Segmentation of Stochastic Images using ... - Jacobs University

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
6.2.1 Γ-Convergence of the Stochastic Ambrosio-Tortorelli Model
Ambrosio and Tortorelli [14] showed the Γ-convergence of their model towards the Mumford-Shah
model. It is possible to extend this result to show the Γ-convergence of the stochastic extension of
the Ambrosio-Tortorelli model towards a stochastic Mumford-Shah model. For the formulation of
the result, we use the stochastic analog of the space D h,n from [14], the space D h,n ⊗ L 2 (Ω), which
contains admissible functions for the energies. In the notation of [14], n is the space dimension and
h = 1/ √ ε. Thus, letting the scale of the phase field ε tend to zero is equivalent to letting h → ∞.
Theorem 6.1. The stochastic Ambrosio-Tortorelli model E(E AT ) Γ-converges to the stochastic
Mumford-Shah model E(E MS ) as ε → 0. More precisely let (u h ,φ h ) ∈ D h,n ⊗ L 2 (Ω) be a sequence
that converges to (u,φ) in D h,n ⊗ L 2 (Ω). Then we have
∫
∫
E MS (u(ω),K(ω))dω ≤ liminf E AT (u h (ω),φ h (ω))dω (6.17)
h→∞
Ω
and for every (u,φ) there exists a sequence (u h ,φ h ) ∈ D h,n converging to (u,φ) such that
∫
∫
E MS (u(ω),K(ω))dω ≥ limsup E AT (u h (ω),φ h (ω))dω . (6.18)
Ω
h→∞ Ω
In both inequalities, the edge set K is defined accordingly as the discontinuity set of u.
Proof. We begin the proof by citing a famous theorem for the interchange of a limit process and
integration, Fatou’s lemma (see [32]):
Theorem 6.2 (Fatou’s lemma). For a sequence of nonnegative measurable functions f n ,
∫
∫
liminf f n ≤ liminf f n . (6.19)
We have to show that we can interchange the limit process and the integration. Let us assume that
this interchange is possible (all requirements of Fatou’s lemma are satisfied). Then, we have
∫
∫
∫
liminf E AT (u h ,φ h )dω ≥ liminf E AT (u h (ω),φ h (ω))dω = E MS (u(ω),K(ω))dω (6.20)
h→∞ Ω
Ω h→∞ Ω
and by using the reverse of Fatou’s lemma we get
∫
∫
∫
limsup
h→∞
Ω
E AT (u h ,φ h )dω ≤
Ω
limsupE AT (u h (ω),φ h (ω))dω =
h→∞
Ω
Ω
E MS (u(ω),K(ω))dω , (6.21)
because for every realization ω ∈ Ω we have the Γ-convergence of the Ambrosio-Tortorelli model to
the Mumford-Shah model initially proved by Ambrosio and Tortorelli [14]. Thus, we have to show
that the interchange of the limit process and the integration is possible.
The existence of the deterministic series ensures the existence of a series for which the limit
superior is less than the Γ-limit. For every ω ∈ Ω we choose the deterministic series constructed by
Ambrosio and Tortorelli [14]. The inequality is ensured because Fatou’s lemma yields
∫
∫
∫
limsup
h→∞
Ω
E AT (u h ,φ h )dω ≤
Ω
limsupE AT (u h (ω),φ h (ω))dω =
h→∞
Ω
E MS (u(ω),K(ω))dω . (6.22)
We justify the applicability of Fatou’s lemma in the following.
To use Fatou’s lemma we have to show that E AT is nonnegative and measurable. The first condition
is trivially ensured, because E AT is the sum of integrals of positive (squared) functions and thus
nonnegative. The second condition is also ensured, because of the following theorem from [142]:
Theorem 6.3. Any lower semicontinuous function f is measurable.
Following [14], the functional E AT is semicontinuous when we use the space D h,n . Thus, the
Ambrosio-Tortorelli functional is nonnegative and measurable and Fatou’s lemma can be applied.
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