IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i21p4077-d960767.html
   My bibliography  Save this article

On an Application of Lattice Integral Transforms in Image Processing

Author

Listed:
  • Michal Holčapek

    (CE IT4I—IRAFM, University of Ostrava, 701 03 Ostrava, Czech Republic)

  • Viec Bui Quoc

    (CE IT4I—IRAFM, University of Ostrava, 701 03 Ostrava, Czech Republic)

  • Petr Ferbas

    (Varroc Lighting Systems, 702 00 Ostrava, Czech Republic)

Abstract

The lattice integral transforms have been introduced to generalize lower and upper fuzzy transforms for lattice-valued functions that are used to approximate original functions from below and above. They are defined in complete analogy with classical integral transforms, particularly, the product of a lattice-valued function and a fuzzy relation called the integral kernel is integrated by a Sugeno-like fuzzy integral. In the article, we first investigate the conditions under which lattice integral transforms preserve (reverse) constant functions, which appears to be a fundamental presumption for a successful approximation of lattice-valued functions. Further, we show how the lattice integral transforms can be applied in image processing, more specifically, in non-linear filtering, compression/decompression, and opening/closing of images. We demonstrate that the filters based on integral transforms generalize the popular median filter as well as minimum and maximum filters, and also opening and closing defined using fuzzy morphological erosion and dilation. We illustrate the proposed methods in various selected images.

Suggested Citation

  • Michal Holčapek & Viec Bui Quoc & Petr Ferbas, 2022. "On an Application of Lattice Integral Transforms in Image Processing," Mathematics, MDPI, vol. 10(21), pages 1-30, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4077-:d:960767
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/21/4077/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/21/4077/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michel Grabisch & Jean-Luc Marichal & Radko Mesiar & Endre Pap, 2009. "Aggregation functions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00445120, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luca Anzilli & Silvio Giove, 2020. "Multi-criteria and medical diagnosis for application to health insurance systems: a general approach through non-additive measures," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 559-582, December.
    2. Michel Grabisch & Jean-Luc Marichal & Radko Mesiar & Endre Pap, 2011. "Aggregation functions: construction methods, conjunctive, disjunctive and mixed classes," Post-Print hal-00539032, HAL.
    3. Ya-Qiang Xu & Le-Sheng Jin & Zhen-Song Chen & Ronald R. Yager & Jana Špirková & Martin Kalina & Surajit Borkotokey, 2022. "Weight Vector Generation in Multi-Criteria Decision-Making with Basic Uncertain Information," Mathematics, MDPI, vol. 10(4), pages 1-11, February.
    4. Pollesch, N.L. & Dale, V.H., 2016. "Normalization in sustainability assessment: Methods and implications," Ecological Economics, Elsevier, vol. 130(C), pages 195-208.
    5. Gia Sirbiladze & Otar Badagadze, 2017. "Intuitionistic Fuzzy Probabilistic Aggregation Operators Based on the Choquet Integral: Application in Multicriteria Decision-Making," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 245-279, January.
    6. Peter Reichert & Klemens Niederberger & Peter Rey & Urs Helg & Susanne Haertel-Borer, 2019. "The need for unconventional value aggregation techniques: experiences from eliciting stakeholder preferences in environmental management," EURO Journal on Decision Processes, Springer;EURO - The Association of European Operational Research Societies, vol. 7(3), pages 197-219, November.
    7. Genest Christian & Scherer Matthias, 2020. "The gentleman copulist: An interview with Carlo Sempi," Dependence Modeling, De Gruyter, vol. 8(1), pages 34-44, January.
    8. GRABISCH, Michel & LABREUCHE, Christophe & RIDAOUI, Mustapha, 2019. "On importance indices in multicriteria decision making," European Journal of Operational Research, Elsevier, vol. 277(1), pages 269-283.
    9. Michel Grabisch & Agnieszka Rusinowska, 2010. "Iterating influence between players in a social network," Documents de travail du Centre d'Economie de la Sorbonne 10089, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    10. Grabisch, Michel & Rusinowska, Agnieszka, 2013. "A model of influence based on aggregation functions," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 316-330.
    11. Vicenç Torra, 2022. "Andness Directedness for t-Norms and t-Conorms," Mathematics, MDPI, vol. 10(9), pages 1-10, May.
    12. Chunqiao Tan & Xiaohong Chen, 2016. "Generalized Archimedean Intuitionistic Fuzzy Averaging Aggregation Operators and their Application to Multicriteria Decision-Making," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 311-352, March.
    13. José Luis García-Lapresta & Casilda Lasso de la Vega & Ricardo Alberto Marques Pereira & Ana Marta Urrutia, 2010. "A class of poverty measures induced by the dual decomposition of aggregation functions," Working Papers 160, ECINEQ, Society for the Study of Economic Inequality.
    14. Bonifacio Llamazares, 2019. "An Analysis of Winsorized Weighted Means," Group Decision and Negotiation, Springer, vol. 28(5), pages 907-933, October.
    15. Karlson Pfannschmidt & Pritha Gupta & Bjorn Haddenhorst & Eyke Hullermeier, 2019. "Learning Context-Dependent Choice Functions," Papers 1901.10860, arXiv.org, revised Oct 2021.
    16. Radko Mesiar & Andrea Stupňanová, 2021. "Directional Shift-Stable Functions," Mathematics, MDPI, vol. 9(10), pages 1-12, May.
    17. Labreuche, Christophe & Grabisch, Michel, 2018. "Using multiple reference levels in Multi-Criteria Decision aid: The Generalized-Additive Independence model and the Choquet integral approaches," European Journal of Operational Research, Elsevier, vol. 267(2), pages 598-611.
    18. Fabrizio Durante & Juan Fernández-Sánchez & Wolfgang Trutschnig & Manuel Úbeda-Flores, 2020. "On the Size of Subclasses of Quasi-Copulas and Their Dedekind–MacNeille Completion," Mathematics, MDPI, vol. 8(12), pages 1-11, December.
    19. Belles-Sampera, Jaume & Merigó, José M. & Guillén, Montserrat & Santolino, Miguel, 2013. "The connection between distortion risk measures and ordered weighted averaging operators," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 411-420.
    20. Miroslav D. Vujičić & Uglješa Stankov & Vanja Pavluković & Ivana Štajner-Papuga & Sanja Kovačić & Jovana Čikić & Nemanja Milenković & Tamara Zelenović Vasiljević, 2023. "Prepare for Impact! A Methodological Approach for Comprehensive Impact Evaluation of European Capital of Culture: The Case of Novi Sad 2022," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 165(2), pages 715-736, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4077-:d:960767. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.