Unsupervised segmentation of images using bi-dimensional pairwise Markov chains model

A. Joumad; A. El Moutaouakkil; A. Nasroallah; O. Boutkhoum; Mejdl Safran; Sultan Alfarhood; Imran Ashraf; A. Joumad; A. El Moutaouakkil; A. Nasroallah; O. Boutkhoum; Mejdl Safran; Sultan Alfarhood; Imran Ashraf

doi:10.3934/math.20241498

AIMS Mathematics

2024, Volume 9, Issue 11: 31057-31086. doi: 10.3934/math.20241498

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Research article

Unsupervised segmentation of images using bi-dimensional pairwise Markov chains model

1.
Department of informatics, Chouaib Doukkali University, Faculty of Sciences, B. P. 299-24000, El Jadida, Morocco
2.
Department of mathematics, Cadi Ayyad University, Faculty of Sciences Semlalia, B. P. 2390, Marrakesh, Morocco
3.
Department of Computer Science, College of Computer and Information Sciences, King Saud University, P.O.Box 51178, Riyadh 11543, Saudi Arabia
4.
Department of Information and Communication Engineering, Yeungnam University, Gyeongsan 38541, Republic of Korea

Received: 30 July 2024 Revised: 25 September 2024 Accepted: 26 September 2024 Published: 31 October 2024
MSC : 62C10, 62H30

The pair-wise Markov chain (PMC) model serves as an extension to the hidden Markov chain (HMC) model and has been widely used in unsupervised restoration tasks associated with reconstructing the hidden data. In fact, the PMC model can treat fairly complicated situations for which application of Bayesian restoration estimators such as maximum A Posteriori (MAP), or maximal Posterior mode (MPM) remains possible. The major novelty in this work is to construct a PMC model with observational data in two dimensions, and subsequently adapt the estimation algorithms, as well as, image restoration methods for that context. Often, the transformation of an image from a two-dimensional format to a one-dimensional sequence occurs via Hilbert-Peano scan (HPS), whereas in the proposed model, the second component of the observed process takes over this role to exceed the situation of pixel missing information after transformation for a to be segmented image. To reconstruct the hidden process, we used the MPM decision criterion after estimating the model's parameters with two algorithms: Stochastic expectation maximization (SEM) and iterative conditional estimation (ICE). In this study, experimental, numerical, and visual results are shown to demonstrate the superiority of the proposed model over the classical PMC for unsupervised restorations.
- Bayesian restoration,
- hidden Markov chain model,
- pairwise Markov chain model,
- unsupervised segmentation,
- Hilbert-Peano scan,
- ICE algorithm,
- SEM algorithm,
- bi-dimensional parwise Markov chain
Citation: A. Joumad, A. El Moutaouakkil, A. Nasroallah, O. Boutkhoum, Mejdl Safran, Sultan Alfarhood, Imran Ashraf. Unsupervised segmentation of images using bi-dimensional pairwise Markov chains model[J]. AIMS Mathematics, 2024, 9(11): 31057-31086. doi: 10.3934/math.20241498

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Abstract

The pair-wise Markov chain (PMC) model serves as an extension to the hidden Markov chain (HMC) model and has been widely used in unsupervised restoration tasks associated with reconstructing the hidden data. In fact, the PMC model can treat fairly complicated situations for which application of Bayesian restoration estimators such as maximum A Posteriori (MAP), or maximal Posterior mode (MPM) remains possible. The major novelty in this work is to construct a PMC model with observational data in two dimensions, and subsequently adapt the estimation algorithms, as well as, image restoration methods for that context. Often, the transformation of an image from a two-dimensional format to a one-dimensional sequence occurs via Hilbert-Peano scan (HPS), whereas in the proposed model, the second component of the observed process takes over this role to exceed the situation of pixel missing information after transformation for a to be segmented image. To reconstruct the hidden process, we used the MPM decision criterion after estimating the model's parameters with two algorithms: Stochastic expectation maximization (SEM) and iterative conditional estimation (ICE). In this study, experimental, numerical, and visual results are shown to demonstrate the superiority of the proposed model over the classical PMC for unsupervised restorations.

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