Minimal Stochastic Complexity Image partitioning with nonparametric statistical model

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Minimal Stochastic Complexity Image partitioning
with nonparametric statistical model
G. Delyon, Ph. Réfrégier and F. Galland
MaxEnt 2006, Paris
Physics Image Processing
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Overview
1. Introduction to image segmentation
2. Stochastic Complexity and application to
parametric statistical techniques
3. Novelty of the work nonparametric
statistical technique
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Overview
1. Introduction to image segmentation
2. Stochastic Complexity and application to
parametric statistical techniques
3. Novelty of the work nonparametric
statistical technique
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Noisy image segmentation
Goal to automatically recover the shape of
objects in noisy images
ESA 1993 Distribution Spot Image
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agricultural fields (Bourges, France)
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Noisy image segmentation
In general one needs to - choose a probability
model for the grey level fluctuations
Example SAR image gt Gamma Probability Density
Function (PDF)
- adjust parameters to control the regularity of
the boundaries
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Overview
1. Introduction to image segmentation
2. Stochastic Complexity and application to
parametric statistical techniques
3. Novelty of the work nonparametric
statistical technique
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Introduction to Stochastic Complexity
For example to fit data one can choose a
polynomial of order M

• Choice of the order M of the polynomial
• trade-off between the complexity of the model
  and the error between the data and the model

Measurements
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Stochastic complexity
Model complexity
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Application to segmentation in homogenous regions
Best segmentation in the stochastic complexity
sense ltgt segmentation which allows one to get
the best compression with a Shannon code.

• To apply this principle one thus needs
• to choose a probabilistic image model
• to determine the code length associated to a
  given segmentation
• to find the segmentation which minimizes this
  length

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Polygonal grid
Polygonal grid
R (unknown) regions
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F. Galland, N. Bertaux and Ph. Réfrégier. IEEE
IP, pp. 995-1006 September 2003.
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Parametric statistical models
Probability Density Functions (PDF) of the grey
levels are assumed to belong to the exponential
family (for example Gaussian or Gamma or
Poisson, etc).

• gt - no parameter
• - no EM technique since the explicit
  expression of the profile
• likelihood is known in most of the PDF in
  the exponential family

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Parametric statistical models
Segmentation result obtained with the algorithm
adapted to Gamma PDF
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Limitations of the parametric approach

• Constraints
• knowledge of the PDF family in each region
  strong hypothesis,
• images for which a parametric description
  (Gaussian, Gamma, Poisson, ) of the grey levels
  fluctuations is not available.

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Overview
1. Introduction to image segmentation
2. Stochastic Complexity and application to
parametric statistical techniques
3. Novelty of the work nonparametric
statistical technique
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Nonparametric statistical model
Estimation of the Probability Density Functions
(PDF) of the grey levels with irregular step
functions
Region B
Region A
Pb(j)
Pa(j)
s
s
aj1
aj
Grey levels
aj1
Grey levels
aj
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Determimation of the aj
Start with a regular histogram sampling
Nj
Nj
Histogram
Histogram
s
s
Grey levels
Fusion of two steps if it allows one to decrease D
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Optimization procedure
Stage 3
Stage 1
Stage 2
SAR image segmentation result obtained with the
step functions. Image courtesy of the
NASA/JPL-Caltech AIRSAR system.
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Relevance of the proposed approach
Comparison with standard approaches (Mumford and
Shah energy function or Zhu and Yuille criterion)
DE external energy
DI internal energy
l regularization parameter
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Relevance of the proposed approach
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Relevance of the proposed approach
l 0.3
l 0.5
l 0.75
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Relevance of the proposed approach
l 0.2
l 0.5
l 0.85
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Relevance of the proposed approach

• 0.5
• is the optimal value

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Illustration of the result on a real image
SAR image provided by the CNES and the ESA
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Comparison with parametric approaches
G. Delyon and Ph. Réfrégier, IEEE Geoscience, 44
(7), pp. 1954-1961, July 2006.
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Comparison with parametric approaches
Segmentation results obtained on the hue
component of the HSV representation
G. Delyon, F. Galland and Ph. Réfrégier, IEEE IP,
15 (10), October 2006, to appear.
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Conclusion

• The Stochastic Complexity allows one to
• - write the likelihood and the regularization
  terms in the same unit (number of bits) so that
  the weighting is optimal considering a quality
  segmentation criterion,
• - define a segmentation technique based on
  the minimization of a criterion without parameter
  to be tuned by the user,
• - take into account the grey levels
  fluctuations without making hypotheses on the PDF
  family of the noise.

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Polygonal grid
To encode a segmented image, one needs to encode
the grid, the pdf parameters, and the pixel
grey levels in each region
Grid encoding
PDF parameters encoding
Pixels grey levels encoding
( ML estimate)
No free parameter in the criterion to optimize
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