Page 80 - IJOCTA-15-3

C. D’Apice et .al. / IJOCTA, Vol.15, No.3, pp.449-463 (2025)
a Gaussian kernel, and with a given exponent δ ∈ (1, 2), and
2
1 |x| m+1
G σ (x) = √ exp − 2 , σ > 0, Ξ ε = (c, φ) ∈ R × BV (Ω) :
2
2πσ 2σ
c = (c 0 , c 1 , . . . , c m ), c j ≥ 0, j = 0, . . . , m . (12)
and σ > 0 is a tuned small positive value. h i h i
As for the functions χ A τ (·), χ E τ (·) ∈
j ε j ε
∞
For the simplicity, we assume that the vector C (R), they have the following representations:
∞
2
field θ ∈ C (Ω; R ) vanishes along the boundary
χ A τ = H ε (l 1 − τ − φ),
∂Ω, that is, 0 ε
h i
f
M (x) = I ∀ x ∈ ∂Ω. (6) χ A τ j = H ε (φ − l j − τ), j = 1, . . . .m,
ε
h i
χ E τ = H ε (φ − l j + τ), j = 1, . . . .m,
j ε
As for the total variation of the measure
f
M Du, we define it by the rule where H ε (z) = [η ε ∗ H] (z) stands for the smooth
approximation of the Heaviside step function
Z
n 1, z≥0 o
f
f
|M Du|(Ω) := |M Du| H(z) = through the mollification. For
0,z<0
Ω
Z the practical implementations, it would be enough
n
1
2
f
= sup u div(M φ) dx : φ ∈ C (Ω; R ), to set
0

Ω  0, z < −ε,
o
|φ(x)| ≤ 1 ∀ x ∈ Ω) . (7) H ε (z) = z + 1 1 + 1 sin πz , −ε ≤ z ≤ ε,
2ε 2 π ε
1, z > ε.

(13)
Since the existence of minimizers to the Considering that the slope of H ε (z) is sufficiently
problem (3)–(4) remains an open issue nowa- near to H(z), the Heaviside function for ε > 0
1
days, according to, the following family of two- small enough, and that
parametric approximated problems should be
2
H ε (·) ∈ C (R),
used loc (14)
Z = ε, ∀ ε > 0,
f ∥H(·) − H ε (·)∥ 1
L (R)
J ε,τ (c, φ) = (f − c 0 ) log χ A τ 0 ε dx+
Ω c 0 the function H ε (·) is seen as a H(z) pointwise ap-
proximation.

m−1 Z h i h i
X f
1
(f − c j ) log χ A τ − χ A τ dx The function δ ε ∈ C (R) is defined, establish-
j j+1 0
Ω c j ε ε
j=1 ing

Z
f
+ (f − c m ) log χ A m ε dx dH ε (z)
τ
Ω c m δ 1,ε (z) = dz

1 0, |z| > ε,
+ ∥Ψ(l 0 − φ)∥ 1 + ∥Ψ(φ − l m+1 )∥ 1 = ∀ z ∈ R, (15)
L (Ω)
L (Ω)
ε 1 1 + cos πz , |z| ≤ ε,
2ε ε
f
+ ε|M Dφ|(Ω) as an approximation of δ(z), that is, the Dirac
ε
function.
m Z
i
X h
f
+ α M D χ E j τ → inf , (8) The following result has recently been estab-

ε
j=1 Ω 2ε ε φ∈Ξ ε lished in: 1
where τ and ε are small parameters, which vary
within strictly decreasing sequences of positive Theorem 1. Let f : Ω → R be a given grayscale
numbers converging to 0, image satisfying properties (2). Then, for each
A τ  ε ∈ (0, 1) and τ > 0 small enough, the con-
0
= {x ∈ Ω : φ(x) ≤ l 1 − τ} , 
 strained minimization problem (8)–(12) admits at

τ

A = {x ∈ Ω : φ(x) ≥ l j + τ} ,  0 0
j least one solution. Moreover, if (c , φ ) ∈ Ξ ε
(9) ε,τ ε,τ
E j τ = {x ∈ Ω : φ(x) ≥ l j − τ} ,  is a local minimizer to problem (8)–(12), then


 R 0
j = 1, . . . , m,  0 0 Ω fH ε (l 1 − φ ε,τ − τ) dx
c ε,τ,0 = c 0 (φ ε,τ ) = R ,
2 Ω H ε (l 1 − φ 0 ε,τ − τ) dx
Ω 2ε = x ∈ R : dist (x, Ω) ≤ 2ε ,
(10)
f 2 0 0
M ε (x) = I − (1 − ε) θ(x) ⊗ θ(x) , c = c j (φ ) =
ε,τ,j ε,τ
R
2−τ f H ε (φ 0 − l j − τ) − H ε (φ 0 − l j+1 − τ) dx
z , if z ≥ 0, Ω ε,τ ε,τ
Ψ(z) = (11) R ,
0, if z < 0, H ε (φ 0 − l j − τ) − H ε (φ 0 − l j+1 − τ) dx
Ω ε,τ ε,τ
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