Processing images on square and hexagonal grids - a comparison

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

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Grid overlays reduce bias in mental representations of topographic maps

Abstracts of the ICA ◽

10.5194/ica-abs-1-196-2019 ◽

2019 ◽

Vol 1 ◽

pp. 1-1

Author(s):

Lars Kuchinke ◽

Julian Keil ◽

Dennis Edler ◽

Anne-Kathrin Bestgen ◽

Frank Dickmann

Keyword(s):

Spatial Information ◽

Mental Representations ◽

Empirical Studies ◽

Topographic Maps ◽

Object Location Memory ◽

Object Locations ◽

Gaze Patterns ◽

Map Design ◽

Cognitive Principles ◽

Hexagonal Grids

Abstract. Reading spatial information from topographic maps to form mental representations that guide spatial orientation and navigation is a rather complex cognitive process. Perceptual and knowledge-driven processes interact to support the map reader in building these mental representations. The resulting cognitive maps are not one-to-one mappings of the spatial information and known to be distorted systematically. It is assumed that spatial information is hierarchically organized in these mental models. We are interested in how map design based on cognitive principles supports memory formation and leads to less distorted mental representations.Based on the results of empirical studies we are able to show that overlaid grids in these maps address the hierarchical nature of these mental representations of map space. When map users are asked to learn object locations in a map the availability of overlaid grid layers improve object location memory. This effect is independent of the shape of these grid patterns (square grids or hexagonal grids) and, moreover, can be shown to be effective even in situations where the grids are interrupted by other maps layers (i.e. so-called illusory grids).These results seem best explained by the formation of less distorted mental representations based on the availability of superordinate hierarchical information and the application of Gestalt principles by the map user. Thus again, point to the interaction between perceptual and knowledge-driven processes in the formation of these mental representations of map space. This assumption receives further support by eye-tracking data that reveal that grids do not only attract attention towards their own location but also seem to structure the gaze patterns in relation to the relevant object locations that are not necessarily located close to a grid line.

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SPATIAL PATTERNS OF BIODIVERSITY-ASSESSING VEGETATION USING HEXAGONAL GRIDS

Biology & Environment Proceedings of the Royal Irish Academy ◽

10.1353/bae.2006.0003 ◽

2006 ◽

Vol 106B (3) ◽

pp. 401-411

Author(s):

Gerald Jurasinski ◽

Carl Beierkuhnlein

Keyword(s):

Spatial Patterns ◽

Hexagonal Grids

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Lower Bounds for Identifying Codes in some Infinite Grids

The Electronic Journal of Combinatorics ◽

10.37236/394 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 6

Author(s):

Ryan Martin ◽

Brendon Stanton

Keyword(s):

Lower Bounds ◽

Finite Graph ◽

Infinite Graphs ◽

Locally Finite ◽

Identifying Codes ◽

Identifying Code ◽

Hexagonal Grids ◽

Closed Neighborhood ◽

Definition Of ◽

Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

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Spot addressing for microarray images structured in hexagonal grids

Computer Methods and Programs in Biomedicine ◽

10.1016/j.cmpb.2011.08.001 ◽

2012 ◽

Vol 106 (1) ◽

pp. 1-13 ◽

Cited By ~ 8

Author(s):

Nikolaos Giannakeas ◽

Fanis Kalatzis ◽

Markos G. Tsipouras ◽

Dimitrios I. Fotiadis

Keyword(s):

Hexagonal Grids

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(ℓ, k)-ROUTING ON PLANE GRIDS

Journal of Interconnection Networks ◽

10.1142/s0219265909002431 ◽

2009 ◽

Vol 10 (01n02) ◽

pp. 27-57

Author(s):

FLORIAN HUC ◽

IGNASI SAU ◽

JANEZ ŽEROVNIK

Keyword(s):

Communication Networks ◽

Routing Algorithms ◽

Full Duplex ◽

Packet Routing ◽

Routing Problem ◽

Shortest Path Routing ◽

Running Time ◽

Half Duplex ◽

Hexagonal Grids ◽

Permutation Routing

The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the (ℓ, k)-routing problem, each node can send at most ℓ packets and receive at most k packets. Permutation routing is the particular case ℓ = k = 1. In the r-central routing problem, all nodes at distance at most r from a fixed node v want to send a packet to v. In this article we study the permutation routing, the r-central routing and the general (ℓ, k)-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the store-and-forward Δ-port model, and we consider both full and half-duplex networks. We first survey the existing results in the literature about packet routing, with special emphasis on (ℓ, k)-routing on plane grids. Our main contributions are the following: 1. Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. 2. Tight r-central routing algorithms on triangular and hexagonal grids. 3. Tight (k, k)-routing algorithms on square, triangular and hexagonal grids. 4. Good approximation algorithms (in terms of running time) for (ℓ, k)-routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. These algorithms are all completely distributed, i.e., can be implemented independently at each node. Finally, we also formulate the (ℓ, k)-routing problem as a WEIGHTED EDGE COLORING problem on bipartite graphs.

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Estimation of geometrically necessary dislocation density from filtered EBSD data by a local linear adaptation of smoothing splines

Journal of Applied Crystallography ◽

10.1107/s1600576719004035 ◽

2019 ◽

Vol 52 (3) ◽

pp. 548-563 ◽

Cited By ~ 6

Author(s):

Anthony Seret ◽

Charbel Moussa ◽

Marc Bernacki ◽

Javier Signorelli ◽

Nathalie Bozzolo

Keyword(s):

Electron Backscatter Diffraction ◽

Smoothing Splines ◽

Data Sets ◽

Square Grid ◽

Orientation Gradient ◽

Geometrically Necessary Dislocation ◽

Ebsd Data ◽

Local Linear ◽

Density Values ◽

Hexagonal Grids

An implementation of smoothing splines is proposed to reduce orientation noise in electron backscatter diffraction (EBSD) data, and subsequently estimate more accurate geometrically necessary dislocation (GND) densities. The local linear adaptation of smoothing splines (LLASS) filter has two advantages over classical implementations of smoothing splines: (1) it allows for an intuitive calibration of the fitting versus smoothing trade-off and (2) it can be applied directly and in the same manner to both square and hexagonal grids, and to 2D as well as to 3D EBSD data sets. Furthermore, the LLASS filter calculates the filtered orientation gradient, which is actually at the core of the method and which is subsequently used to calculate the GND density. The LLASS filter is applied on a simulated low-misorientation-angle boundary corrupted by artificial orientation noise (on a square grid), and on experimental EBSD data of a compressed Ni-base superalloy (acquired on a square grid) and of a dual austenitic/martensitic steel (acquired on an hexagonal grid). The LLASS filter leads to lower GND density values as compared to raw EBSD data sets, as a result of orientation noise being reduced, while preserving true GND structures. In addition, the results are compared with those of filters available in theMTEXtoolbox.

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Hierarchical Hexagonal Clustering and Indexing

Symmetry ◽

10.3390/sym11060731 ◽

2019 ◽

Vol 11 (6) ◽

pp. 731 ◽

Cited By ~ 5

Author(s):

Vojtěch Uher ◽

Petr Gajdoš ◽

Václav Snášel ◽

Yu-Chi Lai ◽

Michal Radecký

Keyword(s):

Data Analysis ◽

Dimensional Space ◽

Memory Representation ◽

Convex Shape ◽

Uniform Distance ◽

Straightforward Method ◽

Uniform Grids ◽

Indexing Method ◽

Hexagonal Grids ◽

Pattern Shape

Space-filling curves (SFCs) represent an efficient and straightforward method for sparse-space indexing to transform an n-dimensional space into a one-dimensional representation. This is often applied for multidimensional point indexing which brings a better perspective for data analysis, visualization and queries. SFCs are involved in many areas such as big data analysis and visualization, image decomposition, computer graphics and geographic information systems (GISs). The indexing methods subdivide the space into logic clusters of close points and they differ in various parameters including the cluster order, the distance metrics, and the pattern shape. Beside the simple and highly preferred triangular and square uniform grids, the hexagonal uniform grids have gained high interest especially in areas such as GISs, image processing and data visualization for the uniform distance between cells and high effectiveness of circle coverage. While the linearization of hexagons is an obvious approach for memory representation, it seems there is no hexagonal SFC indexing method generally used in practice. The main limitation of hexagons lies in lacking infinite decomposition into sub-hexagons and similarity of tiles on different levels of hierarchy. Our research aims at defining a fast and robust hexagonal SFC method. The Gosper fractal is utilized to preserve the benefits of hexagonal grids and to efficiently and hierarchically linearize points in a hexagonal grid while solving the non-convex shape and recursive transformation issues of the fractal. A comparison to other SFCs and grids is conducted to verify the robustness and effectiveness of our hexagonal method.

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