Information Theoretic Image Thresholding

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Information Theoretic Image Thresholding

• Laura Frost
• Supervisors Dr Peter Tischer
• Dr Sid Ray

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Aims of Project

• Investigate mixture modelling as approximation to
  images histogram
• Investigate relative / objective criterion for
  assessment of thresholding results

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Thresholding

• Good for images with fairly distinct, homogeneous
  regions
• Region of uncertainty object boundaries
• Two important properties for mixture modelling
  thresholding

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Mixture Modelling

• Approximate complex distribution with component
  distributions
• Can describe components easily
• Classify data in this case pixels

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An example
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Need to ask

• Of thresholding
• How good is a threshold?
• Of mixture modelling
• When is a mixture model a good model?

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And so keep in mind

• A good mixture model implies a good threshold
• A good threshold does not necessary imply a good
  mixture model

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Methodology 1

• Test / extend / possibly improve iterative
  mixture modelling
• Examine Kullback-Leibler measure as possible
  relative / objective criterion

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Methodology 2

• Test mixture modelling image histograms using
  Snob
• Examine Minimum Message Length as possible
  objective criterion

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Iterative Mixture Modelling

• Fit mixture model at each grey-level
• Select grey-level that produces best model
• Good for bi-level thresholding

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Implementation

• Based on work completed by David Bertolo (Honours
  2001, Monash)
• Distributions
• Poisson (1 parameter)
• Gaussian, Rayleigh (2 parameters)

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Implementation

• Improve threshold selection use intersection of
  components
• Account for overlap

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Testing

• Synthetic and natural images
• Synthetic images created with specific
  distributions
• Test accuracy of model fitting
• Give lower bound for Kullback-Leibler measure
  assessment

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Synthetic Images
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Results Iterative Mixture Modelling

• Component parameters correct for synthetic images
• Component parameters for natural images (esp.
  outliers at boundaries)
• Subjective assessment of segmented image

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Results Iterative Mixture Modelling

• Examined five information measures
• Entropy of image H(p)
• Entropy of mixture model H(q)
• Kullback-Leibler (KL) measure I(pq)
• KL relative to entropy of image I(pq) / H(p)
• KL relative to entropy of model I(pq) / H(q)

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Results 2, 3, 4 components

• Good fit for synthetic images
• Gaussian (?)
• Poisson (?)
• Rayleigh opposite (?)
• Rayleigh right (?)

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Results 2, 3, 4 components

• Dealt with outliers at boundaries
• Gaussian (?)
• Poisson (?)
• Rayleigh opposite (?)
• Rayleigh right (?)

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Results 2, 3, 4 components

• Overall, segmented images good quality
• Gaussian (?)
• Poisson (?)
• Rayleigh opposite (?)
• Rayleigh right (?)
• Except for images with outliers

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Results 2, 3, 4 components

• Time unreasonable for 4 components (complexity
  increases exponentially)
• Poisson takes 4 times longer than other
  distributions

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Matches example
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3 Gaussian mixture model
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3 Poisson mixture model
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3 Rayleigh mixture model
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Results 2, 3, 4 components

• Gaussians
• H(p) lt H (q) for all images
• Poissons
• H(p) gt H (q) for all successfully thresholded
  images
• Rayleighs
• H(p) H(q)

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Results 2, 3, 4 components

• I(pq) decreased as no. components increased to
  be expected
• I(pq) / H(p)
• I(pq) / H(q)

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A relative criterion

• Is there value in comparing models of different
  complexities?
• From these results, probably not
• But comparing models of similar complexities
  looks ok

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Mixture modelling with Snob

• Problem overfitting data on natural and
  synthetic images
• Eg, 512 x 512 image has 262 144 pixels to
  classify
• Cheaper for Snob to make more classes

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Sampling

• Randomly sampling data at different rates
• Snob finding very good classes!
• Sampling image alumina.gif at 100 pixels (from
  total 65 536)

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Alumina example

• Over many runs found two classes
• 0.20 N(91.80, 23.20)
• 0.80 N(206.80, 11.40)
• Compare to Iterative method
• 0.19 N(94.30, 26.56)
• 0.81 N(206.21, 11.80)

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Alumina example

• Message length 4.94 bpp
• Not all images so successful at just 100 pixels
• All seem to be ok at about 500 pixels

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Snob and Thresholding

• Sampling at such small rates, Snob handling
  missing data very well!
• Since need to sample at such small rates, did not
  compare Poissons as hoped
• More work needs to be done, but looks promising

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An Objective Criterion

• Takes complexity of mixture model into account
  when calculating message length
• Message Length a very good candidate for use as
  an objective criterion for thresholding

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Aims of Project

• Investigate mixture modelling as approximation to
  images histogram
• Investigate relative / objective criterion for
  assessment of thresholding results

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Conclusion

• Iterative Method
• Consistent results
• Optimal no. of components trial and error
• Complexity
• Snob
• Problem with overfitting large no. of data
  points
• Sampling at very tiny rates working well

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Evaluation Criteria

• Kullback-Leibler measure
• Relative to Entropy
• Relative to model complexity
• Minimum Message Length
• Promising as objective criterion

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Future Work

• Better way to initially assign pixels to classes
• Modify Snob to do this
• More testing with Snob
• Addition of more distributions to Iterative method