Optimal Geometric Disks Covering using Tessellable Regular Polygons

Donkoh, Elvis K.; Opoku, Alex Akwasi

doi:10.5539/jmr.v8n2p25

Cited by 6 publications

(5 citation statements)

References 3 publications

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“…Given this constraint, the number of measurement points in area B is minimized via hexagonal sampling (see e.g. Donkoh and Opoku 2016). To support our hypothesis (i.e.…”

Section: Case 1: Soil Water Permeability Prediction Based On Soil Typementioning

confidence: 91%

Estimating the prediction performance of spatial models via spatial k-fold cross validation

Pohjankukka

Pahikkala

Nevalainen

et al. 2017

International Journal of Geographical Information Science

125

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In machine learning one often assumes the data are independent when evaluating model performance. However, this rarely holds in practise. Geographic information data sets are an example where the data points have stronger dependencies among each other the closer they are geographically. This phenomenon known as spatial autocorrelation (SAC) causes the standard cross validation (CV) methods to produce optimistically biased prediction performance estimates for spatial models, which can result in increased costs and accidents in practical applications. To overcome this problem we propose a modified version of the CV method called spatial k-fold cross validation (SKCV), which provides a useful estimate for model prediction performance without optimistic bias due to SAC. We test SKCV with three real world cases involving open natural data showing that the estimates produced by the ordinary CV are up to 40% more optimistic than those of SKCV. Both regression and classification cases are considered in our experiments. In addition, we will show how the SKCV method can be applied as a criterion for selecting data sampling density for new research area.

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“…Given this constraint, the number of measurement points in area B is minimized via hexagonal sampling (see e.g. Donkoh and Opoku 2016). To support our hypothesis (i.e.…”

Section: Case 1: Soil Water Permeability Prediction Based On Soil Typementioning

confidence: 91%

Estimating the prediction performance of spatial models via spatial k-fold cross validation

Pohjankukka

Pahikkala

Nevalainen

et al. 2017

International Journal of Geographical Information Science

125

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“…Next, the papers from each class are separated based on the specific application. The search and selection results are given in Table II for circular and Table III [3], [5], [6], [8][9][10][11][12][13][14][15], [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], [33-42] 37 Total #papers 42 We further categorized the papers from each application in sub-classes depending on whether authors provide formulas on how to calculate the total number of embedded hexagonal cells, their vertices or edges.…”

Section: Paper Search and Selection Resultsmentioning

confidence: 99%

“…Another advantage of the hexagonal grid is the possibility to fully cover the Region of Interest (ROI) without void or gaps. The increased area usage is shown with hexagons compared to other polygon shapes [17,18].…”

Section: Introductionmentioning

confidence: 99%

A Review of Embedding Hexagonal Cells in the Circular and Hexagonal Region of Interest

Prvan¹,

Ožegović²,

Mišura³

2019

IJACSA

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Hexagonal cells are applied in various fields of research. They exhibit many advantages, and one of the most important is their possibility to be closely packed and to form a hexagonal grid that fully covers the Region of Interest (ROI) without overlaps or gaps. ROI can be of various geometrical shapes, but this paper deals with the circular or hexagonal ROI approximations. The main purpose of our research is to provide a short review on the literature concerning the hexagonal grid, summarizing the existing state-of-the-art approaches on embedding hexagonal cells in the targeted ROI shapes and offering application-specific advantages. We report on formulas and algebraic expressions given in the existing researches that are used for calculating the number of embedded inner hexagonal cells or their vertices and/or edges. We contribute by integrating all researches in one place, finding a connection between previously unrelated applications concerning the use of embedded hexagonal grid and extracting commonality between previous researches on whether it provides the formulas on calculating the inner hexagon cells. In case only the number of edges or vertices is provided for the targeted application, we derive formulas for calculating the number of inner hexagons. Therefore, our survey results with the overview on solving the problem of embedding hexagonal cells in the desired circular or hexagonal ROI. The contribution of the review is the following: first it provides the existing and the derived formulas for calculating the embedded hexagons and second, it provides a theoretical background that is necessary to encourage further research. Namely, our main motivation, that is the geometrical design of the one of the world's largest CERN particle detectors, Compact Muon Solenoid (CMS) is analyzed as a source for the future research directions.

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“…When producing a single sensor, it is known that SE is maximal if hexagon shape is used with respect to triangle or square [17,18]. Since hexagon is the closest to being circular with the largest area, it has been shown to provide minimized silicon waste with respect to other geometrical shapes [19].…”

Section: Introductionmentioning

confidence: 99%

On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

Prvan

Ožegović

Mišura

2019

Mathematical Problems in Engineering

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In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean {3,122} semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE.

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Optimal Geometric Disks Covering using Tessellable Regular Polygons

Cited by 6 publications

References 3 publications

Estimating the prediction performance of spatial models via spatial k-fold cross validation

Estimating the prediction performance of spatial models via spatial k-fold cross validation

A Review of Embedding Hexagonal Cells in the Circular and Hexagonal Region of Interest

On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

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