Outline

1
Outline

• Texture modeling - continued
• Markov Random Field models
• Fractals

2
Some Texture Examples
3
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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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

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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Autocorrelation Features

• 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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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

8
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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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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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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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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Markov Random Fields cont.

• Markov random fields
• Let FF1, ...., 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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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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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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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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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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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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Markov Random Fields cont.

• The ?-model
• The energy function is of the form
• with

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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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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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Markov Random Fields cont.

• Texture synthesis
• Generate samples from the Gibbs distributions
• Two sampling techniques
• Metropolis sampler
• Gibbs sampler

22
Markov Random Fields cont.
23
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

24
Fractals An Example