Page 78 - IJOCTA-15-3

C. D’Apice et .al. / IJOCTA, Vol.15, No.3, pp.449-463 (2025)
∗
Ω j , f | , are defined as the segmentation problem (consult the model
Ω j
of Mumford-Shah 11 or the approaches proposed
1 Z 12–19
∗
|f | (x) = ⟨f⟩ := f(s) ds, by along with the others). We refer to the
Ω j Ω j |Ω j | 17
Ω j recent study for an overview of the state of the
∀ j = 1, . . . , K, ∀ x ∈ Ω j , art in this topic.
and the 1-D Hausdorff measure of the set To sort out these issues, we involve an
S K
j=1 ∂Ω j should be as small as possible. anisotropic specification of the standard active
contour model. Apparently, the latter was first
Hereinafter, the decomposition of Ω satis-
introduced in 20 to transform the image segmen-
fying properties (a)–(d) is denoted with f-
tation problem into a minimization problem in a
decomposition.
suitable functional space, and it offered a fresh
The role of the function f : Ω → R is
perspective on image segmentation, which was
explained by pointing out that after the f-
decomposition the new objects {Ω j } K should referred to as the focus and hot spot of analy-
j=1 sis in. 21 The subject of image segmentation has
have homogeneous values of f (see 2,3 ). To char- seen the proposal of numerous good algorithms
acterize a measure of the variability of f in each based on active contour models in recent years
subdomain Ω j , one can calculate the coefficient (we recommend that readers consult the latest
of variation, that is, it is the ratio of the stan- publications 22–28 and the bibliography therein).
dard deviation of f(x) within Ω j to its mean However, their correct application to the NDVI-
⟨f⟩ . Furthermore, if we consider applications decomposition of agricultural fields remains an
Ω j
in agriculture it might be reasonable to regard open problem nowadays.
the NDVI-characteristic as the primary feature
The main idea, that was realized in the
of the zone of interest Ω, that is, in this case,
new setting of variational issue for the f-
f(x) = NDVI(x) for all x ∈ Ω. However, a
decomposition, is that we associate with a given
thorough analysis of this issue shows that it is
a difficult task to understand NDVI characteris- objective function f : Ω → R the constrained op-
tics quantitatively over Ω. By presuming that the timization problem as follows:
vegetation data generated from the NDVI is uni- J(c, φ) = T(c, φ)
formly and smoothly distributed within the spe-
m Z
cific crop fields, numerous studies have restricted + α X |M Dχ | → inf , (1)
f
this interpretation. 4,5 If they attempt to use the Ω {φ(x)>l j } φ∈Ξ
j=1
NDVI on heterogeneous canopies, for example, m+1
plantations that have weeds, mixed soil, and vari- where T : R × BV (Ω) → R is a fidelity term,
{l 0 , l 1 , . . . , l m+1 } is a given collection of distinct
ous crop mixtures where the vegetation has vary-
level values, χ is the characteristic func-
ing NDVI characteristics because of spatial vari- {φ(x)>l j }
ability, this assumption is violated. tion of the set {x ∈ Ω : φ(x) > l j }, and the
set of feasible solutions Ξ is defined as follows:
Accordingly, the data acquired in this man- m+1
(c, φ) ∈ Ξ if and only if (c, φ) ∈ R × BV (Ω)
ner is a valuable source of information that may
and
be applied to the management of environmental
resources. 6,7 Specifically, it enables us to deter- l 0 ≤ φ(x) ≤ l m+1 a.e. in Ω,
mine the crop’s health and find out whether there c = (c 0 , c 1 , . . . , c m ), c j ≥ 0, j = 0, . . . , m.
are trouble spots within the designated fields or f
areas where the crops require nutrients or water. As for the matrix-value function M (·) : Ω →
R 2×2 , it is the so-called anisotropic diffusion ten-
A similar point of view is recommended as crucial
in. 3,6 Some methods for assessing such charac- sor that avoids the emergence of subdomains with
teristics have recently been proposed in. 8–10 The f discontinuity zones or locations where this func-
tion has a tendency to change quickly.
most stringent barrier to creating such a decom-
position, however, is that these subdomains can- Thus, the main idea, which we push forward,
not include even minor portions of various fields is that the segmentation problem should be given
with plausibly distinct crops, nor should they as a constrained minimization problem with a
overlap or contain any fragments of field borders. special anisotropic cost functional, where the “ef-
Because of this, the segmentation challenge indi- fect of anisotropy”we associate with the structure
29
cated above is fairly difficult. topology of f-distribution (see ). With that in
Moreover, the principal distinguishing char- mind, we utilize the main characteristic of the
acteristic of f-decomposition is the impossibil- given function f — the normal unit vector field
2
ity to reduce it to the well-known settings of θ : Ω → R to level sets of f. As a result,
450