Algorithm Design

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Chapter 7 Network Flow
Thus we see that the maximization of q(A, B) is the same problem as the
minimization of the quantity
j~B
(i,13aE
IANIi,I|I=I
As for the missing source and the sink, we work by analogy with our constructions
in previous sections: We create a new "super-source" s to represent
the foreground, and a new "super-sink" t to represent the background. This
also gives us a way to deal with the values ai and b i that reside at the nodes
(whereas minimum cuts can only handle numbers associated with edges).
Specifically, we will attach each of s and t to every pixel, and use a~ and bi to
define appropriate capacities on the edges between pixel i and the source-and
sink respectively.
Finally, to take care of the undirected edges, we model each neighboring
pair (i, j) with two directed edges, (i, j) and (], i), as we did in the undirected
Disjoint Paths Problem. We will see that this works very well here too, since
in any s-t cut, at most one of these two oppositely directed edges Can cross
from the s-side to the t-side of the cut (for if one does, then the other must go
from the t-side to the s-side).
Specifically, we define the following flow network G’ = (V’, E t) shown in
Figure 7.18(b). The node set V t consists of the set V of pixels, together with
two additional nodes s and t. For each neighboring pair of pixels i and j, we
add directed edges (i,j) and (j, i), each with capacity p~j. For each pixel i, we
add an edge (s, i) with capacity a~ and an edge (i, t) with capacity hi.
Now, an s-t cut (A, B) corresponds to a partition of the pixels into sets A
and B. Let’s consider how the capacity of the cut c(A, B) relates to the quantity
q~(A, B) that we are trying to minimize. We can group the edges that cross the
cut (A, B) into three natural categories.
o Edges (s, j), where j ~ B; this edge contributes aj to the capacity of the
cut.
o Edges (i, t), where i ~ A; this edge contributes bi to the capacity of the
cut.
o Edges (i,)) where i ~ A andj ~ B; this edge contributes Pii to the capacity
of the cut.
Figure 7.19 illustrates what each of these three kinds of edges looks like relative
to a cut, on an example with four pixels.
Pij.
b.
7.10 Image Segmentation
Figure 7.19 An s-~ cut on a graph constructed from four pLxels. Note how the three
types of terms in the expression for q’(A, B) are captured by the cut.
If we add up the contributions of these three kinds of edges, we get
=q(,B). ’A
(i,])~
IAnli,jll~l
So everything fits together perfectly. The flow network is set up so that the
capacity of the cut (A, B) exactly measures the quantity q’(A, B): The three
kinds of edges crossing the cut (A, B), as we have just defined them (edges
from the source, edges to the sink, and edges involving neither the source nor
the sink), correspond to the three kinds of terms in the expression for q’(A, B).
Thus, if we want to minimize q’(A, B) (since we have argued earlier that
this is equivalent to maximizing q(A, B)), we just have to find a cut of minimum
capaciW. And this latter problem, of course, is something that we know how
to solve efficiently.
Thus, through solving this minimum-cut problem, we have an optimal
algorithm in our model of foreground/background segmentation.
(7.55) The solution to the Segmentation Problem can be obtained by a
minimum-cut algorithm in the graph G’ constructed above. For a minimum
cut (A’, B~), the partition (A, B) obtained by deleting s* and t* maximizes the
segmentation value q(A, B).
P~
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