scholarly journals Distribution of Distances and Triangles in a Point Set and Algorithms for Computing the Largest Common Point Sets

1998 ◽  
Vol 20 (3) ◽  
pp. 307-331 ◽  
Author(s):  
T. Akutsu ◽  
H. Tamaki ◽  
T. Tokuyama
Keyword(s):  
Common Point ◽  
Point Sets ◽  
Point Set

Finding Largest Common Point Sets

2017 ◽  
Vol 27 (03) ◽  
pp. 177-185 ◽  
Author(s):  
Juyoung Yon ◽  
Siu-Wing Cheng ◽  
Otfried Cheong ◽  
Antoine Vigneron
Keyword(s):  
Input Parameter ◽  
Common Point ◽  
Transformation Formula ◽  
Point Sets ◽  
Subset Size ◽  
Running Time ◽  
Additional Input ◽  
Point Set ◽  
Rigid Motions ◽  
The Given

Let [Formula: see text] and [Formula: see text] be two discrete point sets in [Formula: see text] of sizes [Formula: see text] and [Formula: see text], respectively, and let [Formula: see text] be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets [Formula: see text] and [Formula: see text] such that [Formula: see text] and there exists a transformation [Formula: see text] that makes the bottleneck distance between [Formula: see text] and [Formula: see text] at most [Formula: see text]. We present two algorithms that solve a relaxed version of this problem under translations in [Formula: see text] and under rigid motions in the plane, and that takes an additional input parameter [Formula: see text]. Let [Formula: see text] be the largest subset size achievable for the given [Formula: see text]. Our algorithm finds subsets [Formula: see text] and [Formula: see text] of size [Formula: see text] and a transformation [Formula: see text] such that the bottleneck distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text]. For rigid motions in the plane, the running time is [Formula: see text]. For translations in [Formula: see text], the running time is [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

scholarly journals Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoyun Wang ◽  
Xianquan Zhang
Keyword(s):  
Pattern Matching ◽  
Complete Graph ◽  
Point Pattern ◽  
Point Sets ◽  
Matching Problem ◽  
Matching Algorithm ◽  
Planar Point ◽  
Point Pattern Matching ◽  
Point Set ◽  
Pattern Matching Algorithm

Point pattern matching is an important topic of computer vision and pattern recognition. In this paper, we propose a point pattern matching algorithm for two planar point sets under Euclidean transform. We view a point set as a complete graph, establish the relation between the point set and the complete graph, and solve the point pattern matching problem by finding congruent complete graphs. Experiments are conducted to show the effectiveness and robustness of the proposed algorithm.

scholarly journals INVOLUTIONS OF GRAPH LINK EXTERIORS WHOSE FIXED POINT SETS ARE CLOSED SURFACES

2010 ◽  
Vol 81 (2) ◽  
pp. 298-303 ◽  
Author(s):  
TORU IKEDA
Keyword(s):  
Fixed Point ◽  
Lower Estimate ◽  
Closed Surface ◽  
Point Sets ◽  
Fixed Point Set ◽  
Closed Surfaces ◽  
Point Set ◽  
Graph Link ◽  
Fixed Point Sets

AbstractA link L in S3 possibly admits an involution of the exterior E(L) with fixed point set a closed surface, which is not extendable to an involution of S3. In this paper, we focus on the case of graph links and show that the genus of the surface provides a lower estimate of the number of link components.

Adaptive Slicing of Moving Least Squares Surfaces: Toward Direct Manufacturing of Point Set Surfaces

2007 ◽  
Author(s):  
Pinghai Yang ◽  
Xiaoping Qian
Keyword(s):  
Least Squares ◽  
Point Cloud ◽  
Moving Least Squares ◽  
Point Sets ◽  
New Approach ◽  
Adaptive Slicing ◽  
3D Sensing ◽  
Point Set ◽  
Point Set Surfaces ◽  
Model Conversion

Rapid advancement of 3D sensing techniques has lead to dense and accurate point cloud of an object to be readily available. The growing use of such scanned point sets in product design, analysis and manufacturing necessitates research on direct processing of point set surfaces. In this paper, we present an approach that enables the direct layered manufacturing of point set surfaces. This new approach is based on adaptive slicing of moving least squares (MLS) surfaces. Salient features of this new approach include: 1) it bypasses the laborious surface reconstruction and avoids model conversion induced accuracy loss; 2) the resulting layer thickness and layer contours are adaptive to local curvature and thus it leads to better surface quality and more efficient fabrication; 3) the MLS surface naturally smoothes the point cloud and allows up-sampling and down-sampling, and thus it is robust even for noisy or sparse point sets. Experimental results of the slicing algorithm on both synthetic and scanned point sets are presented.

scholarly journals Counting Triangulations of Planar Point Sets

10.37236/557 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Line Graphs ◽  
Point Sets ◽  
Planar Point ◽  
Straight Line ◽  
Point Set ◽  
Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

scholarly journals Packing Plane Perfect Matchings into a Point Set

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Ahmad Biniaz ◽  
Prosenjit Bose ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Perfect Matchings ◽  
Point Sets ◽  
Exact Answer ◽  
International Audience ◽  
Point Set

International audience Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of &#x230A;log<sub>2</sub>$n$&#x230B;$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

Conceptual Design of Freeform Surfaces From Unstructured Point Sets Using Neural Network Regression

2011 ◽  
Author(s):  
Mehmet Ersin Yumer ◽  
Levent Burak Kara
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Locally Linear Embedding ◽  
Point Sets ◽  
3D Space ◽  
Closed Surfaces ◽  
Test Models ◽  
Point Set ◽  
Linear Embedding ◽  
Trained Network

This paper presents a new point set surfacing method that employs neural networks for regression. Our technique takes as input unstructured and possibly noisy point sets representing two-manifolds in R3. To facilitate parametrization, the set is first embedded in R2 using neighborhood preserving locally linear embedding. A neural network is then constructed and trained that learns a mapping between the embedded 2D parametric coordinates and the corresponding 3D space coordinates. The trained network is then used to generate a tessellation that spans the parametric space, thereby producing a surface in the original space. This approach enables the surfacing of noisy and non-uniformly distributed point sets, and can be applied to open or closed surfaces. We show the utility of the proposed method on a number of test models, as well as its application to freeform surface creation in virtual reality environments.

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