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Optimum Cuts in Graphs by General Fuzzy Connectedness with Local Band Constraints

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Abstract

The goal of this work is to describe an efficient algorithm for finding a binary segmentation of an image such that the indicated object satisfies a novel high-level prior, called local band, LB, constraint; the returned segmentation is optimal, with respect to an appropriate graph-cut measure, among all segmentations satisfying the given LB constraint. The new algorithm has two stages: expanding the number of edges of a standard edge-weighted graph of an image; applying to this new weighted graph an algorithm known as an oriented image foresting transform, OIFT. In our theoretical investigation, we prove that OIFT algorithm belongs to a class of general fuzzy connectedness algorithms and so has several good theoretical properties, like robustness for seed placement. The extension of the graph constructed in the first stage ensures, as we prove, that the resulted object indeed satisfies the given LB constraint. We also notice that this graph construction is flexible enough to allow combining it with other high-level constraints. Finally, we experimentally demonstrate that the LB constraint gives competitive results as compared to geodesic star convexity, boundary band, and hedgehog shape prior, all implemented within OIFT framework and applied to various scenarios involving natural and medical images.

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Notes

  1. In fact, we can use \(h(\omega (s,t))\) in place of \(e^{-\omega (s,t)}\) when h is any strictly decreasing function from \({\mathbb {R}}\) into \([0,\infty )\).

  2. Shortly, \(F_L:={\bar{\omega }}\cdot \chi _{X_L}\), where \(\chi _{X_L}:{{\mathcal {A}}}\rightarrow \{0,1\}\) is the characteristic function of \(X_L\).

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, CEP 05508-090, Brazil

    Caio de Moraes Braz & Paulo A. V. Miranda

  2. Department of Mathematics, West Virginia University, Morgantown, USA

    Krzysztof Chris Ciesielski

  3. Instituto de Ciência e Tecnologia, São José dos Campos, SP, Brazil

    Fábio A. M. Cappabianco

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  1. Caio de Moraes Braz
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  2. Paulo A. V. Miranda
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  3. Krzysztof Chris Ciesielski
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  4. Fábio A. M. Cappabianco
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Correspondence to Paulo A. V. Miranda.

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Thanks to CNPq (313554/2018-8, 486988/2013-9, FINEP 1266/13), FAPESP (2014/12236-1, 2016/21591-5), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and NAP eScience - PRP - USP for funding.

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de Moraes Braz, C., Miranda, P.A., Ciesielski, K.C. et al. Optimum Cuts in Graphs by General Fuzzy Connectedness with Local Band Constraints. J Math Imaging Vis 62, 659–672 (2020). https://doi.org/10.1007/s10851-020-00953-w

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