Segmentation via
Graph Cuts
Balakrishna Thiagarajan
Computer Science and Engineering
State University of New York at Buffalo
Introduction
Gestalt approach
(From) Normalized Cuts and Image Segmentation
Image Segmentation
 Segmentation refers to the process of
partitioning a digital image into multiple
segments.
 The goal of segmentation is to simplify and/or
change the representation of an image into
something that is more meaningful and easier
to analyze.
 Similarity (gray level difference), Proximity
and continuity are important factors in
segmentation.
Graph representation of Image
 Set of points of the feature space represented
as a weighted, undirected graph, G = (V, E).
 The points of the feature space are the nodes
of the graph.
 Edge between every pair of nodes.
Graph representation of Image
 Weight on each edge, w(i, j), is a function of
the similarity between the nodes i and j.
 Partition the set of vertices into disjoint sets
where similarity within the sets is high and
across the sets is low.
 Therefore, Segmentation is equivalent to
Graph partition.
Graph representation of Image
Cut:
 A cut in a graph is a set of edges whose
removal disconnects the graph.
 A minimum cut is a cut with a minimum
number of edges. It is denoted by S. In the
weighted graph,
cut(A,B) = ∑ w(u,v)
u E A,v E B
where A & B are two partitioned disjoint sets
Segmentation by Min Cut method
 By Min Cut method, the graph is partitioned
into clusters.
 Each cluster is considered as an image
segment.
 Min Cut method uses the HCS (Highly
Connected Subgraphs) Algorithm to find the
clusters.
Segmentation by Min Cut method
 Edge Connectivity is the minimum number of
edges whose removal results in a
disconnected graph. It is denoted by k(G).
 For a graph with vertices n > 1 to be highly
connected if its edge-connectivity k(G) > n/2.
 A highly connected subgraph (HCS) is an
induced subgraph H in G such that H is highly
connected.
Segmentation by Min Cut method
Example:
A
B
Not HCS!
D
No. of nodes = 5
C
Edge Connectivity = 1
E
Segmentation by Min Cut method
Example continued:
A
B
HCS!
D
No. of nodes = 4
C
Edge Connectivity = 3
HCS Algorithm - Min Cut method
HCS(G(V,E))
begin
(H, H’,C)  MINCUT(G)
if G is highly connected
then return (G)
else
HCS(H)
HCS(H’)
end if
end
HCS Algorithm - Min Cut method
 The procedure MINCUT(G) returns H, H’ and
C where C is the minimum cut which
separates G into the subgraphs H and H’.
 Procedure HCS returns a graph in case it
identifies it as a cluster.
 Single vertices are not considered clusters
and are grouped into singletons set S.
HCS Algorithm - Min Cut method
Example
HCS Algorithm - Min Cut method
Example Continued
HCS Algorithm - Min Cut method
Example Continued
Cluster 2
Cluster 1
Cluster 3
HCS Algorithm - Min Cut method
 The HCS Algorithm returns clusters which are
nothing but image segments.
 The problem with HCS Algorithm – Min Cut
method is that there is a chance that many
singletons may be separated.
 To overcome this we use normalized-cut for
image segmentation.
HCS Algorithm – Main Problem
Example
HCS Algorithm – Main Problem
Example – Another possible cut
Max Flow - Min Cut Method
 Theorem
In graph G, the maximum source-to-sink flow
possible is equal to the capacity of the
minimum cut in G.
(L. R. Foulds, Graph Theory Applications, 1992 Springer-Verlag
New York Inc., 247-248)
Max Flow - Min Cut Method
 Min Cut can be made by using Ford-
Fulkerson Algorithm to find the max-flow in
the graph.
 In the graph temporarily add Source ‘s’ and
Sink ‘t’.
 Consider the weight associated with each
each node as a flow capacity.
 Starting from zero flow, increase the flow
gradually by finding a path from s to t along
which more flow can be sent, until a max-flow
is achieved.
Max Flow - Min Cut Method
f=0
0/2
0/2
0/1
s
0/2
1
1
t
0/3
s
2
t
2
1
0/2
(a)
2
(d)
f=0
2
1
1
1
t
2
f=3
1
2
3
s
2
2
2
2
(b)
2
1
t
1
1
1
1
(e)
1
1
2
1
1
0/2
s
f=2
1
f=1
1/2
1
1
f=3
2/2
1/1
1
s
t
3
2
2
2
(c)
s
t
1/3
2/2
1/2
1/2
(f)
Max Flow - Min Cut Method
 After the max-flow is found, the minimum cut
is by
S = {All vertices reachable from s}
T=G\S
1
1
1
s
2
2
2
1
1
1
t
1
1
Max Flow - Min Cut Method Illustration
(From) Image Segmentation Using Minimal Graph Cuts by Anders P. Eriksson, Olof Barr
and Kalle Astr•om
Normalized Cut
 Normalized Cut (Ncut) :
Ncut(A,B) = cut(A,B)
+ cut(A,B)
assoc(A,V) assoc(B,V)
where, assoc(A,V) = ∑ w(u,t)
u E A, t E V
 The cut is made such that the normalized cut
value between the two sets A & B is
minimum.
Normalized Cut
 Minimum cut criteria favors cutting small sets
of isolated nodes in the graph
(From) Normalized Cuts and Image Segmentation –
Shi & Malik
Normalized Cut
 Exact solution to minimizing normalized cut is
an NP-complete problem.
 However, approximate discrete solutions can
be found efficiently.
 Normalized cut criterion can be computed
efficiently by solving a generalized eigenvalue
problem.
Normalized Cut – Physical
Interpretation
 Think of the weighted graph as a spring mass
system.
 Graph nodes  physical masses
 Graph edges  springs
 Graph edge weight  spring stiffness
 Total incoming edge weights  mass of the
node
Normalized Cut – Physical
Interpretation
 Imagine giving a hard shake to this spring-
mass system, forcing the nodes to oscillate in
the direction perpendicular to the image
plane.
 Nodes that have stronger spring connections
among them will likely oscillate together.
 Eventually, the group will “pop” off from the
image plane.
Normalized Cut
(From) Normalized Cuts and Image
Segmentation - Shi & Malik
Average Cut
 Average Cut:
Averagecut(A,B) = cut(A,B)
|A|
+ cut(A,B)
|B|
 The average cut measure is proportional to
the total link weight cut, normalized by the
sizes of the partitions.
 The problem with Average Cut is that one
cannot ensure the two partitions computed
will have tight within-group similarity.
Comparison
 Different types of graph cuts - MinCut,
Average Cut, and Normalized Cut were
analysed by Padmanabhan Soundararajan
and Sudeep Sarkar.
 Totally 50 images were used. 5 different
image classes with 10 images in each class.
Comparison
 The 5 classes of images were
1. natural objects in indoor backgrounds
2. natural objects in outdoor backgrounds
3. man-made objects in indoor backgrounds
4. man-made objects in outdoor backgrounds
5. aerial images that contained man-made
objects in natural surroundings.
Comparison
 The best possible performances on each
image for the three different cut measures
were not significantly different from each
other.
 The performance with the three measures is
worst for man made objects in indoor
surroundings and for aerial images.
 The normalized cut based partition took
significantly larger time to compute than the
average and the mincut.
References
J. Shi and J. Malik, “Normalized Cuts and Image
Segmentation,” IEEE Trans. Pattern Analysis and
Machine Intelligence, vol. 22, no. 8, pp. 888-905,
Aug. 2000.
2. Erez Hartuv and Ron Shamir, “A Clustering
Algorithm based on Graph Connectivity”, Mar. 1999.
3. Padmanabhan Soundararajan and Sudeep Sark,
“Analysis of MinCut, Average Cut, and Normalized
Cut Measures”, May 2000.
4. Anders P. Eriksson, Olof Barr and Kalle Astrom,”
Image Segmentation Using Minimal Graph Cuts”.
1.
Thank You!!