Segmentation of Stochastic Images using ... - Jacobs University

Chapter 2 Image Segmentation and Limitations
where ∇v i and ∇v j are the normalized gradients at position i respectively j, i.e. ∇v k = ∇u k /|∇u k |.
Eq. (2.26) is close to one if the gradients are aligned (than the scalar product ∇v i · ∇v j is close to
one) and close to zero if the gradients are not aligned. The set η s contains s pixels in the direction
perpendicular to the image gradient and the parameter ε dc controls the influence of the gradient.
For the numerical experiments we used ε dc = 0.25 and four pixels in directions perpendicular to the
image gradient, i.e. |η s | = 4.
Edge Continuity
To avoid the breaking up of edges, Erdem et al. [49] proposed to use an additional feedback measure,
which lowers the diffusivity around detected edges, to allow a growth of the detected edges. Using a
simplified version of the original model [49], the feedback measure is
(c h ) i =
1
1 + φ i − φ 2
i
, (2.27)
where φ is the phase field. We use a slight modification of the above feedback measure by adding an
additional scale factor α, allowing us to weight the deviation of the phase field from 0 and 1:
1
(c h ) i =
1 + α ( )
φ i − φi
2 . (2.28)
In the numerical experiments, we set α = 10. Thus, the diffusivity decreases in regions, where the
phase field is away from zero and one, i.e. regions, where the edge detection is in an intermediate state
between smoothing the structure away and building up a sharp edge. Fig. 2.6 shows a comparison
between Ambrosio-Tortorelli segmentation with and without the additional factor c.
2.4 Level Sets for Image Segmentation
The Ambrosio-Tortorelli segmentation approach introduces a new quantity besides the image and a
smooth representation: the phase field. This phase field approximates the edge set in the Ambrosio-
Tortorelli approach, but even with the above modifications there is no guarantee to obtain a connected
phase field in the end.
Level set based segmentation methods use another viewpoint. They place a closed curve somewhere
in the image and try to adjust this initial curve to the edges, respectively the object boundaries
in the image. This approach assures that the final segmentation result is a closed contour. For the
representation of the contour, there are methods for an explicit [76] or an implicit [29,31,40,96,138]
representation available in the literature. A famous method for the explicit representation is the snake
model [76], but explicit representations have drawbacks: parametrization of the curve, distribution
of the nodes describing the curve, dependence of the result on the parametrization, etc. To avoid
these shortcomings we focus on the implicit representation based on level sets in the following.
Dervieux and Thomasset [40] and Osher and Sethian [121, 138] developed the level set method.
The main idea is the implicit representation of a curve by embedding a curve C 0 ⊂ IR n into a higherdimensional
function φ : IR n+1 → IR and to identify the zero level set of φ with the curve, i.e.
C 0 = {x ∈ IR n : φ(0,x) = 0} . (2.29)
It is possible to describe the motion of the curve by the motion of the level sets of φ. At any time t > 0,
we get the curve C(t) back from the level set representation via C(t) = {x ∈ IR n : φ(t,x) = 0}. Using
this concept, we describe the motion of the curve using the level set equation [138]
φ t + F|∇φ| = 0 , (2.30)
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