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Markov random field: A brief introduction (2)
Tzu-Cheng Jen
Institute of Electronics, NCTU
2007-07-25
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Outline
Markov random field: Review
Edge-preserving regularization in image processing
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Markov random field: Review
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Prior knowledge
In order to explain the concept of the MRF, we first introduce following definition:
1. i: Site (Pixel) 2. fi: The value at site i (Intensity)
3. S: Set of sites (Image)
4. Ni: The neighboring site of i (1st order neighborhood of i: f2, f4, f5, f7 )
5. Ci: Clique, the subset of S and the element in this subset must be neighboring
f1 f2 f3
f4 fi f5
f6 f7 f8
A 3x3 imagined image
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Markov random field (MRF)
View the 2D image f as the collection of the random variables (Random field)
A random field is said to be Markov random field if it satisfies following properties
Red: Neighboring site
1 2 8
{ }
1 2 8 2 4 5 7
(1) ( ) 0, (Positivity)
( ) ( , ,.... , ) 0
(2) ( | ) ( | ) (Markovianity)
( | , ,.... ) ( | , , , )
i
i S i i Ni
i i
P f f
P f P f f f f
P f f P f f
P f f f f P f f f f f
Ff1 f2 f3
f4 fi f5
f6 f7 f8
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Gibbs random field (GRF) and Gibbs distribution
A random field is said to be a Gibbs random field if and only if its configuration f obeys Gibbs distribution, that is:
f1 f2 f3
f4 fi f5
f6 f7 f8
A 3x3 imagined image
1 2
1 2 '{ } { , '}
( ) ( ) ( ) ( , ) .....c i i ic C i C i i C
U f V f V f V f f
1( )1
1 2 8( ) ( , ,.... , )U f
TiP f P f f f f Z e
U(f): Energy function; T: Temperature Vi(f): Clique potential
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Markov-Gibbs equivalence
Hammersley-Clifford theorem: A random field F is an MRF if and only if F is a GRF
Red: Neighboring site
f1 f2 f3
f4 fi f5
f6 f7 f8
{ }
{ }
1( )1
1 2 8
( | ) ( | )
( | ) ( | )
=> ( ) ( , ,.... , )
f is Gibbs field
i S i i Ni
i S i i Ni
U fT
i
P f f P f f
P f f P f f
P f P f f f f Z e
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Edge-preserving regularization in image processing
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MAP formulation for denoising problem
Noisy signal d denoised signal f
d = f + N(0, σ)
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MAP formulation for denoising problem
A signal denoising problem could be modeled as the MAP estimation problem, that is,
arg max{ ( | )}
By Baye's rule:
arg max{ ( | ) ( )}
:
:
f
f
f p f d
f p d f p f
f Unknown data
d Observed data
(Prior model)
(Observation
model)
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MAP formulation for denoising problem
Assume the observation is the true signal plus the independent Gaussian noise, that is
Assume the unknown data f is MRF, the prior model is:
2 2
1
( ) / 2( | )
2 2
1 1( | )
2 2
m
i i ii
f dU d f
m m
i ii m i m
p d f e e
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11 ( )( )1 1( )i i
i
f fU f TTP f Z e Z e
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MAP formulation for denoising problem
Substitute above information into the MAP estimator, we could get:
22
121 1
arg max{ ( | )} arg min{ ( | ) ( )}
( )arg min{ ( ) }
2
f f
m mi i
f i ii i
f p f d U d f U f
f df f
Observation model (Similarity measure)
Prior model (Reconstruction constrain, Regularization)
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The solver of the optimization problem: Gradient descent algorithm
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Simulation results for denoising problem
Simulation resultSimulation result
Processed profiles are blurred !
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Discussion for the phenomenon of blur (1)
From the potential function point of view:
,
^2
, ,, , ( '
')
''
,,
,arg min ( ) arg min{(1 )* ( ) }( )*i j
i j i jf fi j i j i j N
i j i jf E f w d gf fw f
Quadratic function is used as potential function g=x2
Simulation result
1st derivative
Energy
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Discussion for the phenomenon of blur (2)
From the optimization process point of view (gradient descent algorithm):
,
,
( 1) ( ) ( ) ( ) ( ), , , , , ', '
( ', ')
( ) ( ) ( ) ( ), , , , ', '
( ', ')
*(2(1 )*( ) * '( ))
= *(2(1 )*( ) * 2 ( ))
i j
i j
t t t t ti j i j i j i j i j i j
i j N
t t t ti j i j i j i j i j
i j N
f f step w f d w g f f
f step w f d w f f
Update equation of gradient descent:
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Edge-preserving regularization
S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell, 6, 721-741, 1984.
S.Z. Li, “On Discontinuity-adaptive smoothness priors in computer vision,” IEEE Trans. Pattern Anal. Mach. Intell, June, 1995.
Pierre Charbonnier et al, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Processing, Feb, 1997.
S.Z. Li, “Markov random field modeling in computer vision,“ Springer, 1995
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MRF with pixel process and line process (Geman and Geman)
Lattice of pixel site: SP Labeling value: fi
p (real value)
Lattice of line site: SE Labeling value: fii’
E (only 0 or 1)
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MRF with pixel process and line process (Geman and Geman)
Based on the concept of line process, we could modify the original restoration problem as follows:
, ,
1( , )( | )
, ,
2
2,1
, arg max{ ( , | )} arg max{ ( | , )* ( , )}
arg max{ ( | )* ( , )} arg max{ * }
( )arg min{ ( , )}
2
P E P E
P EP
P E P E
P E
P E P E P E P E
f f f f
U f fP P E U d f Tf f f f
mP Ei i
f fi
f f p f f d P d f f P f f
P d f P f f e e
f dU f f
Goal: Find realizations fp and fE such that edge-preserving regularization could be achieved
?
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MRF with pixel process and line process (Geman and Geman)
Define the prior:
Substitute above information into the MAP estimator, we could get:
2' ' '
'
( , ) ( ) (1 )P
P E P P E Ei i ii ii
i Nii S
U f f f f f f
, ,
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' ' '2,1 '
, arg max{ ( | )} arg min{ ( | ) ( )}
( )arg min{ ( ) (1 ) }
2
P E P E
P E
P
P E
f f f f
mP P E Ei ii i ii iif f
i i Nii S
f f p f d U d f U f
f df f f f
The above optimization problem is a combination of real and combinatorial problem
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MRF with pixel process and line process (Geman and Geman)
Blake and Zisserman covert previous restoration problem into real minimization problem by introducing truncated quadratic function as potential function
'
^
2
( ')
arg min ( )
arg min{(1 )* ( })) (*i
f
i if
ii i
i
P
i
P
N
g
f E f
w d f fw f
Truncated quadratic function
1st derivative alphaalpha
Energy
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MRF with pixel process and line process (Geman and Geman)
Simulation results
Original image Degraded image Restoration result (1000 iterations)
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MRF with pixel process and line process (Geman and Geman)
Simulation results
Original image
Degraded image
Restoration result (1000 iterations)
Restoration result (100 iterations)
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Discontinuity-adaptive regularization (S. Z. Li)
Revisit the gradient descent algorithm
,
( 1) ( ) ( ) ( ) ( ), , , , , ', '
( ', ')
( 1) ( ) ( ) ( ) ( ) ( ) ( ), , , , , ', ' , ', '
( ', ')
*(2(1 )*( ) * '( ))
g'( )=2 h ( )
*(2(1 )*( ) * 2*( )* ( )
i j
t t t t ti j i j i j i j i j i j
i j N
t t t t t t ti j i j i j i j i j i j i j i j
i j
f f step w f d w g f f
Set
f f step w f d w f f h f f
,
)i jN
Adjust it adaptively !
Derivative or compensator
Weight or interaction function
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Discontinuity-adaptive regularization (S. Z. Li)
For edge-preserving regularization, interaction function hr should satisfy following property:
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Discontinuity-adaptive regularization (S. Z. Li)
Possible choices for interaction function hr
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Discontinuity-adaptive regularization (S. Z. Li)
Simulation results (1D)
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Discontinuity-adaptive regularization (S. Z. Li)
Simulation results (2D)
Original image Edge-preserving restoration
Restoration without preserving edge
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Discontinuity-adaptive regularization (Pierre Charbonnier et al )
Pierre Charbonnier et al impose following conditions on potential function φfor edge preserving regularization
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Discontinuity-adaptive regularization (Pierre Charbonnier et al )
Possible choices for potential function φ
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Other related techniques for edge-preserving regularization
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell, July, 1990.
,
^2
, , , ', ', , ( ', ')
(1 )arg min ( ) arg min{ * ( ) * ( )}i j
i j i j i j i jf fi j i j i j N
f E f d f g f fw w
Dropping observation model (w=1) when evaluating f
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Other related techniques for edge-preserving regularization
L.I. Rudin, S. Osher, E. Fatemi (1992): Nonlinear Total Variation Based Noise Removal Algorithms, Physica D, 60(1-4), 259-268.
,
^2
, ', ,, ,
, '( ', ')
arg min ( ) arg min{(1 | |)* ( ) * }i j
i j i jf fi
i j ij j i j N
ji
f E f w fd f w f
Replace the quadratic potential function with absolute value function
1st derivative
Quadratic function versus Absolute value function
Energy
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