Page 83 - IJOCTA-15-3

Computational aspects of the crop field segmentation problem based on anisotropic active contour model
the form problem:

Z
f 0 !
f

∂ V 2 φ 0 ε,τ φ |M Dφ 0 ε,τ | + 0 f M ε ∇φ ε,τ
ε
f
Ω −L(φ ε,τ ) div M ε |M ε ∇φ 0 |
∂J (w, φ)| 0 + V 1 φ 0 ∋ 0. (32) ε,τ
w=φ ε,τ ε,τ 0 ′
+V 1 φ = 0 in D (Ω), (39)
ε,τ
f !
M ε ∇φ 0 ε,τ
M ε f f , ν Ω = 0 a.e. on ∂Ω. (40)
Following Anzellotti, 32 for an open bounded |M ε ∇φ 0 ε,τ |
2
set with Lipschitz boundary Ω ⊂ R , we denote
∞ 2 2
X 2 (Ω) = z ∈ L (Ω; R ) : div(z) ∈ L (Ω)
(33)
to give a correct interpretation of the Anzelotti 4. Numerical approximation of the
pairing (z, Du) with u ∈ BV (Ω). In this case, if optimality conditions
u ∈ BV (Ω) and z ∈ X 2 (Ω), then for every test
∞
function ψ ∈ C (Ω), we have In this section, we briefly describe the numeri-
0
cal scheme and a procedure to solve the Euler–
Z
⟨(z, Du) , ψ⟩ := − u div(ψz) dx = Lagrange system (39)–(40) numerically. Since for
Ω implementations, it is reasonable to define the so-
Z Z
lution of the problem (39)–(40) using a “gradi-
− uψ div(z) dx − u(z, ∇ψ) dx. (34)
Ω Ω ent descent”strategy, denoted by S a given circle
and by d(x, S) the Euclidean distance of the point
x ∈ Ω from the circle S, we suggest starting with
the initial level function φ 0 ∈ C(Ω) that can be
19
Arguing as in, ?Proposition 4.1.]Vese it can be defined as follows (see also the other variants in )
shown that ξ ∈ ∂J (w, φ)| 0 if and only if 
w=φ ε,τ  +d(x, S), x ∈ inside S,
there exists a vector field z ∈ X 2 (Ω) (see (33)) φ 0 (x) = 0, x ∈ S, (41)
such that the following conditions hold:  −d(x, S), x ∈ outside S.
∥z∥ ∞ 2 ≤ 1, Next, for numerical purposes, we calculate u ε,τ ,
L (Ω;R )

the continuous approximation of the BV solution
f
f
z, M Dφ 0 = M Dφ 0 as measures,

ε ε,τ ε ε,τ 0
φ ε,τ , with ε > 0 and τ > 0 small enough, as a

f
− div L(φ 0 ε,τ )M z = ξ in Ω, solution of the problem
ε
!
f
h i ∂φ f M ε ∇φ
f
L(φ 0 ε,τ )M z, ν Ω = 0 a.e. on ∂Ω. ∂t = L(φ) div M ε |M ε ∇φ| − V 1 (φ)
ε
f
Then taking into account the Anzelotti represen-
in (0, ∞) × Ω, (42)
tation (34) and the fact that
!
f
M ε ∇φ
M f = 0 on (0, ∞) × ∂Ω, (43)
f
− div L(φ 0 )M z ε f , ν Ω
ε,τ ε |M ε ∇φ|

f

= −V 2 φ 0 ε,τ M Dφ 0 ε,τ , z φ(0, ·) = φ 0 (·) in Ω. (44)
ε
The second point that we realize in this algo-
f
− L(φ 0 ε,τ ) div M z rithm is to consider the adaptive version of the
ε
19 34
f
f

= −V 2 φ 0 ε,τ M Dφ 0 − L(φ 0 ε,τ ) div M z model (42)–(44). Following and, we substi-

ε
ε
ε,τ
tute the multipliers δ ε (−φ 0 ε,τ + l j+1 + τ) in the
we can finally rewrite (32) as follows external force V 1 (φ) with 1. It has been recently
shown in 34 that, in this way, the external force
f
−L(φ 0 ε,τ ) div M z + V 1 φ 0 ε,τ ∋ 0, (35) without δ ε (−φ 0 + l j+1 + τ) can take action on
ε
h i ε,τ
f
L(φ 0 )M z, ν Ω = 0 a.e. on ∂Ω, (36) φ everywhere in Ω (not simply on the prescribed
ε,τ ε
level sets of φ), and so can keep φ moving more

f
f
z, M Dφ 0 ε,τ = M Dφ 0 , (37) quickly towards the desired segment edges. As a

ε
ε,τ
ε
result, the adaptive version of the external force
∥z∥ ∞ 2 ≤ 1. (38) V 1 (φ) takes the form
L (Ω;R )
If in addition φ 0 ε,τ is differentiable in the weak f !
sense, then the Euler–Lagrange system (35)–(38) V 1 (φ) = − f − c 0 ε,τ,0 log 0
b
assumes the form of the following boundary value c ε,τ,0
455