Chapter 3 contd'

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Chapter 3 contd.

• Adjacency, Histograms, Thresholding

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RAGs(Region Adjacency Graphs)
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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?
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But how do we obtain binary images?
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Histograms Thresholding
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Gray to binary

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

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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).

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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)?

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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,

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Histogram
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Something is missing here!
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Example of histogram
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Example of histogram
We can even analyze the histogram just as we
analyze images. One common measure is entropy
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Example of histogram
We can even analyze the histogram just as we
analyze images. One common measure is entropy
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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?

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Example histograms
Same subject but different images and histograms
(because of difference in contrast).
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Example of different thresholds
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So how can we determine the threshold
automatically?
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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

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Otsus method
Best choices for t.
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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.

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Otsus methodprobability of being in each group
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Otsus methodmean of individual groups
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Otsus methodvariance of individual groups
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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

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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

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Something is missing here!
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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)

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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.

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K-Means for Optimal Thresholding

• What are the features?

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K-Means for Optimal Thresholding

• What are the features?
• Individual pixel gray values

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K-Means for Optimal Thresholding

• What value for K should be used?

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K-Means for Optimal Thresholding

• What value for K should be used?
• K2 to be like Otsus method.

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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.

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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.

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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?