scispace - formally typeset
Search or ask a question

Showing papers on "Regular polygon published in 2018"


Book ChapterDOI
16 Sep 2018
TL;DR: This work proposes to localize cell nuclei via star-convex polygons, which are a much better shape representation as compared to bounding boxes and thus do not need shape refinement.
Abstract: Automatic detection and segmentation of cells and nuclei in microscopy images is important for many biological applications. Recent successful learning-based approaches include per-pixel cell segmentation with subsequent pixel grouping, or localization of bounding boxes with subsequent shape refinement. In situations of crowded cells, these can be prone to segmentation errors, such as falsely merging bordering cells or suppressing valid cell instances due to the poor approximation with bounding boxes. To overcome these issues, we propose to localize cell nuclei via star-convex polygons, which are a much better shape representation as compared to bounding boxes and thus do not need shape refinement. To that end, we train a convolutional neural network that predicts for every pixel a polygon for the cell instance at that position. We demonstrate the merits of our approach on two synthetic datasets and one challenging dataset of diverse fluorescence microscopy images.

521 citations


Book ChapterDOI
TL;DR: In this paper, the authors propose to localize cell nuclei via star-convex polygons, which are a much better shape representation as compared to bounding boxes and thus do not need shape refinement.
Abstract: Automatic detection and segmentation of cells and nuclei in microscopy images is important for many biological applications. Recent successful learning-based approaches include per-pixel cell segmentation with subsequent pixel grouping, or localization of bounding boxes with subsequent shape refinement. In situations of crowded cells, these can be prone to segmentation errors, such as falsely merging bordering cells or suppressing valid cell instances due to the poor approximation with bounding boxes. To overcome these issues, we propose to localize cell nuclei via star-convex polygons, which are a much better shape representation as compared to bounding boxes and thus do not need shape refinement. To that end, we train a convolutional neural network that predicts for every pixel a polygon for the cell instance at that position. We demonstrate the merits of our approach on two synthetic datasets and one challenging dataset of diverse fluorescence microscopy images.

382 citations


Posted Content
TL;DR: The uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities was studied in this paper, where it was shown that for embedded two-spheres the singularity is well-posed.
Abstract: In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2\pi/e}+\delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.

52 citations


Book
05 Feb 2018
TL;DR: An algorithm which calculates shortest paths amidst two convex polyhedral obstacles in time and a new kind of Voronoi diagram, calledeper's Voronoa diagram, is introduced and analyzed here.
Abstract: We consider the problem of computing the Euclidean shortest path between two points in three-dimensional space which must avoid the interiors of k given disjoint convex polyhedral obstacles, having altogether n faces. Although this problem is hard to solve when k is arbitrarily large, it had been efficiently solved by Mount [Mo84] (cf. also Sharir and Schorr [SS84]) for k = 1, i.e. in the presence of a single convex polyhedral obstacle, in time O(n2log n). In this paper we consider the generalization of this technique to the cases k = 2 and k > 2. In the first part of this presentation we describe an algorithm which calculates shortest paths amidst two convex polyhedral obstacles in time O(n3a(n)O(a(n)7)log n), where a(n) is the functional inverse of Ackermann's function (and is thus extremely slowly growing). This result is achieved by constructing a new kind of Voronoi diagram, called peeper's Voronoi diagram, which is introduced and analyzed here. In the second part we show that shortest paths amidst k > 2 disjoint convex polyhedral obstacles can be calculated in time polynomial in the total number n of faces of these obstacles (but exponential in the number of obstacles). This is a consequence of the following result: Let K be a 3-D convex polyhedron having n vertices. Then the number of shortest-path edge sequences on K is polynomial in n (specifically O(n7)), where a shortest-path edge sequence x is a sequence of edges of K for which there exist two points X, Y on the surface S of K such that x is the sequence of edges crossed by the shortest path from X to Y along S.

50 citations


Journal ArticleDOI
TL;DR: It is proved that certain infinite families of convex polytopes are the families of graphs with constant fault-tolerant metric dimension.

49 citations


Journal ArticleDOI
TL;DR: In this article, a systematic study of regular sequences of quasi-none-expansive operators in Hilbert space is presented, in particular, in weakly, boundedly, and linearly regular sequence of operators.
Abstract: In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly, and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations, and products of operators. Moreover, in this connection, we show that weak, bounded, and linear regularity lead to weak, strong, and linear convergence, respectively, of various iterative methods. This applies, in particular, to block iterative and string averaging projection methods, which, in principle, are based on the abovementioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems.

37 citations


Book
08 Feb 2018
TL;DR: It is shown that 2n + k tactile probes are sufficient to determine the shape of a convex polygon of n sides selected from a known finite set of polygons.
Abstract: We show that 2n + k tactile probes are sufficient to determine the shape of a convex polygon of n sides selected from a known finite set of polygons. This result improves on the 3n probe algorithm of Cole and Yap (1983) in the finite case. We show k = 3 under the assumptions of Cole and Yap, k = 2 under slightly stronger assumptions, and k = −1 under the assumptions of Schwartz and Sharir (1984).

35 citations


Journal ArticleDOI
TL;DR: In this article, a unified geometric interpretation of extreme points and Arveson boundary points is given for matrix convex sets, and dilation-theoretic formulations for the above notions of extremity points are explored.
Abstract: For matrix convex sets, a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of “Euclidean” extreme points, “matrix” extreme points, and “absolute” extreme points. A seemingly different notion, the “Arveson boundary”, has by contrast a dilation-theoretic flavor. An Arveson boundary point is an analog of a (not necessarily irreducible) boundary representation for an operator system. This article provides and explores dilation-theoretic formulations for the above notions of extreme points. The scalar solution set of a linear matrix inequality (LMI) is known as a spectrahedron. The matricial solution set of an LMI is a free spectrahedron. Spectrahedra (resp. free spectrahedra) lie between general convex sets (resp. matrix convex sets) and convex polyhedra (resp. free polyhedra). As applications of our theorems on extreme points, it is shown that the polar dual of a matrix convex set K is generated, as a matrix convex set, by finitely many Arveson boundary points if and only if K is a free spectrahedron, and if the polar dual of a free spectrahedron K is again a free spectrahedron, then at the scalar level K is a polyhedron.

28 citations


Proceedings ArticleDOI
06 Nov 2018
TL;DR: The solution is, in a well-defined sense, a locally optimal solution to the problem of choosing centers in the plane and choosing an assignment of people to those 2-d centers so as to minimize the sum of squared distances subject to the assignment being balanced.
Abstract: We consider the problem of political redistricting: given the locations of people in a geographical area (e.g. a US state), the goal is to decompose the area into subareas, called districts, so that the populations of the districts are as close as possible and the districts are "compact" and "contiguous," to use the terms referred to in most US state constitutions and/or US Supreme Court rulings. We study a method that outputs a solution in which each district is the intersection of a convex polygon with the geographical area. The average number of sides per polygon is less than six. The polygons tend to be quite compact. Every two districts differ in population by at most one (so we call the solution balanced). In fact, the solution is a centroidal power diagram: each polygon has an associated center in R3 such that • the projection of the center onto the plane z = 0 is the centroid of the locations of people assigned to the polygon, and • for each person assigned to that polygon, the polygon's center is closest among all centers. The polygons are convex because they are the intersections of 3D Voronoi cells with the plane. The solution is, in a well-defined sense, a locally optimal solution to the problem of choosing centers in the plane and choosing an assignment of people to those 2-d centers so as to minimize the sum of squared distances subject to the assignment being balanced. A practical problem with this approach is that, in real-world redistricting, exact locations of people are unknown. Instead, the input consists of polygons (census blocks) and associated populations. A real redistricting must not split census blocks. We therefore propose a second phase that perturbs the solution slightly so it does not split census blocks. In our experiments, the second phase achieves this while preserving perfect population balance. The district polygons are no longer convex at the fine scale because their boundaries must follow the boundaries of census blocks, but at a coarse scale they preserve the shape of the original polygons.

27 citations


Posted Content
TL;DR: In this article, the Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces were obtained for the case of Ω(n+1)-dimensional Euclidean unit ball.
Abstract: In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

27 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the magnitudes of compact sets in Euclidean spaces and showed that the magnitude of an odd-dimensional ball is a rational function of its radius, thus disproving the general form of the Leinster-Willerton conjecture.
Abstract: The notion of the magnitude of a metric space was introduced by Leinster and developed in works by Leinster, Meckes and Willerton, but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small- and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton in the special case of balls in dimension three. In general the magnitude of an odd-dimensional ball is a rational function of its radius, thus disproving the general form of the Leinster-Willerton conjecture. In addition to Fourier-analytic and PDE techniques, the arguments also involve some combinatorial considerations.

Journal ArticleDOI
TL;DR: In this article, the half-space depth for multivariate data with notions from convex and affine geometry is discussed, which is a generalization of a measure of symmetry for convex sets, well studied in geometry.
Abstract: Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

Posted Content
TL;DR: In this paper, it was shown that any geodesic plane in a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume is either closed or dense.
Abstract: Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$ In this paper we show that any geodesic plane in $M^*$ is either closed or dense We also show that only countably many planes are closed There are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on the topology of M

Journal ArticleDOI
TL;DR: In the infinite-time limit, the density approaches a "saturation" limit as mentioned in this paper, where particles of certain shapes are randomly and sequentially placed into an empty space without overlap.
Abstract: Random sequential adsorption (RSA) is a time-dependent packing process, in which particles of certain shapes are randomly and sequentially placed into an empty space without overlap. In the infinite-time limit, the density approaches a "saturation" limit. Although this limit has attracted particular research interest, the majority of past studies could only probe this limit by extrapolation. We have previously found an algorithm to reach this limit using finite computational time for spherical particles and could thus determine the saturation density of spheres with high accuracy. In this paper, we generalize this algorithm to generate saturated RSA packings of two-dimensional polygons. We also calculate the saturation density for regular polygons of three to ten sides and obtain results that are consistent with previous, extrapolation-based studies.

Posted Content
TL;DR: When f(x,\omega) is merely convex but smoothable, by suitable choices of the smoothing, steplength, and batch-size sequences, smoothed VS-APM (or sVS- APM) produces sequences for which expected sub-optimality diminishes at the rate of $\mathcal{O}(1/k)$ with an optimal oracle complexity of $1/\epsilon^2.
Abstract: We consider minimizing $f(x) = \mathbb{E}[f(x,\omega)]$ when $f(x,\omega)$ is possibly nonsmooth and either strongly convex or convex in $x$. (I) Strongly convex. When $f(x,\omega)$ is $\mu-$strongly convex in $x$, we propose a variable sample-size accelerated proximal scheme (VS-APM) and apply it on $f_{\eta}(x)$, the ($\eta$-)Moreau smoothed variant of $\mathbb{E}[f(x,\omega)]$; we term such a scheme as (m-VS-APM). We consider three settings. (a) Bounded domains. In this setting, VS-APM displays linear convergence in inexact gradient steps, each of which requires utilizing an inner (SSG) scheme. Specifically, mVS-APM achieves an optimal oracle complexity in SSG steps; (b) Unbounded domains. In this regime, under a weaker assumption of suitable state-dependent bounds on subgradients, an unaccelerated variant mVS-PM is linearly convergent; (c) Smooth ill-conditioned $f$. When $f$ is $L$-smooth and $\kappa = L/\mu \ggg 1$, we employ mVS-APM where increasingly accurate gradients $ abla_x f_{\eta}(x)$ are obtained by VS-APM. Notably, mVS-APM displays linear convergence and near-optimal complexity in inner proximal evaluations (upto a log factor) compared to VS-APM. But, unlike a direct application of VS-APM, this scheme is characterized by larger steplengths and better empirical behavior; (II) Convex. When $f(x,\omega)$ is merely convex but smoothable, by suitable choices of the smoothing, steplength, and batch-size sequences, smoothed VS-APM (or sVS-APM) produces sequences for which expected sub-optimality diminishes at the rate of $\mathcal{O}(1/k)$ with an optimal oracle complexity of $\mathcal{O}(1/\epsilon^2)$. Finally, sVS-APM and VS-APM produce sequences that converge almost surely to a solution of the original problem.

Journal ArticleDOI
TL;DR: A suite of algorithms to enable depth‐constrained autonomous bathymetric (underwater topography) mapping by an autonomous surface vessel (ASV) are described and introduced for the partitioning of convex polygons to allow efficient path planning for coverage.
Abstract: This paper describes the design, implementation, and testing of a suite of algorithms to enable depth-constrained autonomous bathymetric (underwater topography) mapping by an autonomous surface vessel (ASV). Given a target depth and a bounding polygon, the ASV will find and follow the intersection of the bounding polygon and the depth contour as modeled online with a Gaussian process (GP). This intersection, once mapped, will then be used as a boundary within which a path will be planned for coverage to build a map of the bathymetry. Efficient methods are implemented enabling online fitting, prediction and hyperparameter optimization within the GP framework on a small embedded PC. New algorithms are introduced for the partitioning of convex polygons to allow efficient path planning for coverage. These algorithms are tested both in simulation and in the field with a small twin hull differential thrust vessel built for the task.

Journal ArticleDOI
TL;DR: In this article, it was shown that edge-labeled triangulated polygons with a single reflex vertex can have a disconnected flip graph, which is in sharp contrast with the unlabeled case, where the flip graph is connected for any triangle.
Abstract: Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ ( n log ⁡ n ) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ ( n ) bounds in their respective unlabelled settings. The Ω ( n log ⁡ n ) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O ( log 2 ⁡ n ) , although we no longer have a matching lower bound. Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of O ( n 2 ) on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter Ω ( n 3 ) .

Posted Content
TL;DR: The Gaussian Multi-Bubble Conjecture (GMC) as mentioned in this paper shows that the least Gaussian-weighted perimeter way to decompose ρ into ρ cells of prescribed (positive) Gaussian measure is to use a "simplicial cluster", obtained from the Voronoi cells of ρ equidistant points.
Abstract: We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). The case $q=2$ recovers the classical Gaussian isoperimetric inequality, and the case $q=3$ recovers our recent confirmation of the Gaussian Double-Bubble Conjecture, using a different argument. To handle the case $q > 3$, numerous new ideas and tools are required: we employ higher codimension regularity of minimizing clusters and knowledge of the combinatorial incidence structure of its interfaces, establish integrability properties of the interface curvature for stationary regular clusters, derive new formulae for the second variation of weighted volume and perimeter of such clusters, and construct a family of (approximate) inward vector-fields, having (almost) constant normal component on the interfaces of a given cell. We use all of this information to show that when $2 \leq q \leq n+1$, stable regular clusters must have convex polyhedral cells, which is the main new ingredient in this work (this was not known even in the $q=3$ case).

Journal ArticleDOI
TL;DR: This paper extends the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates.
Abstract: Summary The conventional approach to construct quadratic elements for an n sided polygon will yield n(n + 1)/2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates. This approach ensures 2n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker-delta property and satisfy constant, linear and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate and can tolerate extreme mesh distortion. This article is protected by copyright. All rights reserved.

Proceedings ArticleDOI
01 Jul 2018
TL;DR: In this article, the authors present a set of re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity, the cake is a one-dimensional interval and each piece must be a contiguous interval.
Abstract: A heterogeneous resource, such as a land-estate, is already divided among several agents in an unfair way. It should be re-divided among the agents in a way that balances fairness with ownership rights. We present re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity --- the cake is a one-dimensional interval and each piece must be a contiguous interval; (c) rectangularity --- the cake is a two-dimensional rectangle or rectilinear polygon and the pieces should be rectangles; (d) convexity --- the cake is a two-dimensional convex polygon and the pieces should be convex. Our re-division protocols have implications on another problem: the price-of-fairness --- the loss of social welfare caused by fairness requirements. Each protocol implies an upper bound on the price-of-fairness with the respective geometric constraints.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in Ωd.
Abstract: We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

Journal ArticleDOI
TL;DR: In this article, the authors studied random lozenge tilings of general non-convex polygonal regions and showed that the pairwise interaction of the non-Convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations.
Abstract: This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model, we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.

Journal ArticleDOI
01 Nov 2018-Order
TL;DR: In this paper, a class of polyhedra associated to marked posets are studied and a combinatorial characterization of the partitions of these partitions is given. But the authors focus on the intersection of these polyhedras with facets in correspondence to the covering relations of the poset.
Abstract: We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand–Tsetlin polytopes and cones, as well as Berenstein–Zelevinsky polytopes—all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.

Posted Content
TL;DR: In this paper, it was shown that any convex body in the plane can be partitioned into convex parts of equal areas and perimeters for any integer ρ ≥ 2.
Abstract: We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $m\ge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible higher-dimensional generalizations and difficulties of extending our technique to equalizing more than one non-additive function.

Journal ArticleDOI
TL;DR: Through similarity measurement experiments of different lakes in multiple representations and matching experiments between two urban area datasets, results showed that the method could distinguish areal entities even if they are represented by different kinds of polygons.
Abstract: Shape similarity measurement model is often used to solve shape-matching problems in geospatial data matching. It is widely used in geospatial data integration, conflation, updating and quality assessment. Many shape similarity measurements apply only to simple polygons. However, areal entities can be represented either by simple polygons, holed polygons or multipolygons in geospatial data. This paper proposes a new shape similarity measurement model that can be used for all kinds of polygons. In this method, convex hulls of polygons are used to extract boundary features of entities and local moment invariants are calculated to extract overall shape features of entities. Combined with convex hull and local moment invariants, polygons can be represented by convex hull moment invariant curves. Then, a shape descriptor is obtained by applying fast Fourier transform to convex hull moment invariant curves, and shape similarity between areal entities is measured by the shape descriptor. Through similarity measurement experiments of different lakes in multiple representations and matching experiments between two urban area datasets, results showed that the method could distinguish areal entities even if they are represented by different kinds of polygons.

Journal ArticleDOI
TL;DR: In this paper, the authors report conjectural realizations for all 2-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes.
Abstract: A k-associahedron is a simplicial complex whose facets, called k-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no k + 1 diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes, and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This article reports on conjectural realizations for all 2-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes. Experiments certify that we obtain fan realizations of 2-associahedra of an n-gon for n ∈ {10, 11, 12, 13}, further ones being out of our computational reach.

Journal ArticleDOI
06 Dec 2018-Symmetry
TL;DR: An integer linear programming (ILP) formulation is provided for the binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally and tightness is shown in the obtained upper bounds.
Abstract: A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

Posted Content
TL;DR: The first result is a new characterization of convex polygons that multi-tile the plane by translations that provides a very efficient criteria to tell whether a convex polygon admits translational multi-tilings.
Abstract: Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficient criteria to tell whether a convex polygon admits translational multi-tilings. As an application, one can easily construct symmetric $(2m)$-gons, for any $m\geq 4$, that do not multi-tile by translations. Finally, we prove a convex polygon which is not a parallelogram only admits periodic multiple tilings, if any.

Posted Content
TL;DR: In this paper, the Vietoris-Rips complex of a regular polygon in the plane with sides is characterized and the homotopy types of the complex are shown to include spheres of all dimensions.
Abstract: Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let $P_n$ be the boundary of a regular polygon in the plane with $n$ sides; we describe the homotopy types of Vietoris-Rips complexes of $P_n$. Indeed, when $n=(k+1)!!$ is an odd double factorial, we provide a complete characterization of the homotopy types and persistent homology of the Vietoris-Rips complexes of $P_n$ up to a scale parameter $r_n$, where $r_n$ approaches the diameter of $P_n$ as $n\to\infty$. Surprisingly, these homotopy types include spheres of all dimensions. Roughly speaking, the number of higher-dimensional spheres appearing is linked to the number of equilateral (but not necessarily equiangular) stars that can be inscribed into $P_n$. As our main tool we use the recently-developed theory of cyclic graphs and winding fractions. Furthermore, we show that the Vietoris-Rips complex of an arbitrarily dense subset of $P_n$ need not be homotopy equivalent to the Vietoris-Rips complex of $P_n$ itself, and indeed, these two complexes can have different homology groups in arbitrarily high dimensions. As an application of our results, we provide a lower bound on the Gromov-Hausdorff distance between $P_n$ and the circle.

Journal ArticleDOI
TL;DR: In this paper, a general method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull is presented.
Abstract: In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.