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Chapter 3 contd.

- Adjacency, Histograms, Thresholding

RAGs(Region Adjacency Graphs)

RAGs (Region Adjacency Graphs)

- Steps
- label image
- scan and enter adjacencies in graph
- (includes containment)

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Define degree of a node. What is special about

nodes with degree 1?

But how do we obtain binary images?

Histograms Thresholding

Gray to binary

- Thresholding
- G ? B
- const int t200
- if (Grcgtt) Brc1
- else Brc0
- How do we choose t?
- Interactively
- Automatically

Gray to binary

- Interactively. How?
- Automatically.
- Many, many, many, , many methods.
- Experimentally (using a priori information).
- Supervised / training methods.
- Unsupervised
- Otsus 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) / (wh)
- wh of pixels in entire image
- What are the range of possible values for p(g)?

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

Example of histogram

We can even analyze the histogram just as we

analyze images. One common measure is entropy

Calculating entropy

- Notes
- p(k) is in 0,1
- If p(k)0 then dont 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?

Example histograms

Same subject but different images and histograms

(because of difference in contrast).

Example of different thresholds

So how can we determine the threshold

automatically?

Otsus method

- Automatic thresholding method
- automatically picks t given an image histogram
- Assume 2 groups are present in the image
- Those that are ltt
- Those that are gtt

Otsus method

Best choices for t.

Otsus method

- For every possible t
- Pick a t.
- Calculate within group variances
- probability of being in group 1
- probability of being in group 2
- determine mean of group 1
- determine mean of group 2
- calculate variance for group 1
- calculate variance for group 2
- calculate weighted sum of group variances and

remember which t gave rise to minimum.

Otsus methodprobability of being in each group

Otsus methodmean of individual groups

Otsus methodvariance of individual groups

Otsus methodweighted sum of group variances

- Calculate for all ts and minimize.
- Demo Otsu.

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Generalized thresholding

- Single range of gray values
- const int t1200
- const int t2500
- if (Grcgtt1 Grcltt2) Brc1
- else Brc0

Even more general thresholding

- Union of ranges of gray values.
- const int t1200, t2500
- const int t31200, t41500
- if (Grcgtt1 Grcltt2) Brc1
- else if (Grcgtt3 Grcltt4) Brc1
- else Brc0

Something is missing here!

K-Means Clustering

- Clustering the process of partitioning a set of

pattern vectors into subsets called clusters. - K number of clusters (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. - Set ic1 (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

j1,,K. - Assign xi to the cluster Cj with the nearest

mean. - ic ic1 update the means to get a new set

m1(ic), m2(ic), mK(ic). - Repeat 3..5 until Cj(ic1) Cj(ic) for all j.

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?
- K2 to be like Otsus method.

Iterative K-Means Clustering Algorithm

- Form 2 clusters from a set of pixel gray values.
- Set ic1 (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 j1,2. - Assign xi to the cluster Cj with the nearest

mean. - ic ic1 update the means to get a new set

m1(ic), m2(ic), mK(ic). - Repeat 3..5 until Cj(ic1) 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.

Otsu vs. K-Means

- Otsus method as presented determines the single

best threshold. - How many objects can it discriminate?
- Suggest a modification to discriminate more.
- How is Otsus method similar to K-Means?