Computing Minimum Area Homologies

Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a simila...

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Bibliographic Details
Published in:Computer graphics forum Vol. 34; no. 6; pp. 13 - 21
Main Authors: Chambers, Erin Wolf, Vejdemo-Johansson, Mikael
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.09.2015
Subjects:
Algorithms
Analysis
Chains
Clustering algorithms
Computation
computational geometry
Computer graphics
curves and surfaces
Efficient implementation
Feasibility
Geometry
Homology
Homotopies
I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling-Geometric algorithms
I.5.3 [Pattern Recognition]: Clustering-Similarity measures
Image processing systems
languages and systems
languages and systems; I.5.3 [Pattern Recognition]: Clustering–Similarity measures
Linear algebra
Linear algebra tools
Mathematical analysis
Object modelling
Pattern recognition
Pattern recognition systems
Similarity
Similarity measure
Similarity measures
Studies
Topological manifolds
weird math
I.5.3 [Pattern Recognition]: Clustering-Similarity measures
languages and systems
ISSN:0167-7055, 1467-8659, 1467-8659
Online Access:Get full text
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Summary:Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.
Bibliography:istex:4460226735616F31DE30722E9F3A42BBAE52AB98
Supplementary Material
ArticleID:CGF12514
ark:/67375/WNG-HPX9266C-F
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SourceType-Scholarly Journals-1
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ISSN:0167-7055
1467-8659
1467-8659
DOI:10.1111/cgf.12514