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Discrete Dynamical System Framework for Construction of Connections between Critical Regions in Lattice Height Data

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Abstract

We define a new mathematical model for the topological study of lattice height data. A discrete multivalued dynamical system framework is used to establish discrete analogies of a Morse function, its gradient field, and its stable and unstable manifolds in order to interpret functions numerically given on finite sets of pixels. We present efficient algorithms detecting critical components of a height function f and displaying connections between them by means of a graph, called the Morse connections graph whose nodes represent the critical components of f and edges show the existence of connecting trajectories between nodes. This graph encodes efficiently the topological structure of the data and makes it easy to manipulate for subsequent processing.

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Authors and Affiliations

  1. Department of Computer Science, Bishop’s University, Sherbrooke, Québec, Canada, J1M 1Z7

    Madjid Allili & Anik Trahan

  2. Département d’informatique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1

    David Corriveau

  3. Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1

    Sara Derivière & Tomasz Kaczynski

Authors
  1. Madjid Allili
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  2. David Corriveau
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  3. Sara Derivière
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  4. Tomasz Kaczynski
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  5. Anik Trahan
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Corresponding author

Correspondence to Sara Derivière.

Additional information

M. Allili and T. Kaczynski supported by the NSERC of Canada discovery grant.

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Allili, M., Corriveau, D., Derivière, S. et al. Discrete Dynamical System Framework for Construction of Connections between Critical Regions in Lattice Height Data. J Math Imaging Vis 28, 99–111 (2007). https://doi.org/10.1007/s10851-007-0010-0

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  • DOI: https://doi.org/10.1007/s10851-007-0010-0

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