Page 38 - IJOCTA-15-4

Kazemi et al. / IJOCTA, Vol.15, No.4, pp.578-593 (2025)
this is the first study to systematically investigate and α-tension field are contrasted in both mathe-
the potential of the α-tension fields on image pro- matical formulation and properties, as well as in
cessing tasks. The proper noise smoothing, as practice. Finally, Section 6 summarizes the pa-
well as, sharp feature extraction capabilities of per, noting implications and directions for future
the α-tension field, stems from its exploitation of work.
higher-order gradient information in addition to
being derived from geometric variational bases as 2. Geometry of tension fields and
opposed to classical linear operators, which either α-tension fields
overly enhance the noise in the minimization pro- The tension field is a fundamental notion in geo-
cess or prevent significant image details from be- metric variational problems, which rely on the
ing evident. As such, it presents a mathematically principles of energy minimization to study the
principled and generalizable framework for ad- structure of mappings between manifolds. These
vanced image processing applications, including are rooted in differential geometry and yield
denoising, edge-preserving smoothing, and fea- strong results on how curvature, topology, and
ture enhancement. Additionally, the authors inte- energy functionals restrict the behavior of map-
grate geometrical theory with the following prac- pings. This part has been dedicated to explor-
tical implementations: this not only encourages ing the geometric background under tension fields
more efficient image-analysis processes but also and their generalizations, which we call α− ten-
sets the stage for future inquiry into real-time and sion fields. We emphasize their intrinsic math-
large-scale image-processing techniques. ematical features and discuss how these notions
9
This work differs from Ref. in terms of the- shed light on the relationship between energy min-
oretical contributions, methodologies, and novel imization and the geometrical structure of map-
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applications. The paper defines (α, f)-harmonic pings.
maps and checks how stable they are under differ- Let M and N be smooth Riemannian mani-
ent geometric conditions, like Ricci and sectional folds, M compact and oriented, and let u : M −→
curvatures. However, it does not use these ideas N be a smooth map. We can write the energy
in real-world image-processing tasks, so there is functional of u as:
Z
a gap between theory and practice. On the other 1 2
E(u) = | du | dvol g , (1)
hand, our manuscript addresses this limitation by 2 M
proposing the 2-tension field (α = 2) as a novel 2
where | du | is the norm of du with respect to
operator for image processing at the intersection
the metrics on M and N, and dvol g is the volume
of geometric theory and tangible applications. We
form on M. Critical points of E(u) are called har-
use adaptive local gradient adjustments and fi-
monic maps, and they solve the Euler – Lagrange
nite difference methods to discretize the 2-tension
equation:
field. This works around the problems that tra-
ditional operators (like Laplacian) have with be- τ(u) := traceg∇du = 0, (2)
ing too smooth while keeping structural details. where ∇ is the induced connection on the pull-
In addition, our work analyzes advanced applica- back bundle u −1 TN, trace g is the trace with re-
tions like image denoising, edge preservation, and spect to the metric g on M and τ(u) is the tension
feature enhancement, which were not present in field of u. Noting that the tension field measures
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the literature. 9 the distance of u from a totally geodesic map. 5
The rest of this paper is organized as follows: Geometrically, the tension field describes the
Section 2 gives a general description of the geome- amount of unbalance or stress in the map u. This
try of tension fields, generalizations of the geome- reflects the dependence of how the geometry of
try of tension fields, and there we introduce the α- the domain manifold M interacts with the geom-
tension field. It explores the mathematical under- etry of the target manifold N. Specifically:
pinnings of these concepts and their role in geo- 1. If τ(u) = 0. The mapping u is harmonic,
metric variational problems. Section 3 discusses that is, it satisfies the condition that min-
the use of tension fields to handle certain prob- imizes the energy functional, so that the
lems in image processing, including edge detec- tension is balanced at all points. 5
tion, denoising, and local contrast enhancement. 2. The points where τ(u) ̸= 0 represent the
Section 4 is devoted to the utility of the α-tension regions where the map is either stretch-
field τ α (I) in image processing, emphasizing its ing or compressing excessively. These ar-
benefits in denoising, edge preservation, and fea- eas typically correspond to locations with
ture enhancement over conventional approaches. high curvature or substantial variations in
This leads us to Section 5, where the tension field the mapping. 4
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