1. G. Delyon, Ph. Réfrégier and F. Galland Physics & Image Processing
Minimal Stochastic Complexity Image partitioning with nonparametric statistical model
G. Delyon, Ph. Réfrégier and F. Galland
MaxEnt 2006, Paris
Physics & Image Processing

2. Overview 1. Introduction to image segmentation
2. Stochastic Complexity and application to parametric statistical techniques
3. Novelty of the work: nonparametric statistical technique 2 9 9 11 10 9 10 10 10 9 9 9

3. Overview 1. Introduction to image segmentation
2. Stochastic Complexity and application to parametric statistical techniques
3. Novelty of the work: nonparametric statistical technique 3 9 9 11 10 9 10 10 10 9 9 9

4. Noisy image segmentation
Goal: to automatically recover the shape of objects in noisy images
Forest
Wheat
Corn
Fields
Town
Water
® ESA 1993 Distribution
Spot Image 4
agricultural fields (Bourges, France) 4 4 4 4 2 4

5. Noisy image segmentation
In general one needs to:
- choose a probability model for the grey level fluctuations
Example: SAR image => Gamma Probability Density Function (PDF)
- adjust parameters to control the regularity of the boundaries
Not enough regularization
Too much regularization
Satisfactory regularization 5 4 4 4 2 4 4

6. Overview 1. Introduction to image segmentation
2. Stochastic Complexity and application to parametric statistical techniques
3. Novelty of the work: nonparametric statistical technique 6 9 9 11 10 9 10 10 10 9 9 9

7. Introduction to Stochastic Complexity
For example to fit data one can choose a polynomial of order M
Models
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 7 4 4 4 4 4 2

8. Stochastic complexity
DGL = Bit number needed to describe the data
DM = Bit number needed to describe the model
(PDF and other parameters).
Stochastic Complexity
Stochastic complexity =
Selected model
Model complexity 8 4 4 4 4 2 4

9. Application to segmentation in homogenous regions
Best segmentation in the stochastic complexity sense <=> 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 9 9 9 10 9 11 9 10 10 10 9 9

10. Polygonal grid G
Polygonal grid
R (unknown) regions
= Number of bits needed to code the grey levels knowing the partition and the PDF in each region
= Number of bits needed to code the PDF in each
region
= Number of bits needed to code the grid
Encoding of the model 10
F. Galland, N. Bertaux and Ph. Réfrégier. IEEE IP, pp September 2003. 9 9 11 10 9 10 10 10 9 9 9

11. 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).
= number of bits needed to code the grey levels of
each homogenous region
~ opposite of the log-likelihood
Unknown parameters of the PDF : estimated with a maximum likelihood approach
~ opposite of the profile likelihood
=> - no parameter
- no EM technique since the explicit expression of the profile
likelihood is known in most of the PDF in the exponential family 11 4 4 4 4 4 2

12. Parametric statistical models
Segmentation result obtained with the algorithm adapted to Gamma PDF
fusion
move
remove
fusion
move
remove 12 9 9 11 10 9 10 10 10 9 9 9

13. Limitations of the parametric approach
region B
SAR-polarimetric image, parametric technique adapted to Gamma noise
Constraints:
knowledge of the PDF family in each region = strong hypothesis,
images for which a parametric description (Gaussian, Gamma, Poisson, …) of the grey level’s fluctuations is not available.
The parametric model is not adapted.
The nonparametric model is necessary. 13 4 4 4 2 4 4

14. Overview 1. Introduction to image segmentation
2. Stochastic Complexity and application to parametric statistical techniques
3. Novelty of the work: nonparametric statistical technique 14 9 9 11 10 9 10 10 10 9 9 9

15. 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 aj
aj+1 aj
aj+1
Grey levels
Grey levels aj s 1
Rj (s)
aj+1 15 4 4 4 4 4 2

16. 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 16 4 4 4 4 2 4

17. Optimization procedure
Stage 1
Stage 3
Stage 2
SAR image segmentation result obtained with the step functions. Image courtesy of the NASA/JPL-Caltech AIRSAR system. 17

18. 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
Application to the Stochastic Complexity:
Likelihood term
Regularization term
New criterion: 18 4 4 4 4 2 4

19. Relevance of the proposed approach
NMP
0.4
0.2
0.6
0.8
100
3000 30
300
1000
Gaussian noise
l = 0.1
l = 0.5
l = 0.8
Criterion: 19 9 9 11 10 9 10 10 10 9 9 9

20. Relevance of the proposed approach
NMP
0.4
0.2
0.6
0.8
100
3000 30
300
1000
Gamma noise
l = 0.3
l = 0.5
l = 0.75
Criterion: 20 9 9 11 9 10 10 10 10 9 9 9

21. Relevance of the proposed approach
NMP
0.4
0.2
0.6
0.8
100
3000 30
300
1000
Poisson noise
l = 0.2
l = 0.5
l = 0.85
Criterion: 21 9 9 11 9 10 10 10 10 9 9 9

22. Relevance of the proposed approach
NMP
0.4
0.2
0.6
0.8
100
3000 30
300
1000
Gaussian noise
Gamma noise
Poisson noise
= 0.5
is the optimal value 22 9 9 10 9 11 10 10 10 9 9 9

23. Illustration of the result on a real image
SAR image
From Ukraine
l = 0.2
l = 0.8
l = 0.5
Criterion:
SAR image provided by the CNES and the ESA 23 9 9 10 9 11 10 10 10 9 9 9

24. Comparison with parametric approaches
SAR image courtesy of the CNES.
Algorithm adapted to Gamma PDF
Step functions
Algorithm adapted to Gamma PDF
SAR image courtesy of the NASA/JPL-Caltech AIRSAR system.
Step functions
G. Delyon and Ph. Réfrégier, IEEE Geoscience, 44 (7), pp , July 2006. 24

25. Comparison with parametric approaches
Hue image
Hue component
Color image
algorithm adapted to Gaussian PDF
step functions
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. 25 4 4 4 4 4 2

26. 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 level’s fluctuations without making hypotheses on the PDF family of the noise. 26 4 4 4 4 2 4

27. 27 9 9 11 10 9 9 10 10 10 9 9

28. PDF parameters encoding
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 28 9 9 9 10 11 10 10 10 9 9 9