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The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Abstract

Markov random fields (MRF) and the Potts model have many applications in different areas. Especially, conditional random fields (CRF) and Potts model have been used in connection with classifiers. In this work, we focus on the Potts model and use image segmentation and data classification as examples to show some new techniques and fast algorithms for this model. We survey different piecewise constant representation techniques. Many of these representations can be interpreted as min-cut and max-flow problems on some special graphs. We will concentrate especially on the continuous setting and formulate continuous min-cut and max-flow models. When the min-cut/max-flow models are discretized, they give corresponding discrete min-cut/max-flow models on grids. Using these connections, we are able to turn the non-convex Potts model into some simple convex minimization problems with solutions that can be obtained by properly designed fast algorithms. In this survey, we will start by introducing some widely studied variational segmentation models and the classical level-set approaches to solve them. Then, we will describe three different piecewise constant representations for the general Potts model and their corresponding convex relaxations and fast algorithms. In the end, we will also generalize the method to a graph setting for high-dimensional data classifications. This survey presents the different techniques and algorithms in an integrated and self-contained manner.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Xuecheng Tai

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Lingfeng Li

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Lingfeng Li

  4. Norwegian Defence Research Establishment (FFI), Kjeller, Norway

    Egil Bae

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  1. Xuecheng Tai
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  2. Lingfeng Li
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  3. Egil Bae
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Correspondence to Xuecheng Tai .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Dept Appl Math & Theoretical Phys, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics & Stat, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younces

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    Xue-Cheng Tai

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    Carola-Bibiane Schönlieb

  3. No affiliation provided

    Ke Chen

  4. No affiliation provided

    Laurent Younes

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Tai, X., Li, L., Bae, E. (2021). The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_90-1

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  • DOI: https://doi.org/10.1007/978-3-030-03009-4_90-1

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  1. Latest

    The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey
    Published:
    08 October 2023

    DOI: https://doi.org/10.1007/978-3-030-03009-4_90-3

  2. The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey
    Published:
    16 September 2021

    DOI: https://doi.org/10.1007/978-3-030-03009-4_90-2

  3. Original

    The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey
    Published:
    28 May 2021

    DOI: https://doi.org/10.1007/978-3-030-03009-4_90-1

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