Page 84 - IJOCTA-15-3

C. D’Apice et .al. / IJOCTA, Vol.15, No.3, pp.449-463 (2025)

f where (x, y) stands for image pixel and N x ×N y is
0

+ f − c ε,τ,m log
c 0 the original image size at the grid G H , we define
ε,τ,m
m−1 ! the following discrete notations
X 0 f
n
x
+ f − c log ∆ φ = ± φ n − φ n ,
ε,τ,j c 0 ± ij i±1,j ij
j=1 ε,τ,j y n n n
∆ φ = ± φ i,j±1 − φ ij ,
± ij
[H ε (−φ + l j+1 + τ) − H ε (φ − l j − τ)]
m(a, b) = minmod (a, b) =
1−τ
2 − τ (φ − l m+1 ) , if φ − l m+1 > 0,
+ sgn a + sgn b
ε 0, otherwise, min (|a|, |b|) ,
2
2 − τ (l 0 − φ) 1−τ , if l 0 − φ > 0, n
− (45) where φ i,j denotes the approximation of
ε 0, otherwise. φ k+1 (n∆t, i, j). Then, the numerical approxima-
We also need to specify the boundary condi- tion of the principle components of the boundary
tion on ∂Ω associated with (43). To obtain a well- value problem (47)–(49) takes the form
posed problem, we change the natural boundary ∂φ n φ n+1 − φ n
i,j
i,j
condition (43) just slightly, namely ≈ ,
∂t ∆t
i,j
 
f
M ε ∇φ   f 2  n
f
ε
 , ν Ω  = 0 a.e. on ∂Ω. M ε ∇φ
M q   
f
2
ε + |M ε ∇φ| 2 div  q  ≈
f
2
ε + |M ε ∇φ| 2
q i,j
f
Because the denominator ε + |M ε ∇φ| 2 is x n y n
∆ X + ∆ Y ,
f − i,j − i,j
strictly positive, the matrix function M ε (x) for X n = aP n + bQ , Y n = bP n + cQ ,
n
n
each x ∈ Ω is symmetric, and the discrete gradi- i,j i,j i,j i,j i,j i,j
2
ents are bounded in the norm, we can pass to the a b = M f ,
following boundary condition instead of (43) b c ε
n
f
f
M ∇φ, M ν Ω = 0 on ∂Ω, (46) P i,j =
ε
ε
x
∆ φ n
where ν Ω is a unit normal to ∂Ω. However, in + i,j ,
r
view of the assumption (6), this condition can be x n 2 y n y n 2
2
ε + ∆ φ + m ∆ φ , ∆ φ
rewritten as follows: + i,j + i,j − i,j
∂φ Q n =
= 0 on ∂Ω. i,j
∂ν Ω y n
∆ φ
+ i,j
.
Consequently, we obtain the subsequent prob- y 2 2
r
x
n
x
2
lem for numerical simulation ε + ∆ φ n + m ∆ φ , ∆ φ n
+ i,j
+ i,j
− i,j
 
2
f
∂φ M ε ∇φ
 − V 1 (φ)
 
b
= L(φ) div  q
∂t f As a result, utilizing the formulas given above,
2
ε + |M ε ∇φ| 2
we arrive at the following numerical scheme asso-
in (0, ∞) × Ω, (47) ciated with the initial boundary problem (47)–
∂φ (49):
= 0 on (0, ∞) × ∂Ω, (48)
∂ν Ω n+1 n n
φ = φ i,j + L(φ )
i,j
φ(0, ·) = φ 0 (·) in Ω, (49) i,j
n
n
n
× ∆ x − X i,j + ∆ y − Y i,j ∆t − V 1 (φ )∆t
b
l,j
where the functions L(φ) and V 1 (φ) are defined
b
∀i = 1, . . . , N x , ∀ j = 1, . . . , N y ,
by (27) and (45), respectively.
∀ n = 0, 1, . . . (50)
Quasi-linear partial differential equations can
be solved in a variety of ways (for a list of meth- with the initial conditions
ods, check the references 35,36 ). Finite differences φ 0 = (φ 0 ) ,
i,j i,j (51)
techniques and an explicit scheme of the forward ∀i = 1, . . . , N x , ∀ j = 1, . . . , N y ,
Euler method are maybe the best choices because
and boundary conditions
we are working with pixels in image processing. n n n n
Let ∆t be a time step size. Then setting φ 0,j = φ 1,j , φ N x,j = φ N x−1,j ,
n
t = n∆t, n = 0, 1, 2, . . . , v n = v , φ n = φ n , (52)
l,0
l,N y −1
l,1
l,N y
∀l = 1, . . . , N x , ∀ j = 1, . . . , N y .
x = i (1 ≤ i ≤ N x ), y = j (1 ≤ j ≤ N y ),
456