Page 77 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.449-463 (2025)
https://doi.org/10.36922/IJOCTA025060018

RESEARCH ARTICLE

Computational aspects of the crop field segmentation problem
based on anisotropic active contour model

2
1
Ciro D’Apice , Umberto De Maio , Peter I. Kogut 3,4 , and Rosanna Manzo 5*
1
Department of Management Innovation Systems, University of Salerno, Fisciano, Italy
2
Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”,
Naples, Italy
3
Department of Differential Equations, Oles Honchar Dnipro National University, Dnipro, Ukraine
4
EOS Data Analytics Ukraine, Dnipro, Ukraine
5
Department of Political and Communication Sciences, University of Salerno, Fisciano, Italy
cdapice@unisa.it, udemaio@unina.it, p.kogut@i.ua, peter.kogut@eosda.com, rmanzo@unisa.it
ARTICLE INFO ABSTRACT
Article History:
Received: February 4, 2025 In this paper, we analyze the numerical aspects of the practical implementation
1st revised: February 11, 2025 of the generalized active contour model, that has been recently proposed in the
2nd revised: April 4, 2025 literature, for extracting agricultural crop fields with a high level of inhomo-
Accepted: April 11, 2025 geneity from satellite data. We also derive the corresponding Euler–Lagrange
Published Online: May 20, 2025 equation and discuss its relaxation method.
Keywords:
Segmentation problem
Piecewise smooth approximation
Optimality conditions
Active contour model
Euler-Lagrange equation
AMS Classification 2010:
26A33; 34A08; 35H15; 34K50

1. Introduction to a given objective agricultural index f : Ω → R.
Namely, this decomposition has to inherit the fol-
This paper deals with the numerical aspects of lowing properties:
the implementation of the generalized active con-
tour model, which has been recently proposed in, 1
for extracting agricultural crop fields with a high
level of inhomogeneity from satellite data. These
investigations have been justified by various ap- (a) {Ω i } K are open subsets of Ω with
i=1
∗
plications, such as in satellite image segmenta- nonempty interior, whereas Ω stands for
tion under remote sensing of the Earth’s surface. the boundaries between different Ω j ;
2
Fixed a field Ω ⊂ R , one of the fundamental (b) within each Ω i the objective agricultural
problems in agriculture is to give a disjunctive index f varies smoothly and/or slowly;
decomposition into a finite quantity of nonempty (c) the function f cannot vary discontinu-
∗
subsets Ω = Ω 1 ∪ Ω 2 ∪ · · · ∪ Ω K ∪ Ω such that a ously inside each Ω i ;
distinctive value of some agricultural index (PVI, (d) the decomposition is optimal in the fol-
NDVI, LAI, etc.) could be associated to each of lowing sense: the approximation of f by a
∗
these subsets. The characteristic feature of the piece-wise constant function f : Ω → R is
∗
decomposition is that it has to be closely related such that the restrictions f to the pieces
*Corresponding Author
449