1. Outline Texture modeling - continued Markov Random Field models
Fractals

2. Visual Perception Modeling
Some Texture Examples
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Visual Perception Modeling

3. Visual Perception Modeling
Texture Modeling
Texture modeling is to find feature statistics that characterize perceptual appearance of textures
There are two major computational issues
What kinds of feature statistics shall we use?
How to verify the sufficiency or goodness of chosen feature statistics?
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Visual Perception Modeling

4. Texture Modeling – cont.
The structures of images
The structures in images are due to the inter-pixel relationships
The key issue is how to characterize the relationships
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Visual Perception Modeling

5. Co-occurrence Matrices
Gray-level co-occurrence matrix
One of the early texture models
Was widely used
Suppose that there are G different gray values in a texture image I
For a given displacement vector (dx, dy), the entry (i, j) of the co-occurrence matrix Pd is
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Visual Perception Modeling

6. Autocorrelation Features
Many textures have repetitive nature of texture elements
The autocorrelation function can be used to assess the amount of regularity as well as the fineness/coarseness of the texture present in the image
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Visual Perception Modeling

7. Visual Perception Modeling
Geometrical Models
Geometrical models
Applies to textures with texture elements
First texture elements are extracted
Then one can compute the statistics of local elements or extract the placement rule that describes the texture
Voronoi tessellation features
Structural methods
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Visual Perception Modeling

8. Visual Perception Modeling
Markov Random Fields
Markov random fields
Have been popular for image modeling, including textures
Able to capture the local contextual information in an image
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Visual Perception Modeling

9. Markov Random Fields – cont.
Sites
Let S index a discrete set of m sites
S = {1, ...., m}
A site represents a point or a region in the Euclidean space
Such as an image pixel
Labels
A label is an event that may happen to a site
Such as pixel values
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Visual Perception Modeling

10. Markov Random Fields – cont.
Labeling problem
Assign a label from the label set L to each of the sites in S
Also a mapping from S  L
A labeling is called a configuration
In texture modeling, a configuration is a texture image
The set of all possible configurations is called the configuration space 
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Visual Perception Modeling

11. Markov Random Fields – cont.
Neighborhood systems
The sites in S are related to one another via a neighborhood
A neighborhood system for S is defined as
The neighborhood relationship has the following properties
A site is not a neighbor to itself
The neighborhood relationship is mutual
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Visual Perception Modeling

12. Markov Random Fields – cont.
Let F={F1, ...., Fm} be a family of random variables defined on the set S in which each random variable Fi takes a value from L
F is said to be a Markov random field on S with respect to a neighborhood system N if an only if the following two conditions are satisfied:
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Visual Perception Modeling

13. Markov Random Fields – cont.
Homogenous MRFs
If P(fi | fNi) is regardless of the relative position of site i in S
How to specify a Markov random field
Conditional probabilities P(fi | fNi)
Joint probability P(f)
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Visual Perception Modeling

14. Markov Random Fields – cont.
Gibbs random fields
A set of random variables F is said to be a Gibbs random field on S with respect to N if and only if its configurations obey a Gibbs distribution
and
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Visual Perception Modeling

15. Markov Random Fields – cont.
Cliques
A clique c for (S, N) is defined as a subset of sites in S and it consists of
A single site
A pair of neighboring sites
A triple of neighboring sites
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Visual Perception Modeling

16. Markov Random Fields – cont.
Markov-Gibbs equivalence
Hammersley-Clifford theorm
F is an Markov random field on S respect to N if and only if F is a Gibbs random field on S with respect to N
Practical value of the theorem
It provides a simple way to specify the joint probability by specifying the clique potentials
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Visual Perception Modeling

17. Markov Random Fields – cont.
Markov random field models for textures
Homogeneity of Markov random fields is assumed
A texture type is characterized by a set of parameters associated with clique types
Texture images can be generated (synthesized) by sampling from the Markov random field model
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Visual Perception Modeling

18. Markov Random Fields – cont.
The -model
The energy function is of the form
with
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Visual Perception Modeling

19. Markov Random Fields – cont.
Parameter estimation
Parameters are generally estimated using Maximum-Likelihood estimator or Maximum-A-Posterior estimator
Computationally, the partition function can not be evaluated
Markov chain Monte Carlo is often used to estimate the partition function by generating typical samples from the distribution
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Visual Perception Modeling

20. Markov Random Fields – cont.
Pseudo-likelihood
Instead of maximizing P(f), the joint probability, we maximize the products of conditional probabilities
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Visual Perception Modeling

21. Markov Random Fields – cont.
Texture synthesis
Generate samples from the Gibbs distributions
Two sampling techniques
Metropolis sampler
Gibbs sampler
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Visual Perception Modeling

22. Markov Random Fields – cont.
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Visual Perception Modeling

23. Visual Perception Modeling
Fractals
Fractals
Many natural surfaces have a statistical quality of roughness and self-similarity at different scales
Fractals are very useful in modeling self-similarity
Texture features based on fractals
Fractal dimension
Lacunarity
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Visual Perception Modeling

24. Visual Perception Modeling
Fractals – An Example
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Visual Perception Modeling