Computer Graphics and Image Processing (CIS-601)

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Computer Graphics and Image Processing (CIS-601)
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Image Segmentation by Clustering (Using
Mahalanobis Distance)

• - Manjit Chintapalli

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What is Image Segmentation ?

• To humans, an image is not just a random
  collection of pixels it is a meaningful
  arrangement of regions and objects.
• There also exits a variety of images natural
  scenes, paintings, etc. Despite the large
  variations of these images, humans have no
  problem to interpret them.

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Image Segmentation ? (Contd..)

• Image segmentation is the first step in image
  analysis and pattern recognition.
• It is a critical and essential component of image
  analysis system, is one of the most difficult
  tasks in image processing, and determines the
  quality of the final result of analysis.

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The Process

• Image segmentation is the process of dividing an
  image into different regions such that each
  region is homogeneous.

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Image Segmentation methods
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Image segmentation methods can be categorized as
follows (this is not an exhaustive list)

• Histogram thresholding
• Edge-based approaches
• Region-based approaches
• Hybrid consider both edges and regions.

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Continued

• Segmentation of texture by PCA
• Segmentation of texture with moments

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Color Image Segmentation

• Image segmentation based on color
• We use clustering to segment an image according
  to color features

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Clustering ?

• Clustering is a common step to get a segmentation
• In the data space, clusters are regarded as
  regions of similar data points

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Clustering.
Similar data points grouped together into
clusters.
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Clustering

• Most popular clustering algorithms suffer from
  two major drawbacks
• First, the number of clusters is predefined,
  which makes them inadequate for batch processing
  of huge image databases
• Secondly, the clusters are represented by their
  centroid and built using an Euclidean distance
  therefore inducing generally an hyperspheric
  cluster shape, which makes them unable to capture
  the real structure of the data.
• This is especially true in the case of color
  clustering where clusters are arbitrarily shaped

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Clustering Algorithms

• K-means
• K-medoids
• Hierarchical Clustering
• There are many other algorithms used for
  clustering.

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K-means Clustering Algorithm

• Step 1 Choose K cluster centers
• c¹(1),c² (1),c³(1)..
• Step 2 At the kth iterative step distribute the
  samples among the K cluster
  domains, using the relation
• for all x in S(j), if x Cj (k) lt x Cj
  (k)
• Step 3 Compute the new cluster centers

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Algorithm continued

• Step 4 If the algorithm has converged and the
  procedure is terminated. Otherwise go to
  Step 2
• S(j) contains data points in RGB space
• x is the data point in an iteration
• C(j) is the center of the cluster

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How the problem was approached ?

• First the images are read from the directory.
• Then each image is transformed into a 3-D RGB
  space.
• K-means clustering using Mahanalobis distance and
  Euclidean distance was applied.

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Continued..

• Next, the mean color of each cluster is
  calculated.
• Then it is transformed back from 3-D space.
• Result Segmented image

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Mahalanobis Distance

• M.D. is a very useful way of determining the
  similarity of a set of values from an
  unknown sample to a set of values measured
  from a collection of known samples
• Superior to Euclidean distance because it takes
  distribution of the points (correlations) into
  account
• Traditionally to classify observations into
  different groups

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Mahalanobis vs. other classical statistical
approaches

• It takes into account not only the average value
  but also its variance and the covariance of the
  variables measured
• It compensates for interactions (covariance)
  between variables
• It is dimensionless

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Mahalanobis Distance

• Dt(x) (x mt) S-1t (x mt)

Mahalanobis Distance
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Euclidean Distance

• The Euclidean distance is the straight-line
  distance between two pixels
• Euc dist v((x1 - x2)² (y1 - y2)²) , where
  (x1,y1) (x2,y2) are two pixel points or two
  data points

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After executing the program

• Original Images

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More images
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During execution

• The images are transformed into 3-D RGB space
• K-means clustering is applied which groups data
  points with similar color into clusters

Example
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Continued

• The data points with minimum mahalanobis distance
  are grouped together into clusters.
• After grouping into clusters, the mean color of
  the cluster is taken and mapped back into the
  image

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Continued

• The program was executed with different number of
  clusters
• The same program was run with Euclidean distance
  for distance between data points

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Segmented Images
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Original Image
Segmented with 6 clusters

• Segmented with 4 clusters

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Original Image
2 clusters
4 Clusters
6 Clusters
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Original Image
2 Clusters
4 Clusters
6 Clusters
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Mahalanobis vs. Euclidean
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Original Image
4 clusters Mahalanobis
4 clusters Euclidean
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Original Image
More detailed description using Mahalanobis
6 Clusters Mahalanobis
6 Clusters Euclidean
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Original Image
Image more clear than Euclidean
6 Clusters Mahalanobis
6 Clusters Euclidean
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Mahalanobis vs. Euclidean

• Result
• Using Mahalanobis distance in the k-means
  clustering proved to be better than Euclidean
  distance.
• This is because the Mahalanobis considers the
  covariance between variables.

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Conclusion

• The program runs with all jpeg images and takes a
  lot of time to run for larger images.
• This is due to the calculation of the inverse of
  the covariance matrix for the Mahalanobis
  distance.
• The results show that Image segmentation using
  Mahalanobis distance works and is better than
  Euclidean distance.

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The End