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Variable density preserving topology grids and the digital models for the plane

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Alexander Evako

Discrete Mathematics · Submitted 15 Feb 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object.

how it is stated · asserted 1.00 read the record →

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arXiv ID1302.3809 (version 1)
Pith ID4AG3PZRS, Pith's permanent record of this paper and its review
CitationAlexander Evako. "Variable density preserving topology grids and the digital models for the plane." arXiv:1302.3809 (2013). https://pith.science/paper/4AG3PZRS
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,899 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.12 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. We show that elements of an LCL tiling can be of an arbitrary shape and size. This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. contribution 0.98 0.91 0.06 asserted 0.92
s1 We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. result 0.89 0.93 0.37 asserted 1.00
s2 We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. result 0.99 0.93 0.34 asserted 1.00
s3 We show that elements of an LCL tiling can be of an arbitrary shape and size. result 0.95 0.83 0.07 asserted 0.98
s4 This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. interpretation 0.97 0.59 0.10 asserted 1.00
s5 Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions. method 0.70 0.92 0.16 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.65 · evidence from proof or derivation 1.00.

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  • references and citing sentences
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  • S sections
  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. We show that elements of an LCL tiling can be of an arbitrary shape and size. This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions.

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