Segmentation of Stochastic Images using ... - Jacobs University

2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation
minimizer is a challenging problem, cf. [35]. Since the functional is not differentiable, the estimation
of minimizers based on the Euler-Lagrange equations is impossible. Instead, researchers proposed
regularized approximations (see [17]). The following paragraph summarizes one of these methods,
proposed by Ambrosio and Tortorelli [14].
Remark 4. All components of the Mumford-Shah functional are essential to get a segmentation of
the image u, i.e. it is impossible to omit one of the components to end up with a mathematically and
numerically easier problem. If we omit the first component, we have no control over the difference
between the image and the smooth approximation and u = 0, K = /0 minimize the remaining parts. We
obtain another trivial solution if we omit the second component: Now u = u 0 and K = /0 minimize the
functional. When omitting the last component, K = D minimizes the functional. Thus, the Mumford-
Shah functional contains the minimal number of components necessary for segmentation, and it is
essential to discretize them well to get meaningful numerical solutions.
2.3.1 Ambrosio-Tortorelli Segmentation
As already mentioned, the Ambrosio-Tortorelli segmentation [14] is a kind of regularization of the
Mumford-Shah functional. Ambrosio-Tortorelli segmentation uses a function φ : D → IR, the phase
field, instead of the edge set K. The phase field is a smooth indicator function of the edge set K. It
is zero on the edge set K and goes smoothly to one away from the edge set. An additional variable ε
controls the width of the transition zone. When ε goes to zero, the phase field goes to the characteristic
function of the edge set. In a following section, we will recap that the Ambrosio-Tortorelli
energy converges in the Γ-sense to the Mumford-Shah energy [14].
The idea of the Ambrosio-Tortorelli segmentation for a given initial image u 0 is to find a phase
field φ and a smooth image u minimizing the energy
where
E AT [u,φ] := Efid,u ε [u] + Eε reg,u[u,φ] + Ephase ε [φ] , (2.15)
E ε fid,u [u] = ∫
D
∫
Ereg,u[u,φ] ε =
∫
Ephase ε [φ] =
D
D
1
2 (u − u 0) 2 dx
µ ( φ 2 )
+ k ε |∇u| 2 dx
(
νε|∇φ| 2 + ν 4ε (1 − φ)2) dx .
(2.16)
The first energy, the fidelity energy, ensures closeness of the smoothed image to the original u 0 . The
second energy, the regularization energy, measures smoothness of u apart from areas where φ is
small (the edges), and enforces φ to be small close to edges. The parameter k ε ensures coerciveness
of the differential operator and existence of solutions, because φ 2 may vanish. The third energy,
the phase energy, drives the phase field towards one and ensures small edge sets via the term |∇φ| 2 .
The parameter ε controls the scale of the detected edges, µ the amount of detected edges, and ν the
behavior of the phase field. k ε is a small regularization parameter.
The relation between the first two components of the Ambrosio-Tortorelli and the Mumford-Shah
energy are obvious. The third component, the phase energy, is a combination of a term forcing φ to
be one and the term ∫ D ε|∇φ|2 . In the limit ε → 0, it can be shown to be equal to H d−1 (K) by using
the co-area formula [105].
A minimizer of this energy is an image that is flat away from edges and a phase field, which
is close to zero at edges only. To obtain a minimizer of an energy, a widely used technique is to
solve the Euler-Lagrange equations resulting from this energy. For the computation of the Euler-
Lagrange equations, we have to compute the first variation of the above energies using the Gâteaux
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