Chapter 3 cont’d.
Adjacency, Histograms, &
Thresholding
RAGs
(Region Adjacency Graphs)
Define graph

G=(V,E)

where V is a set of vertices or nodes

and E is a set of edges
Define graph

G=(V,E)

where V is a set of vertices or nodes

and E is a set of edges
 represented
of vertices
by either unordered or ordered pairs
RAGs (Region Adjacency Graphs)
Steps:
1.
2.
label image
scan and enter adjacencies in graph
(RAGs also represent containment.)
0 (background)
1
2
3
-1
-3
-2
(Personally, I’d draw it this way!)
Define degree of a node.
What is special about nodes with
degree 1?
But how do we obtain
binary images (from gray or
color images)?
Histograms & Thresholding
Gray to binary
Thresholding
GB
const int t = 200;
if (G[r][c] > t) B[r][c] = 1;
else
B[r][c] = 0;

How do we choose t?
1.
2.
interactively
automatically
Gray to binary
1.
Interactively. How?
2.
Automatically.

Many, many, many, …, many methods.
a)
Experimentally (using a priori information).
Supervised / training methods.
Unsupervised
b)
c)

Otsu’s method (among many, many, many, many, … other
methods).
Histogram

“Probability” of a given gray value in an image.

h(g) = count of pixels w/ gray value equal
to g.

p(g) = h(g) / (w*h)
w*h = # of pixels in entire image
 What are the range of possible values for
p(g)?

Histogram

“Probability” of a given gray value in an image.

h(g) = count of pixels w/ gray value equal
to g. What data type is used for counts?

p(g) = h(g) / (w*h)
w*h = # of pixels in entire image
 What are the range of possible values for
p(g)? So what data type is p(g)? What
happens when h(g) is divided by w*h?

Histogram
Note: Sometimes we need to group gray
values together in our histogram into “bins”
or “buckets.”
E.g., we have 10 bins in our histogram and
100 possible different gray values. So we
put 0..9 into bin 0, 10..19 into bin 1, …
Histogram
Something is missing here!
Example of histogram
Example of histogram
We can even analyze the histogram just as we
analyze images.
One common measure is entropy:
Entropy

Ice melting in a warm room is a
common example of “entropy
increasing”, described in 1862 by
Rudolf Clausius as an increase in
the disgregation of the molecules of
the body of ice.

from
http://en.wikipedia.org/wiki/Entropy
Entropy
“My greatest concern was what to call it. I thought of calling it
‘information’, but the word was overly used, so I decided to
call it ‘uncertainty’. When I discussed it with John von
Neumann, he had a better idea. Von Neumann told me, ‘You
should call it entropy, for two reasons. In the first place your
uncertainty function has been used in statistical mechanics
under that name, so it already has a name. In the second
place, and more important, nobody knows what entropy really
is, so in a debate you will always have the advantage.”
– Conversation between Claude Shannon and John von
Neumann regarding what name to give to the “measure of
uncertainty” or attenuation in phone-line signals (1949)
Example of histogram
We can even analyze the histogram just as we
analyze images!
One common measure is entropy:
Calculating entropy
Notes:
1.
2.
3.
4.
p(k) is in [0,1]
If p(k)=0 then don’t
calculate log(p(k)).
Why?
My calculator only
has log base 10.
How do I calculate
log base 2?
Why ‘-’ to the left of
the summation?
Let’s calculate some histogram
entropy values.
Say we have 3 bits per gray value.
So our histogram has 8 bins.
 Calculate the entropy for the following
histograms (image size is 10x10):

0.08
1.
99
0
0
0
0
0
0
1
0.08
2.
99
0
1
0
0
0
0
0
2.92
3.
20
20
10
10
10
10
10
10
1.00
4.
50
0
0
50
0
0
0
0
2.00
5.
25
0
25
0
25
0
25
0
most disorder
Example histograms
Same subject
but different
images and
histograms
(because of a
difference in
contrast).
Example of
different
thresholds
So how can we determine
the threshold value
automatically?
Example automatic thresholding
methods
1.
Otsu’s method
2.
K-means clustering
Otsu’s method
Otsu’s method

Automatic thresholding method


automatically picks “best” threshold t given
an image histogram
Assumes 2 groups are present in the
image:
1.
2.
Those that are <= t.
Those that are > t.
Otsu’s method
Best choices for t.
Otsu’s method
For every possible t:
A.
B.
Calculate within group variances:
1.
probability of being in group 1; probability of being in group 2
2.
determine mean of group 1; determine mean of group 2
3.
calculate variance for group 1; calculate variance for group 2
4.
calculate weighted sum of group variances
Remember which t gave rise to minimum.
Otsu’s method:
probability of being in each group
t
q1 t    p i 
i 0
q2 t  
max
 pi 
i t 1
Otsu’s method:
mean of individual groups
t
1 t    i p i  / q1 t 
i 0
2 t  
max
 i pi  / q t 
i t 1
2
Otsu’s method:
variance of individual groups
t
 t    i  1 t  p i  / q1 t 
2
1
 t  
2
2
2
i 0
max
 i   t  pi  / q t 
i  t 1
2
2
2
Otsu’s method:
weighted sum of group variances
 t   q1 t  t   q2 t  t 
2
W

2
1
Calculate for all t’s and minimize.
min 

2
2
Demo Otsu.
2
W
t  | 0  t  max 
Demo of Otsu’s method
before
Demo of Otsu’s method
Otsu’s report
Demo of Otsu’s method
Otsu’s
threshold
Generalized thresholding
Generalized thresholding

Single range of gray values
const int t1 = 200;
const int t2 = 500;
if (G[r][c] > t1 && G[r][c] < t2) B[r][c] = 1;
else
B[r][c] = 0;
Even more general thresholding

Union of ranges of gray values.
const int t1 = 200, t2 = 500;
const int t3 =1200, t4 =1500;
if
(G[r][c] > t1 && G[r][c] < t2) B[r][c] = 1;
else if (G[r][c] > t3 && G[r][c] < t4) B[r][c] = 1;
else
B[r][c] = 0;
Something is missing
here!
K-means clustering
K-Means Clustering

In statistics and machine learning, k-means
clustering is a method of cluster analysis which
aims to partition n observations into k clusters in
which each observation belongs to the cluster
with the nearest mean. It is similar to the
expectation-maximization algorithm for mixtures
of Gaussians in that they both attempt to find the
centers of natural clusters in the data.

from wikipedia
K-Means Clustering
Clustering = the process of partitioning a
set of pattern vectors into subsets called
clusters.
 K = number of clusters (must be known in
advance).
 Not an exhaustive search so it may not
find the globally optimal solution.
 (see section 10.1.1)

Iterative K-Means Clustering
Algorithm
Form K-means clusters from a set of nD feature vectors.
1.
2.
3.
4.
5.
6.
Set ic=1 (iteration count).
Choose randomly a set of K means m1(1), m2(1), …
mK(1).
For each vector xi compute D(xi,mj(ic)) for each
j=1,…,K.
Assign xi to the cluster Cj with the nearest mean.
ic =ic+1; update the means to get a new set m1(ic),
m2(ic), … mK(ic).
Repeat 3..5 until Cj(ic+1) = Cj(ic) for all j.
K-Means Clustering Example
0. Let K=3.
1. Randomly (may not necessarily be actual data
points) choose 3 means (i.e., cluster centers).
- figure from wikipedia
K-Means Clustering Example
1. Randomly (may not necessarily be actual data
points) choose 3 means (i.e., cluster centers).
2. Assign each point to nearest cluster center
(mean).
- figure from wikipedia
K-Means Clustering Example
2. Assign each point to nearest cluster center
(mean).
3. Calculate centroid of each cluster center
(mean). These will become the new centers.
- figure from wikipedia
K-Means Clustering Example
3. Calculate centroid of each cluster center
(mean). These will become the new centers.
4. Repeat steps 2 and 3 until convergence.
- figure from wikipedia
K-Means for Optimal Thresholding

What are the features?
K-Means for Optimal Thresholding

What are the features?

Individual pixel gray values
K-Means for Optimal Thresholding

What value for K should be used?
K-Means for Optimal Thresholding

What value for K should be used?

K=2 to be like Otsu’s method.
Iterative K-Means Clustering
Algorithm
Form 2 clusters from a set of pixel gray values.
1.
2.
3.
4.
5.
6.
Set ic=1 (iteration count).
Choose 2 random gray values as our initial K means,
m1(1), and m2(1).
For each pixel gray value xi compute fabs(xi,mj(ic)) for
each j=1,2.
Assign xi to the cluster Cj with the nearest mean.
ic =ic+1; update the means to get a new set m1(ic),
m2(ic), … mK(ic).
Repeat 3..5 until Cj(ic+1) = Cj(ic) for all j.
Iterative K-Means Clustering
Algorithm
Form 2 clusters from a set of pixel gray values.
1.
2.
Set ic=1 (iteration count).
Choose 2 random gray values as our initial K means,
m1(1), and m2(1).
This can be derived from the original
image or from the histogram.
3.
4.
5.
6.
For each pixel gray value xi compute fabs(xi,mj(ic)) for
each j=1,2.
Assign xi to the cluster Cj with the nearest mean.
ic =ic+1; update the means to get a new set m1(ic),
m2(ic), … mK(ic).
Repeat 3..5 until Cj(ic+1) = Cj(ic) for all j.
Iterative K-Means Clustering
Algorithm
Example.
m1(1)=260.83, m2(1)=539.00
m1(2)=39.37, m2(2)=1045.65
m1(3)=52.29, m2(3)=1098.63
m1(4)=54.71, m2(4)=1106.28
m1(5)=55.04, m2(5)=1107.24
m1(6)=55.10, m2(6)=1107.44
m1(7)=55.10, m2(7)=1107.44
.
.
.
Demo K-Means.
Demo of K Means method
before
Demo of K Means method
K Means
reports for
K=2 & K=3
Demo of K Means method
K Means
t=56
Demo of K Means method
K Means
t=54
Demo of K Means method
K Means
t=128
Otsu vs. K-Means

Otsu’s method as presented determines
the single best threshold.

How many objects can it discriminate?

Suggest a modification to discriminate more.
Otsu vs. K-Means

How is Otsu’s method similar to K-Means?

What does Otsu’s method determine?

What does K-Means determine (for K=2)?
Otsu vs. K-Means

How is Otsu’s method similar to K-Means?

What does Otsu’s method determine?
a

single threshold, t
What does K-Means determine (for K=2)?
 m1
and m2 are cluster means (centers)
 Once m1 and m2 are determined, how can they be
used to determine the threshold?
Otsu vs. K-Means

A final word, . . .


K-Means readily generalizes to:
1.
arbitrary number of classes (K)
2.
can easily be extended to many, many features
(i.e., feature vectors instead of only gray
values/higher dimensions)
K-Means will find a local optimum, but that
may not be the global optimum!