Outline
• Texture modeling - continued
– Markov Random Field models - continued
– Fractals
Texture Modeling
• 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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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
N  {Ni | i  S}
– 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 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:
 Positivity
P( f )  0, f  
 Markoviani ty
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P( f i | f S {i} )  P( f i | f Ni )
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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
P( f ) 
– and
1
e
1
 U ( f1 )
T
e
1
 U( f )
T
f1
U ( f )  Vc ( f )
cC
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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
•
•
•
•
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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.
• Texture models using Markov random fields
– For most existing models, only the single pixel
and pair-wise pixel cliques are considered
• In other words, all other higher-order cliques are zero
– Textures are characterized by a few parameters
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Markov Random Fields – cont.
• Auto Models
– Only the single pixel and pair-wise pixel cliques
can be non-zeros
U ( f )   f i G ( f i )    i ,i  f i f i 
{i ,i}C2
{i}C1
• Auto-logistic models
– Ising models
• Auto-binomial models
• Gaussian Markov random field models
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Markov Random Fields – cont.
• The -model
– The energy function is of the form
6
– with
U( f )  
  ( f
i 1  s ,t  i
i
s
 ft )
1
 () 
1  ( /  ) 2
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Markov Random Fields – cont.
• Parameter estimation
– Parameters are generally estimated using
Maximum-Likelihood estimator or Maximum-APosterior 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
  arg max  P( f s | f  s )
*
sS
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Markov Random Fields – cont.
• Texture synthesis
– Generate samples from the Gibbs distributions
– Two sampling techniques
• Metropolis sampler
• Gibbs sampler
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Markov Random Fields – cont.
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Fractals
• Fractals
– Many natural surfaces have a statistical quality of
roughness and self-similarity at different scales
– Fractals are very useful in modeling selfsimilarity
• Texture features based on fractals
– Fractal dimension
– Lacunarity
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Fractals – An Example
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