Segmentation of Stochastic Images using ... - Jacobs University

2.2 Random Walker Segmentation
Definition The space of functions of bounded variation is
{
∫
}
BV(D) = u ∈ L 1 (D) : |Du|dx < ∞
D
. (2.2)
Following [17] and using the Lebesgue decomposition theorem (see [32]) the derivative of a BVfunction
decomposes into three parts, the absolutely continuous part ∇udx, the jump part D j u and
the Cantor part D c u. This leads us to the definition of the class of special BV-functions:
Definition The class of BV-function for which the Cantor part vanishes, i.e. D c u = 0, is called the
space of special functions of bounded variation (SBV).
Based on the space SBV we define the space of generalized functions of bounded variation (GSBV):
Definition The space GSBV consists of all functions u ∈ L 1 (D) satisfying
i.e. the truncated function belongs to SBV for all T .
∀T > 0 : u T = sign(u)max(T,|u|) ∈ SBV , (2.3)
In particular, the spaces SBV and GSBV are useful, when we introduce Mumford-Shah segmentation
and the related Ambrosio-Tortorelli approximation.
2.2 Random Walker Segmentation
Random walker segmentation performs on a single image u : D → IR defined on the image domain D.
Since random walker segmentation is based on pixel values only, we need no additional assumptions
about the smoothness of the images. The main idea of random walker segmentation is that the user
prescribes a set of seed points for the object and the background. From the remaining unseeded
points, random walks start and the percentage of random walks reaching the object seeds is the
probability of the pixel to belong to the object.
Before we begin to introduce random walker segmentation, we have to define notation for the
graph representation of the image. A graph G is a pair G = (V,E) containing vertices or nodes
v ∈ V and edges e ∈ E ⊂ V ×V. We denote an edge connecting the vertices v i and v j by e i j and
identify e i j with e ji , because we are interested in nondirectional graphs only. Every edge has a
weight w(e i j ) =: w i j that describes the costs for using the edge. A graph containing edge weights is
a weighted graph. The summation of all edge weights for a node i,
d i =
∑
{ j∈V :e i j ∈E}
w(e i j ) , (2.4)
is the degree of the node i.
For random walker segmentation, we identify the image u with a graph G. The pixels of the digital
image are the graph nodes and every pixel (respectively node) is connected to the neighboring nodes
by a weighted edge. Fig. 2.2 shows a graph corresponding to a 3 × 3 image. For random walker
segmentation the graph weights are
w(e i j ) = exp ( −β(g i − g j ) 2) , (2.5)
where (g i − g j ) 2 is the normalized difference between the image intensities at position i and j:
(g i − g j ) 2 =
The parameter β is the only free parameter that the user chooses.
(u i − u j ) 2
max {k,l∈V :ekl ∈E}(u k − u l ) 2 . (2.6)
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