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Region Segmentation Find sets of pixels, such that
All pixels in region i satisfy some constraint of similarity.
nRRR ,,, 21 IR
ii
ji RRji ,
K-Means Choose a fixed number of
clusters
Choose cluster centers and point-cluster allocations to minimize error
can’t do this by search, because there are too many possible allocations.
Algorithm fix cluster centers;
allocate points to closest cluster
fix allocation; compute best cluster centers
x could be any set of features for which we can compute a distance (careful about scaling)
x j i
2
jelements of i'th cluster
iclusters
Image Segmentation by K-Means Select a value of K Select a feature vector for every pixel (color, texture,
position, or combination of these etc.) Define a similarity measure between feature vectors
(Usually Euclidean Distance). Apply K-Means Algorithm. Apply Connected Components Algorithm. Merge any components of size less than some threshold to
an adjacent component that is most similar to it.
K-means clustering using intensity alone and color alone
Image Clusters on intensity Clusters on color
How to find K Use prior knowledge about image. Apply the algorithm for different values of
K and test for goodness of clusters. Analyze Image Histograms.
Agglomerative clustering Assume that each cluster is single pixel (i.e.
every pixel is a cluster itself). Merge Clusters i.e. attach closest to cluster it
is closest to (if possible) Repeat step 2 until no more clusters can be
merged.
Divisive clustering Assume that whole image is a single cluster. Split Clusters along best boundary (if exists) Repeat step 2 until no more clusters can be
split.
Inter-Cluster distance single-link clustering: Minimum distance
between an element of the first cluster and one of the second..
complete-link clustering: Maximum distance between an element of the first cluster and one of the second.
group-average clustering: Average of distances between elements in the clusters.
Segmentation by Split and Merge Start with an initial segmentation (for example by K-
Means). Define a criteria P for goodness of region such that
P( R )=True, if R satisfies the criteria P( R )=False, otherwise
For each region R, split R in four regions (quadrants), if P( R ) = False
Merge any two adjacent regions R and Q if
Repeat until no more clusters can be split or merged.
TrueP RQ
Graph A graph G(V,E) is a triple consisting of a
vertex set V(G) an edge set E(G) and a relation that associates with each edge two vertices called its end points.
Path A path is a sequence of edges e1, e2, e3, …
en. Such that each (for each i>2 & i<n) edge ei is adjacent to e(i+1) and e(i-1). e1 is only adjacent to e2 and en is only adjacent to e(n-1)
Connected & Disconnected Graph
A graph G is connected if there is a path from every vertex to every other vertex in G.
A graph G that is not connected is called disconnected graph.
Minimum CutA cut of a graph G is the set of edges S such that removal of S from G disconnects G.
Minimum cut is the cut of minimum weight, where weight of cut <A,B> is given as
ByAxyxwBAw
,,,
Minimum Cut There can be more than one minimum cut in a
given graph
All minimum cuts of a graph can be found in polynomial time1.
1H. Nagamochi, K. Nishimura and T. Ibaraki, “Computing all small cuts in an undirected network. SIAM J. Discrete Math. 10 (1997) 469-481.
Drawbacks of Minimum Cut Weight of cut is directly proportional to the
number of edges in the cut.
Ideal Cut
Cuts with lesser weightthan the ideal cut
Normalized Cuts1
Normalized cut is defined as
Ncut(A,B) is the measure of dissimilarity of sets A and B.
Minimizing Ncut(A,B) maximizes a measure of similarity within the sets A and B
VyBzVyAx
cut yzw
BAw
yxw
BAwBAN
,,,
,
,
,,
1J. Shi and J. Malik, “Normalized Cuts & Image Segmentation,” IEEE Trans. of PAMI, Aug 2000.
Finding Minimum Normalized-Cut Finding the Minimum Normalized-Cut is
NP-Hard. Polynomial Approximations are generally
used for segmentation
Finding Minimum Normalized-Cut
wherematrix, symmetric NNW
otherwise0
if,22
iNjeejiWXjiFji XXFF
Proximity Spatial
similarity feature Image
ji
ji
XX
FF
wherematrix, diagonal NND j
jiWiiD ,,
It can be shown that
such that
If y is allowed to take real values then the minimization can be done by solving the generalized eigenvalue system
Finding Minimum Normalized-Cut
Dyy
yWDyT
T
y
minmin cutN
0 and ,10 ,,1 D1yTbbiy
DyyWD
Algorithm Compute matrices W & D Solve for eigen vectors with the
smallest eigen values Use the eigen vector with second smallest eigen
value to bipartition the graph Recursively partition the segmented parts if
necessary.
DyyWD
Drawbacks of Minimum Normalized Cut Huge Storage Requirement and time
complexity Bias towards partitioning into equal
segments Have problems with textured backgrounds