Outline
• Texture modeling - continued
– FRAME model for textures
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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FRAME Model
• FRAME model
– Filtering, random field, and maximum entropy
– A well-defined mathematical model for textures
by combining filtering and random field models
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Maximum Entropy
• Maximum entropy
– Is an important principle in statistics for constructing
a probability distribution on a set of random variables
– Suppose the available information is the expectations
of some known functions n(x), that is
E p [ n ( x)]    n ( x) p( x)dx   n for n  1,, N
– Let W be the set of all probability distributions p(x)
which satisfy the constraints
W  { p( x) | E p [ n ( x)]   n , n  1,, N }
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Maximum Entropy – cont.
• Maximum Entropy – continued
– According to the maximum entropy principle, a
good choice of the probability distribution is the
one that has the maximum entropy


p ( x)  arg max   p( x) log p( x)dx
*
subject to
E p [ n ( x)]    n ( x) p( x)dx   n for n  1,, N
 p( x)dx  1
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Maximum Entropy – cont.
• Maximum Entropy – continued
– By Lagrange multipliers, the solution for p(x) is
1
 N

p( x; ) 
exp   n n ( x)
Z ()
 n 1

– where
 N

  (1 , 2 ,, N ) and Z( )   exp -  n n ( x)dx
 n 1

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Maximum Entropy – cont.
• Maximum Entropy – continued
–   (1 , 2 ,, N ) are determined by the
constraints
– But a closed form solution is not available
general
• Numerical solutions
dn
 E p ( x; ) [ n ( x)]   n
dt
n  1, 2,  , N
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Maximum Entropy – cont.
• Maximum Entropy – continued
– The solutions are guaranteed to exist and be
unique by the following properties
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FRAME Model
• Texture modeling
– The features can be anything you want n(x)
– Histograms of filter responses are a good feature
for textures
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FRAME Model – cont.
• The FRAME algorithm
– Initialization
Input a texture image Iobs
Select a group of K filters SK={F(1), F(2), ...., F(K)}
Compute {Hobs(a), a = 1, ....., K}
Initialize (ia )  0, i  1,2,, L a  1,2,, K
Initialize Isyn as a uniform white noise image
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FRAME Model – cont.
• The FRAME algorithm – continued
– The algorithm
Repeat
calculate Hsyn(a), a=1,..., K from Isyn and use it as
E p ( I ; K ,S K ) ( H (a ) )
Update (ia ) by d(ia ) / dt  E p ( I ; ,S ) [ H (a ) ]  H obs(a )
Apply Gibbs sampler to flip Isyn for w sweeps
L
obs(a )
syn (a )
1
until
|
H

H
|   for a  1, 2, , K

i
i
2
K
K
i 1
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FRAME Model – cont.
• The Gibbs sampler
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FRAME Model – cont.
• Filter selection
– In practice, we want a small number of “good”
filters
– One way to do that is to choose filters that carry
the most information
• In other words, minimum entropy
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FRAME Model – cont.
• Filter selection algorithm
– Initialization
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FRAME Model – cont.
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FRAME Model – cont.
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FRAME Model – cont.
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FRAME Model – cont.
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FRAME Model – cont.
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FRAME Model – cont.
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FRAME Model – cont.
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