Page 85 - IJOCTA-15-3

Computational aspects of the crop field segmentation problem based on anisotropic active contour model
Since the set-level function φ(x) should be be a solution of the minimization problem (8)–
subjected by the constraints (see (4)) (12) with ε > 0 and τ > 0 small enough. Let
∞
φ 0 ∈ C (Ω) be the smoothed version as the
l 0 ≤ φ(x) ≤ l m+1 a.e. in Ω, ε,τ ρ
convolution of the 2-D Gaussian kernel G ρ and
in practice this constraint can be incorporated by 0
renormalizing φ n (when φ n < l 0 or φ n > l m+1 ) φ ε,τ . We say that the following decomposition
i,j
i,j
i,j
after each time step. It means that we can omit Ω = Ω 0 ∪ Ω 1 ∪ · · · ∪ Ω m ∪ Ω ∗
the last two terms in the external force V 1 (φ) that is quasi-optimal for the approximation of f by
b
emerge as the penalizing ones. As a result, we can piecewise constant function f ∗ : Ω → R if
modify the iteration procedure (50) as follows:
• the restrictions f ∗ | are defined as
Ω j
y
n

φ n+1 = φ n + L(φ ) ∆ x X n + ∆ Y n ∆t
i,j i,j i,j − i,j − i,j f ∗ | = c 0 , j = 0, . . . , m,
! Ω j ε,τ,j
f 0
0

+ f − c log where the constants c are given by
ε,τ,0 0 ε,τ,j
c
ε,τ,0 (16);
f • the subdomains (or segments) {Ω j } m are
− f − c 0 ε,τ,m log j=0
c 0 ε,τ,m related to the function φ 0 as follows
i,j
m−1 ! n o
X 0 f Ω 0 = x ∈ Ω : φ 0 (x) < l 1 , (57)
− f − c ε,τ,j log 0 ε,τ ρ
j=1 c ε,τ,j n 0 o
Ω j = x ∈ Ω : l j < φ (x) < l j+1 , (58)
n n ε,τ ρ
× H ε (−φ i,j + l j+1 + τ) − H ε (φ i,j − l j − τ) ,
(53) j = 1, . . . , m − 1,
n o
Ω m = x ∈ Ω : φ 0 (x) > l m ; (59)
ε,τ
φ n+1 = l 0 if φ n+1 < l 0 , (54) ρ
i,j i,j
φ n+1 = l m+1 if φ n+1 > l m+1 , (55) • Ω ∗ stands for the boundaries between dif-
i,j
i,j
ferent Ω j , that is,
∀i = 1, . . . , N x , ∀ j = 1, . . . , N y ,
o
m n
[ 0
∀ n = 0, 1, . . . Ω ∗ = x ∈ Ω : φ ε,τ ρ (x) = l j .
Taking into account the following estimates: j=1
2 As immediately follows from this definition,
M ε f ≤ I in the sense of quadratic forms, each segment Ω j of the given f-decomposition can
1 x n n 1 y n be associated with the corresponding coefficient
n
|P | ≤ |∆ φ |, |Q | ≤ |∆ φ |,
i,j
+ i,j
+ i,j
i,j
ε ε of variation CV j , j = 0, . . . , m. This coefficient
m is the measure that quantifies the degree of vari-
n
L(φ ) ≤ ε + α
i,j
ε ability in the set of data by calculating the ratio
and arguing as in ?Chapter 28]Salgado , we can deduce of the standard deviation of f(x) within Ω j to its
the following stability conditions for the proposed mean f ∗ | , that is,
Ω j
approximation scheme s Z
2
m 1
f(x) − f ∗ | (x) dx
∆t 1 + α < . (56) Ω j
ε 2 2 CV j = Ω j
Therefore, the stability of the numerical scheme f ∗ | Ω j
(51),(52),(53)–(55) depends critically on the pa- for all j = 0, . . . , m.
rameter α > 0 being chosen correctly. As for the
Thus, by this definition, the boundaries ∂Ω j
stopping condition, it can be formalized as fol- of the segments Ω j can be defined as the corre-
lows: sponding level sets S j . To identify these sets,

n+1
max max φ − φ ≤ ε. we make use of the so-called marching squares
n
i,j l,j
1≤i≤N x 1≤j≤N y 38,39
method. The main steps of this approach can
Our next intention in this section is to discuss be described as follows:
the algorithm for restoration and visualization of (1) Given an index j ∈ {1, . . . ., m} and a
∞
the level sets smoothed version φ 0 ρ ∈ C (Ω) of the
ε,τ
S j := {x ∈ Ω : φ(x) = l j } , j = 1, 2, . . . , m. level set function φ 0 ε,τ ∈ BV (Ω), create a
binary image B j following the rule
(
1, if φ 0 (x) ≥ l j ,
Definition 1. Let f : Ω → R be a gray-scale im- B j (x) = ε,τ ρ ∀ x ∈ Ω.
age satisfying conditions (2). Let φ 0 ε,τ ∈ BV (Ω) 0, if φ 0 ε,τ ρ (x) < l j ,
457