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Computational aspects of the crop field segmentation problem based on anisotropic active contour model
                    ∗
            Here, S stands for the area of a standard field in  Author contributions
            a given region.                                   Conceptualization: All authors
                After that we define the function φ 0 ε,τ  as a so-  Formal analysis: All authors
            lution of the problem (51), (52), (53)–(55) with  Investigation: All authors
            the given collection {l j } M  . Let {S j } M  be a set
                                   j=1          j=0           Methodology: All authors
            of the corresponding segments that we identify by  Writing-original draft: All authors
            the rules (57)–(59). Then we define the true seg-  Writing-review & editing: All authors
            ments as follows:
                    k=1;                                      Availability of data
                    True_Segment(k):=S_0;
                                                              Not applicable.
                    for j=1:M
                      if area(S_1)<0.10 S*
                         combine{True_Segment(k),S_j};        References
                      else
                                                               1. D’Apice C., Kogut PI, Manzo R. On general-
                     k=k+1;
                                                                  ized active contour model in the anisotropic BV
                     True_Segment(k):=S_j;                        space and its application to satellite remote sens-
                    end                                           ing of agricultural territory. Netw Heterogen Me-
                    Number_of_true_Segments:=k;                   dia. 2025;20(1): 113-142.
                                                               2. Mulla DJ. Twenty five years of remote sensing in
                Here, the multiplier 0.10 emerges as a part of
                                                                  precision agriculture: key advances and remaining
            the field that can be neglected as a zone of inho-
                                                                  knowledge gaps. Biosyst Eng. 2013; 114(4):358-
            mogeneity.
                                                                  371.
                                                               3. Xue J, Su B. Significant remote sensing vegeta-
                                                                  tion indices: a review of developments and appli-
            6. Conclusion                                         cations. Hindawi J Sens. 2017;2017:1-17. (Article
                                                                  ID 1353691)
            The generalized active contour model, proposed
               1
            in, to extract agricultural crop fields with a high  4. Hnatushenko VV, Kogut PI, Uvarov M. Varia-
                                                                  tional approach for rigid co-registration of op-
            degree of inhomogeneity from satellite data has
                                                                  tical/SAR satellite images in agricultural ar-
            been implemented. Setting an optimization prob-       eas. J Comput Appl Math. 2022;400:15. Id
            lem, it was possible to achieve a disjunctive de-     113742.
            composition of a given field into a finite number  5. Khanenko P, Kogut PI, Uvarov M. On varia-
            of non-empty subsets, each of which could be as-      tional problem with nonstandard growth condi-
            sociated with a unique value of an agricultural       tions for the restoration of clouds corrupted satel-
            index.  The Euler–Lagrange equation with the          lite images. In: CEUR Workshop Proceedings, the
            appropriate initial and boundary conditions has       2nd International Workshop on Computational
                                                                  and Information Technologies for Risk-Informed
            been used to express the corresponding optimal-
                                                                  Systems, CITRisk-2021, September 16-17, 2021,
            ity system under certain assumptions. A method
                                                                  Kherson, Ukraine, Vol. 3101; 2021: 6-25.
            for the numerical solution of the Euler–Lagrange
                                                               6. Ivanchuk N, Kogut PI, Martyniuk P. On genera-
            system has been provided along with a numerical
                                                                  tion of daily cloud free images at a high resolution
            scheme. Using a direct approach to the segmenta-
                                                                  level. In: M. Zgurovsky, N. Pankratova, eds. Sys-
            tion technique and the new adaptive one, numer-       tem Analysis and Artificial Intelligence. Studies
            ical simulations utilizing real-life satellite images  in Computational Intelligence, Vol 1107. Cham:
            have been conducted to show the accuracy and          Springer; 2023: 203-232.
            effectiveness of the proposed model.               7. D’Apice C, Kogut PI, Manzo R. A two-level vari-
                                                                  ational algorithm in the Sobolev-Orlicz space to
                                                                  predict daily surface reflectance at LANDSAT
            Acknowledgments
                                                                  high spatial resolution and MODIS temporal fre-
            None.                                                 quency. J Comput Appl Math. 2023;434:115339.
                                                               8. D’Apice C, Kogut PI, Manzo R, Uvarov M. Varia-
                                                                  tional model with nonstandard growth conditions
            Funding                                               for the restoration of satellite optical images using
                                                                  synthetic aperture radar. Eur J Appl Math. 2023;
            None.
                                                                  34(1):77-105.
                                                               9. Hnatushenko VV, Kogut PI, & Uvarov MV. On
            Conflict of interest                                  satellite image Segmentation via piecewise con-
                                                                  stant approximation of selective smoothed target
            The authors declare no conflict of interest.          mapping. Appl Math Comput. 2021;389:125615.
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