Terminal Coordinates arXiv.cs.CG Pub Date : 2021-02-05 Alexei Yu. Uteshev; Elizaveta A. Semenova
Given the coordinates of the terminals $ \{(x_j,y_j)\}_{j=1}^n $ of the full Euclidean Steiner tree, its length equals $$ \left| \sum_{j=1}^n z_j U_j \right| \, , $$ where $ \{z_j:=x_j+ \mathbf i y_j\}_{j=1}^n $ and $ \{U_j\}_{j=1}^n $ are suitably chosen $ 6 $th roots of unity. We also extend this result for the cost of the optimal Weber networks which are topologically equivalent to some full Steiner
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Force-Directed Layout of Order Diagrams using Dimensional Reduction arXiv.cs.CG Pub Date : 2021-02-04 Dominik Dürrschnabel; Gerd Stumme
Order diagrams allow human analysts to understand and analyze structural properties of ordered data. While an experienced expert can create easily readable order diagrams, the automatic generation of those remains a hard task. In this work, we adapt force-directed approaches, which are known to generate aesthetically-pleasing drawings of graphs, to the realm of order diagrams. Our algorithm ReDraw
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Extending Partial Representations of Rectangular Duals with Given Contact Orientations arXiv.cs.CG Pub Date : 2021-02-03 Steven Chaplick; Philipp Kindermann; Jonathan Klawitter; Ignaz Rutter; Alexander Wolff
A rectangular dual of a graph $G$ is a contact representation of $G$ by axis-aligned rectangles such that (i) no four rectangles share a point and (ii) the union of all rectangles is a rectangle. The partial representation extension problem for rectangular duals asks whether a given partial rectangular dual can be extended to a rectangular dual, that is, whether there exists a rectangular dual where
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Length Learning for Planar Euclidean Curves arXiv.cs.CG Pub Date : 2021-02-03 Barak Or; Liam Hazan
In this work, we used deep neural networks (DNNs) to solve a fundamental problem in differential geometry. One can find many closed-form expressions for calculating curvature, length, and other geometric properties in the literature. As we know these concepts, we are highly motivated to reconstruct them by using deep neural networks. In this framework, our goal is to learn geometric properties from
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Closed-form Quadrangulation of N-Sided Patches arXiv.cs.CG Pub Date : 2021-01-27 Marco Tarini
We analyze the problem of quadtragulating a $n$-sided patch, with prescribed numbers of edge-subdivisions at its boundary, with a single internal irregular vertex (none, when $n = 4$) breaking the otherwise fully regular lattice. We derive, in analytical closed-form, (1) the necessary and sufficient conditions that a patch must meet to admit this quadrangulation, and (2) a full description of the resulting
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Poncelet Propellers: Invariant Total Blade Area arXiv.cs.CG Pub Date : 2021-01-25 Dominique Laurain; Daniel Jaud; Dan Reznik
Given a triangle, a trio of circumellipses can be defined, each centered on an excenter. Over the family of Poncelet 3-periodics (triangles) in a concentric ellipse pair (axis-aligned or not), the trio resembles a rotating propeller, where each "blade" has variable area. Amazingly, their total area is invariant, even when the ellipse pair is not axis-aligned. We also prove a closely-related invariant
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Covering a set of line segments with a few squares arXiv.cs.CG Pub Date : 2021-01-25 Joachim Gudmundsson; Mees van de Kerkhof; André van Renssen; Frank Staals; Lionov Wiratma; Sampson Wong
We study three covering problems in the plane. Our original motivation for these problems come from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to four unit-sized, axis-parallel squares. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel
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Recognizing Visibility Graphs of Triangulated Irregular Networks arXiv.cs.CG Pub Date : 2021-01-24 Hossein Boomari Mojtaba Ostovari Alireza Zarei
A Triangulated Irregular Network (TIN) is a data structure that is usually used for representing and storing monotone geographic surfaces, approximately. In this representation, the surface is approximated by a set of triangular faces whose projection on the XY-plane is a triangulation. The visibility graph of a TIN is a graph whose vertices correspond to the vertices of the TIN and there is an edge
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Online Packing to Minimize Area or Perimeter arXiv.cs.CG Pub Date : 2021-01-22 Lorenzo Beretta; Mikkel Abrahamsen
We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90 degrees or translations only. For the
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Geometric Moment Invariants to Motion Blur arXiv.cs.CG Pub Date : 2021-01-21 Hongxiang Hao; Hanlin Mo; Hua Li
In this paper, we focus on removing interference of motion blur by the derivation of motion blur invariants.Unlike earlier work, we don't restore any blurred image. Based on geometric moment and mathematical model of motion blur, we prove that geometric moments of blurred image and original image are linearly related. Depending on this property, we can analyse whether an existing moment-based feature
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Positive Geometries for Barycentric Interpolation arXiv.cs.CG Pub Date : 2021-01-21 Márton Vaitkus
We propose a novel theoretical framework for barycentric interpolation, using concepts recently developed in mathematical physics. Generalized barycentric coordinates are defined similarly to Shepard's method, using positive geometries - subsets which possess a rational function naturally associated to their boundaries. Positive geometries generalize certain properties of simplices and convex polytopes
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On the curvature extrema of special cubic Bézier curves arXiv.cs.CG Pub Date : 2021-01-19 Kenjiro T. Miura; Péter Salvi
It is proved that special cubic B\'ezier curves, generated from quadratic curves by the use of a scalar parameter, have at most one local curvature extremum in the $(0,1)$ parameter interval.
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A Damped Newton Algorithm for Generated Jacobian Equations arXiv.cs.CG Pub Date : 2021-01-20 Anatole GallouëtLJK; Quentin MerigotLMO; Boris ThibertLJK
Generated Jacobian Equations have been introduced by Trudinger [Disc. cont. dyn. sys (2014), pp. 1663-1681] as a generalization of Monge-Amp{\`e}re equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the
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Star Discrepancy Subset Selection: Problem Formulation and Efficient Approaches for Low Dimensions arXiv.cs.CG Pub Date : 2021-01-19 Carola Doerr; Luís Paquete
Motivated by applications in instance selection, we introduce the \emph{star discrepancy subset selection problem}, which consists of finding a subset of \(m\) out of \(n\) points that minimizes the star discrepancy. We introduce two mixed integer linear formulations (MILP) and a combinatorial branch-and-bound (BB) algorithm for this problem and we evaluate our approaches against random subset selection
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Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation arXiv.cs.CG Pub Date : 2021-01-19 Karl Bringmann; André Nusser
Computing the similarity of two point sets is an ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this
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Local Complexity of Polygons arXiv.cs.CG Pub Date : 2021-01-19 Fabian Klute; Meghana M. Reddy; Tillmann Miltzow
Many problems in Discrete and Computational Geometry deal with simple polygons or polygonal regions. Many algorithms and data-structures perform considerably faster, if the underlying polygonal region has low local complexity. One obstacle to make this intuition rigorous, is the lack of a formal definition of local complexity. Here, we give two possible definitions and show how they are related in
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Reliable Spanners: Locality-Sensitive Orderings Strike Back arXiv.cs.CG Pub Date : 2021-01-19 Arnold Filtser; Hung Le
A highly desirable property of networks is robustness to failures. Consider a metric space $(X,d_X)$, a graph $H$ over $X$ is a $\vartheta$-reliable $t$-spanner if, for every set of failed vertices $B\subset X$, there is a superset $B^+\supseteq B$ such that the induced subgraph $H[X\setminus B]$ preserves all the distances between points in $X\setminus B^+$ up to a stretch factor $t$, while the expected
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Computer Aided Formal Design of Swarm Robotics Algorithms arXiv.cs.CG Pub Date : 2021-01-18 Thibaut BalabonskiVALS - LRI; Pierre CourtieuCEDRIC - SYS; Robin PelleVALS - LRI; Lionel RiegVERIMAG - IMAG; Sébastien TixeuilNPA, LINCS; Xavier UrbainDRIM
Previous works on formally studying mobile robotic swarms consider necessary and sufficient system hypotheses enabling to solve theoretical benchmark problems (geometric pattern formation, gathering, scattering, etc.). We argue that formal methods can also help in the early stage of mobile robotic swarms protocol design, to obtain protocols that are correct-by-design, even for problems arising from
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Simultaneous Embedding of Colored Graphs arXiv.cs.CG Pub Date : 2021-01-17 Debajyoti Mondal
A set of colored graphs are compatible, if for every color $i$, the number of vertices of color $i$ is the same in every graph. A simultaneous embedding of $k$ compatibly colored graphs, each with $n$ vertices, consists of $k$ planar polyline drawings of these graphs such that the vertices of the same color are mapped to a common set of vertex locations. We prove that simultaneous embedding of $k\in
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Hypernetworks: From Posets to Geometry arXiv.cs.CG Pub Date : 2021-01-16 Emil Saucan
We show that hypernetworks can be regarded as posets which, in their turn, have a natural interpretation as simplicial complexes and, as such, are endowed with an intrinsic notion of curvature, namely the Forman Ricci curvature, that strongly correlates with the Euler characteristic of the simplicial complex. This approach, inspired by the work of E. Bloch, allows us to canonically associate a simplicial
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Preprocessing Imprecise Points for the Pareto Front arXiv.cs.CG Pub Date : 2021-01-15 Ivor van der Hoog; Irina Kostitsyna; Maarten Löffler; Bettina Speckmann
In the preprocessing model for uncertain data we are given a set of regions R which model the uncertainty associated with an unknown set of points P. In this model there are two phases: a preprocessing phase, in which we have access only to R, followed by a reconstruction phase, in which we have access to points in P at a certain retrieval cost C per point. We study the following algorithmic question:
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Memory-Efficient Modeling and Slicing of Large-Scale Adaptive Lattice Structures arXiv.cs.CG Pub Date : 2021-01-13 Shengjun Liu; Tao Liu; Qiang Zou; Weiming Wang; Eugeni L. Doubrovski; Charlie C. L. Wang
Lattice structures have been widely used in various applications of additive manufacturing due to its superior physical properties. If modeled by triangular meshes, a lattice structure with huge number of struts would consume massive memory. This hinders the use of lattice structures in large-scale applications (e.g., to design the interior structure of a solid with spatially graded material properties)
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A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures arXiv.cs.CG Pub Date : 2021-01-13 Joachim Gudmundsson; Martin P. Seybold
We revisit the randomized incremental construction of the Trapezoidal Search DAG (TSD) for a set of $n$ non-crossing segments, e.g. edges from planar subdivisions. It is well known that this point location structure has ${\cal O}(n)$ expected size and ${\cal O}(n \ln n)$ expected construction time. Our main result is an improved tail bound, with exponential decay, for the size of the TSD: There is
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Space-Efficient Algorithms for Reachability in Geometric Graphs arXiv.cs.CG Pub Date : 2021-01-13 Sujoy Bhore; Rahul Jain
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families -- intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem
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Automated 3D solid reconstruction from 2D CAD using OpenCV arXiv.cs.CG Pub Date : 2021-01-12 Ajay Bangalore Harish; Abhishek Rajendra Prasad
This work demonstrates the application of OpenCV towards feature extraction from 2D engineering drawings. The extracted features are used in the reconstruction of 3D CAD models in SCAD format and generation of 3D point cloud data that is equivalent to LIDAR scan data. Several legacy designs in mechanical and aerospace engineering are available as engineering drawings rather than software generated
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Towards Stratified Space Learning: Linearly Embedded Graphs arXiv.cs.CG Pub Date : 2021-01-12 Yossi Bokor; Katharine Turner; Christopher Williams
In this paper, we consider the simplest class of stratified spaces -- linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract
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Locality Sensitive Hashing for Efficient Similar Polygon Retrieval arXiv.cs.CG Pub Date : 2021-01-12 Haim Kaplan; Jay Tenenbaum
Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval
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On Compatible Matchings arXiv.cs.CG Pub Date : 2021-01-11 Oswin Aichholzer; Alan Arroyo; Zuzana Masárová; Irene Parada; Daniel Perz; Alexander Pilz; Josef Tkadlec; Birgit Vogtenhuber
A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with
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Face-based Volume-of-Fluid interface positioning in arbitrary polyhedra arXiv.cs.CG Pub Date : 2021-01-11 Johannes Kromer; Dieter Bothe
We introduce a fast and robust algorithm for finding a plane $\Gamma$ with given normal $\vec{n}_\Gamma$, which truncates an arbitrary polyhedron $\mathcal{P}$ such that the remaining sub-polyhedron admits a given volume $\alpha|\mathcal{P}|$. In the literature, this is commonly referred to as Volume-of-Fluid (VoF) interface positioning problem. The novelty of our work is twofold: firstly, by recursive
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Sketching Merge Trees arXiv.cs.CG Pub Date : 2021-01-08 Mingzhe Li; Sourabh Palande; Bei Wang
Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. In this paper, we are interested in sketching a set of merge trees. That is, given a set T of merge trees, we would like to find a basis set S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors
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Provably Approximated ICP arXiv.cs.CG Pub Date : 2021-01-10 Ibrahim Jubran; Alaa Maalouf; Ron Kimmel; Dan Feldman
The goal of the \emph{alignment problem} is to align a (given) point cloud $P = \{p_1,\cdots,p_n\}$ to another (observed) point cloud $Q = \{q_1,\cdots,q_n\}$. That is, to compute a rotation matrix $R \in \mathbb{R}^{3 \times 3}$ and a translation vector $t \in \mathbb{R}^{3}$ that minimize the sum of paired distances $\sum_{i=1}^n D(Rp_i-t,q_i)$ for some distance function $D$. A harder version is
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Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise arXiv.cs.CG Pub Date : 2021-01-09 H. Masoomy; B. Askari; M. N. Najafi; S. M. S. Movahed
In this paper, we utilize persistent homology technique to examine the topological properties of the visibility graph constructed from fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm and the standard network in addition to the global properties in the context of topology, will be examined. Our results demonstrate that the distribution of {\it eigenvector}
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Deep Generative Model for Efficient 3D Airfoil Parameterization and Generation arXiv.cs.CG Pub Date : 2021-01-07 Wei Chen; Arun Ramamurthy
In aerodynamic shape optimization, the convergence and computational cost are greatly affected by the representation capacity and compactness of the design space. Previous research has demonstrated that using a deep generative model to parameterize two-dimensional (2D) airfoils achieves high representation capacity/compactness, which significantly benefits shape optimization. In this paper, we propose
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Snappability and singularity-distance of pin-jointed body-bar frameworks arXiv.cs.CG Pub Date : 2021-01-07 Georg Nawratil
It is well-known that there exist rigid frameworks whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these snapping capability based on the total elastic strain energy density of the framework by using the physical concept of Green-Lagrange strain. As this so-called snappability only depends on the intrinsic
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Mesh Total Generalized Variation for Denoising arXiv.cs.CG Pub Date : 2021-01-07 Zheng Liu; YanLei Li; Weina Wang; Ligang Liu; Renjie Chen
Total Generalized Variation (TGV) has recently been proven certainly successful in image processing for preserving sharp features as well as smooth transition variations. However, none of the existing works aims at numerically calculating TGV over triangular meshes. In this paper, we develop a novel numerical framework to discretize the second-order TGV over triangular meshes. Further, we propose a
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Algorithms and Hardness for Multidimensional Range Updates and Queries arXiv.cs.CG Pub Date : 2021-01-06 Joshua Lau; Angus Ritossa
Traditional orthogonal range problems allow queries over a static set of points, each with some value. Dynamic variants allow points to be added or removed, one at a time. To support more powerful updates, we introduce the Grid Range class of data structure problems over integer arrays in one or more dimensions. These problems allow range updates (such as filling all cells in a range with a constant)
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CNN-Driven Quasiconformal Model for Large Deformation Image Registration arXiv.cs.CG Pub Date : 2020-10-30 Ho Law; Gary P. T. Choi; Ka Chun Lam; Lok Ming Lui
Image registration has been widely studied over the past several decades, with numerous applications in science, engineering and medicine. Most of the conventional mathematical models for large deformation image registration rely on prescribed landmarks, which usually require tedious manual labeling and are prone to error. In recent years, there has been a surge of interest in the use of machine learning
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An Improved Approximation for Packing Big Two-Bar Charts arXiv.cs.CG Pub Date : 2021-01-02 Adil Erzin; Vladimir Shenmaier
Recently, we presented a new Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem and 2-D Vector Packing Problem. Earlier, we have proposed several polynomial approximation algorithms. In particular, when each 2-BC has at least one bar of height more than
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