1. Graph Based Image Segmentation
Sandeep Chalasani

2. Image Segmentation
A division of the pixels of an image into disjoint groups (where the groups correspond to objects or parts of objects)
The partition given by a binary pixel affinity function s, where s(i,j)=1 when pixels i and j belong together, and s(i,j)=0 otherwise.

3. Image Segmentation Purely local criteria are inadequate
– Difference along border between A and B is less than differences within C
Criteria based on piecewise constant regions are inadequate
– Will arbitrarily split A into subparts C A B
Image Courtesy

4. Graph Based Formulation
G=(V,E) with vertices corresponding to pixels and edges connecting neighboring pixels
Weight of edge is magnitude of intensity difference between connected pixels
A segmentation, S, is a partition of V such that each C∈S is connected

5. Disjoint Sets Data Structure
A disjoint-set is a collection S={S1, S2,…, Sk} of distinct dynamic sets.
Each set is identified by a member of the set, called representative. Each node points to its parent. Root points to itself.
Rooted trees, each tree is a set, root is the representative. Each node points to its parent. Root points to itself.
An Application of Disjoint-Set is to determine the connected components of an undirected graph.

6. Disjoint Sets Data Structure (cont)
Disjoint set operations:
MAKE-SET(x): create a new set with only x. assume x is not already in some other set.
UNION(x,y): combine the two sets containing x and y into one new set. A new representative is selected.
FIND-SET(x): return the representative of the set containing x. c f h e c f c d d h f
UNION
Set {c,h,e}
Set {f,d}
Image:

7. An Application of Disjoint-Set
Determine the connected components of an undirected graph.
CONNECTED-COMPONENTS(G)
for each vertex v V[G]
do MAKE-SET(v)
for each edge (u,v) E[G]
do if FIND-SET(u)  FIND-SET(v)
then UNION(u,v)
SAME-COMPONENT(u,v)
if FIND-SET(u)=FIND-SET(v)
then return TRUE
else return FALSE

8. Component Difference
Let internal difference of a component be maximum edge weight in its Minimum Spanning Tree
Int(C) = max e∈MST(C,E) w(e)
– Smallest weight such that all pixels of C are
connected by edges of at most that weight
Let difference between two components be minimum edge weight connecting them
Dif(C1,C2) = min vi∈C1, vj∈C2 w((vi,vj))
– Note: infinite if there is no such edge

9. D(C1,C2) =true if Dif (C1;C2) > MInt(C1;C2)
Algorithm
The region comparison is done by evaluating if there is evidence for a boundary between a pair or components by checking if the difference between the components, Dif (C1;C2), is large relative to the internal difference within at least one of the components, Int(C1) and Int(C2).
D(C1,C2) =true if Dif (C1;C2) > MInt(C1;C2)
false otherwise
MInt(C1;C2)=min(Int(C1)+τ(C1);Int(C2)+τ (C2))
The threshold function τ controls the degree to which the difference between two components must be greater than their internal differences in order for there to be evidence of a boundary between them (D to be true).

10. Results
Original Image
τ=150,k=1000
τ=150,k=500
τ=150,k=750

11. Conclusions
The Graph Based Image Segmentation is a highly efficient and cost effective way to perform image segmentation.
The threshold values and the number of minimum vertices that should be present so that a component can be considered as an image segment play an important role in determining the segmentation.