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Nonlinear image processing with α-Tension field: A geometric approach
Let α > 1 be a constant. Given a map u, we only if it is harmonic with respect to the metric ¯g
define the α-energy functional of u by: conformally related to g given by:
Z
2 2 2α−2
2 α
E α (u) := (1+ | du | ) dvol g . (3) ¯ g = {(2α) m−2 (1+ | du | ) m−2 }g. (6)
M
Hence, the critical points of E α are α-harmonic Theorem 2 states an equivalence between α-
maps. The Euler–Lagrange equation related to harmonicity and the harmonicity of u with respect
the α-energy functional E α comes from applying to a suitably conformally rescaled domain metric.
Green’s theorem: The next theorem states the conditions for the ex-
istence of α−harmonic maps in every homotopy
2 α−1
τ α (u) : = (1+ | du | ) τ(u)
class.
2
2 α−2
+ (α − 1)(1+ | du | ) ∇I(∇ | ∇I | )
n
m
Theorem 3. [1] Let u : (M , g) −→ (N , h) be
= 0. (4) a harmonic map, and H be a homotopy class of
The section τ α (u) is called the α-tension field u. Then, there is a smooth metric ¯g on M con-
2
of u. We can then classify the main properties of formally equivalent to g such that u : (M, ¯g) −→
the α-tension fields as (N, h) is α−harmonic for m > 2α.
(1) Nonlinearity: Unlike the classical tension The theme of tension fields and their gener-
field, τ α (u) is intrinsically nonlinear as it alizations, e.g., α-tension fields, has a deep geo-
contains terms of the powers of | du |. metrical interpretation rooted in the balance of
This nonlinearity creates new difficulties energy and the local properties of maps, as mani-
for the analysis of the existence and regu- fested through geometry on manifolds. Although
larity of solutions. 2 the classical tension field incorporates the behav-
(2) Dependence on scaling: The factor ior of harmonic maps, the α-tension field gener-
2 α−1
(1+ | du | ) causes τ α (u) to depend alizes this concept to a more general set of vari-
on the scaling of du. Indeed, the ational problems. A manifold endowed with an
small-gradient maps are treated differ- additional structure (distance, metric, curvature,
ently from the large-gradient maps in etc.) can facilitate the drawing of inferences on
α−energy functional. 3,6 the properties of physical systems underpinning
(3) Geometric flexibility: Varying α al- these mathematical constructs. The α-tension
lows one to interpolate between different field introduces a parameterized family of tension
regimes of energy minimization, making it fields and is a flexible and mathematically potent
α flexible tool for the study of mappings tool to tackle questions across geometry, analysis,
between manifolds. 1 and topology.
Using Equations (3) and (4), we obtain the
result:
3. Application of tension fields in
Theorem 1. [6] Let u : (M, g) −→ (N, h) image processing
be a smooth map between Riemannian manifolds.
The concept of the tension field is the foundation
Then u is α-harmonic if and only if its α-tension
for the mappings between Riemannian manifolds
field vanishes.
have been widely used in image processing. Here,
This is something analogous to the famous we will explore these ideas and how they are ap-
fact that α-harmonicity is equivalent to α-tension plied in image analysis. The tension field is a link
field vanishing. between geometry and optimization, allowing for
advanced techniques in image processing.
3
2
Example 1. Let α = 2 and let u : R −→ R be
Tension field has been widely adapted to show
a smooth map defined by:
its relevance to possible applications in image pro-
u(x 1 , x 2 ) = (3x 2 − x 1 , x 2 − 2x 1 , 5x 2 + 3x 1 ). (5) cessing, where an image can be seen as a function
from a domain (e.g., a 2D image) to a codomain
By (4) and Theorem 1, it can be checked that u is
an α-harmonic map (e.g., pixel intensity values). Denote a 2D image
by I(x, y). By (4), the tension field of I, given by
We show an explicit relation between the α- τ(I) = trace g ∇dI, yields the simplification:
harmonic maps and the harmonic maps by con-
τ(I) = ∆I, (7)
formal deformation:
2
∂ I
Theorem 2. [1] Any smooth map u : where ∆I := ∂x 2 2 + ∂ I 2 , is the Laplacian opera-
∂y
n
m
(M , g) −→ (N , h) is an α-harmonic map if and tor applied on the input image. 10 Computations
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